Распределения , также известные как распределения Шварца или обобщенные функции , являются объектами, которые обобщают классическое понятие функций в математическом анализе . Распределения позволяют дифференцировать функции, производные которых не существуют в классическом смысле. В частности, любая локально интегрируемая функция имеет производную по распределению. Распределения широко используются в теории уравнений с частными производными , где может быть проще установить существование распределительных решений, чем классических решений, или подходящие классические решения могут не существовать. Распределения также важны в физике иинженерия, где многие проблемы естественным образом приводят к дифференциальным уравнениям, решениями или начальными условиями которых являются распределения, такие как дельта- функция Дирака .
функция обычно считается действующим на точки в своем домене , "посылая" точку x в его домене в точку Вместо воздействия на точки теория распределения переосмысливает такие функции, как как действующие на тестовые функции определенным образом. Тестовые функции обычно представляют собой бесконечно дифференцируемые комплекснозначные (или иногда вещественнозначные ) функции с компактной опорой ( ударные функции являются примерами тестовых функций). Многие «стандартные функции» (означающие, например, функцию, которая обычно встречается в курсе математического анализа ), например, непрерывная картамогут быть канонически интерпретированы как действующие на тестовые функции (вместо их обычной интерпретации как действующие на точки своей области) посредством действия, известного как « интегрирование против тестовой функции»; явно это означает, что"действует" на тестовую функцию g , "посылая" g на номер Это новое действие таким образом, является комплексным (или действительным) -значным отображением , обозначаемымчьей областью определения является пространство тестовых функций; у этой карты есть два дополнительных свойства [примечание 1], которые превращают ее в так называемое распределение наРаспределения, которые возникают из "стандартных функций" таким образом, являются прототипами распределений. Но есть много распределений, которые не возникают таким образом, и эти распределения известны как «обобщенные функции». Примеры включают дельта-функцию Дирака или некоторые распределения, которые возникают в результате действия «интегрирования тестовых функций против мер ». Однако, используя различные методы, тем не менее, все еще возможно свести любое произвольное распределение к более простому семейству связанных распределений, которые действительно возникают в результате таких действий интеграции.
В приложениях к физике и технике пространство тестовых функций обычно состоит из гладких функций с компактным носителем , которые определены на некотором заданном непустом открытом подмножестве. Это пространство пробных функций обозначается через или же а распределение на U по определению является линейным функционалом наэто непрерывно, когдадается топология под названием канонической топологии LF . Это приводит к в пространстве (все) распределений на U , обычно обозначается(обратите внимание на штрих ), который по определению является пространством всех распределений на(то есть, это непрерывное двойственное пространство из); именно этим дистрибутивам посвящена основная тема данной статьи.
Есть и другие возможные варианты выбора пространства тестовых функций, которые приводят к другим другим пространствам распределений. Еслито использование функций Шварца [примечание 2] в качестве пробных функций дает начало некоторому подпространствуэлементы которого называются умеренными распределениями . Это важно, потому что они позволяют расширить преобразование Фурье от «стандартных функций» до умеренных распределений. Множество обобщенных образует векторное подпространство в в пространстве распределенийи, таким образом , одним из примеров в пространстве распределений; есть много других пространств дистрибутивов.
Также существуют другие основные классы тестовых функций, которые не являются подмножествамитакие как пространства аналитических тестовых функций , которые производят очень разные классы распределений. Теория таких распределений носит иной характер, чем предыдущая, поскольку не существует аналитических функций с непустым компактным носителем. [примечание 3] Использование аналитических тестовых функций привело к теории гиперфункций Сато .
История
Практическое использование распределений можно проследить до использования функций Грина в 1830-х годах для решения обыкновенных дифференциальных уравнений, но формализовалась лишь намного позже. Согласно Колмогорову и Фомину (1957) , обобщенные функции возникли в работе Сергея Соболева ( 1936 ) по гиперболическим уравнениям с частными производными второго порядка, и эти идеи были развиты в несколько расширенной форме Лораном Шварцем в конце 1940-х годов. Согласно его автобиографии, Шварц ввел термин «распределение» по аналогии с распределением электрического заряда, возможно, включая не только точечные заряды, но и диполи и так далее. Гординг (1997) комментирует, что, хотя идеи в преобразующей книге Шварца (1951) не были полностью новыми, именно широкая атака и убежденность Шварца в том, что распределения будут полезны почти повсюду в анализе, имели значение.
Обозначение
В этой статье будут использоваться следующие обозначения:
- фиксированное положительное целое число и это фиксированное непустое открытое подмножество в евклидовом пространстве
- обозначает натуральные числа .
- будет обозначать неотрицательное целое число или
- Если это функция, тогдабудет обозначать его домен и поддержку в обозначается определяется как замыкание множества в
- Для двух функций , следующие обозначения определяют каноническое спаривание :
- Многоиндексных размера это элемент в (учитывая, что фиксировано, если размер мультииндексов опущен, то предполагается, что размер ). Длина из мультииндекса определяется как и обозначается Мультииндексы особенно полезны при работе с функциями нескольких переменных, в частности, мы вводим следующие обозначения для данного мультииндекса :
- Мы также вводим частичный порядок всех мультииндексов следующим образом: если и только если для всех Когда мы определяем их многоиндексный биномиальный коэффициент как:
- будет обозначать некоторый непустой набор компактных подмножеств (подробно описано ниже).
Определения тестовых функций и распределений
В этом разделе мы формально определим вещественнозначные распределения на U . С небольшими изменениями можно также определять комплексные распределения, и можно заменитьс любым ( паракомпактным ) гладким многообразием .
Обозначение : Предположим
- Позволять обозначим векторное пространство всех к -кратного непрерывно дифференцируемые вещественные функции на U .
- Для любого компактного подмножества позволять а также оба обозначают векторное пространство всех этих функций такой, что
- Обратите внимание, что зависит как от K, так и от U, но мы будем указывать только K , где, в частности, если тогда область является U , а не K . Мы будем использовать обозначения только когда обозначение рискует оказаться двусмысленным.
- Ясно, что каждый содержит постоянную карту 0 , даже если
- Позволять обозначим множество всех такой, что для некоторого компактного подмножества K из U .
- Эквивалентно, это набор всех такой, что имеет компактную опору.
- равно объединению всех в виде колеблется над
- Если - вещественная функция на U , то является элементом если и только если это функция удара . Каждая действительная тестовая функция на всегда также является комплексной тестовой функцией на
Обратите внимание, что для всех и любых компактных подмножеств K и L в U имеем:
Определение : Элементы называются тестовыми функциями на U и называется пространство тестовой функции на U . Мы будем использовать оба а также для обозначения этого пространства.
Распределения на U определяются как непрерывные линейные функционалы накогда это векторное пространство наделено определенной топологией, называемой канонической LF-топологией . Эту топологию, к сожалению, нелегко определить, но, тем не менее, все еще можно охарактеризовать распределения таким образом, чтобы не упоминать каноническую LF-топологию.
Предложение : если T - линейный функционал нато T является распределением тогда и только тогда, когда выполняются следующие эквивалентные условия:
- Для каждого компактного подмножества существуют константы а также такое, что для всех [1]
- Для каждого компактного подмножества существуют константы а также такое, что для всех при поддержке, содержащейся в [2]
- Для любого компактного подмножества и любая последовательность в если сходится равномерно к нулю на для всех мультииндексов , тогда
Приведенные выше характеристики могут быть использованы для определения того, является ли линейный функционал распределением, но более продвинутое использование распределений и тестовых функций (например, приложения к дифференциальным уравнениям ) ограничено, если на них не размещены топологии. а также Чтобы определить пространство распределений, мы должны сначала определить каноническую LF-топологию, которая, в свою очередь, требует, чтобы сначала были определены несколько других локально выпуклых топологических векторных пространств (TVS). Во-первых, ( ненормируемая ) топология на будет определено, то каждый будет наделен топологией подпространства, индуцированной на неми, наконец, ( неметризуемая ) каноническая LF-топология набудет определено. Пространство распределений, определяются как непрерывное сопряженное пространство изтогда наделяется (неметризуемой) сильной дуальной топологией, индуцированнойи каноническая LF-топология (эта топология является обобщением обычной индуцированной операторной нормой топологии, помещенной на непрерывные сопряженные пространства нормированных пространств ). Это, наконец, позволяет рассматривать более сложные понятия, такие как сходимость распределений (как последовательностей, так и сетей), различные (подпространства) пространств распределений и операции с распределениями, включая расширение дифференциальных уравнений на распределения.
- Выбор компактов
Через, будет любой набор компактных подмножеств такой, что (1) и (2) для любого компакта есть некоторые такой, что Наиболее распространенный выбор для находятся:
- Множество всех компактных подмножеств или же
- Множество где и для всех я , а также является относительно компактным непустое открытое подмножество(здесь, «относительно компактные» означает , что закрытие влибо в U, либо компактно).
Мы делаем в направленное множество , определяя если и только если Обратите внимание, что, хотя определения определяемых впоследствии топологий явно ссылаются на в действительности они не зависят от выбора то есть, если а также - любые два таких набора компактных подмножеств то топологии, определенные на а также используя на месте такие же, как те, которые определены с помощью на месте
Топология на C k (U)
Теперь введем полунормы, которые будут определять топологию наРазные авторы иногда используют разные семейства полунорм, поэтому мы перечислим наиболее распространенные семейства ниже. Однако результирующая топология одинакова независимо от того, какое семейство используется.
Предполагать а также - произвольное компактное подмножество Предполагать целое число такое, что [примечание 4] и это мультииндекс с длиной Для определять:в то время как для мы определяем все вышеупомянутые функции как постоянное отображение 0 .
Каждая из вышеперечисленных функций неотрицательна -значные [примечание 5] полунорм на
Каждое из следующих семейств полунорм порождает одну и ту же локально выпуклую векторную топологию на:
Предположение : в дальнейшем мы будем предполагать, что наделен локально выпуклой топологией, определяемой любым (или, что эквивалентно, всеми) семействами полунорм, описанных выше.
В этой топологии становится локально выпуклыми ( нон -normable ) пространство Фреше и все полунормов определенных выше непрерывны на этом пространстве. Все определенные выше полунормы являются непрерывными функциями наВ этой топологии сеть в сходится к тогда и только тогда, когда для каждого мультииндекса с участием и каждый чистая часть частных производных финансовых инструментов равномерно сходится к на [3] Для любоголюбой (фон Нейман) ограниченное подмножество изявляется относительно компактным подмножеством[4] В частности, подмножество ограничен тогда и только тогда, когда он ограничен в для всех [4] Пространствоявляется пространством Монтеля тогда и только тогда, когда[5]
Топология на является верхним пределом топологий подпространств, индуцированных на ТВС поскольку я пробегает неотрицательные целые числа. [3] Подмножество из открыто в этой топологии тогда и только тогда, когда существует такой, что открыт когда наделено топологией подпространств, индуцированной на нем
- Метрика, определяющая топологию
Если семейство компактов удовлетворяет а также для всех тогда полная трансляционно-инвариантная метрика на можно получить, взяв подходящую счетную комбинацию Фреше любого из перечисленных выше семейств. Например, с помощью полунорм результаты в метрике
Часто проще рассмотреть просто полунорм.
Топология на C k (K)
Как и раньше, исправить Напомним, что если любое компактное подмножество тогда
Предположение : для любого компактного подмножества K ⊆ U в дальнейшем будем предполагать, что наделен топологией подпространств, унаследованной от пространства Фреше
Для любого компактного подмножества замкнутое подпространство пространства Фреше и, таким образом, также является пространством Фреше . Для всех компактных удовлетворение обозначим отображение включения черезТогда это отображение является линейным вложением TVS (то есть это линейное отображение, которое также является топологическим вложением ), образ (или «диапазон») которых замкнут в своей области ; иначе говоря, топология на идентична топологии подпространства, от которой наследуется а также является замкнутым подмножеством Интерьер из относительно пустой. [6]
Если конечно, то является банаховым пространством [7] с топологией, определяемой нормой
И когда тогда является даже гильбертовым пространством . [7] Пространствоявляется выделенным пространством Шварца- Монтеля, поэтому, еслито это не нормируемая и , следовательно , не банахово пространство (хотя , как и все остальныеэто пространство Фреше ).
Тривиальные расширения и независимость топологии C k ( K ) от U
Определение зависит от U, поэтому мы позволим обозначим топологическое пространство которое по определению является топологическим подпространством в Предполагать открытое подмножество содержащий Дано его тривиальное продолжение на V по определению является функцией определяется:
чтобы Позволять обозначают карту, которая отправляет функцию в к его тривиальному расширению на V . Это отображение является линейной инъекцией и для каждого компактного подмножества у нас есть где векторное подпространство состоящий из карт с поддержкой, содержащейся в (поскольку также является компактным подмножеством ). Следует, чтоЕсли я ограниченто следующее индуцированное линейное отображение является гомеоморфизмом (и, следовательно, TVS-изоморфизмом):
и, таким образом, следующие две карты (которые, как и предыдущая, определяются как ) являются топологическими вложениями :
(топология на - каноническая топология LF, которая будет определена позже). С использованием мы идентифицируем с его изображением в Так как через это отождествление, также можно рассматривать как подмножество Важно отметить, что топология подпространств наследуется от (когда он рассматривается как подмножество ) идентична топологии подпространства, которую она наследует от (когда вместо этого рассматривается как подмножество через идентификацию). Таким образом, топология нане зависит от открытого подмножества U вкоторый содержит K . [6] Это оправдывает практику письменного вместо
Каноническая LF топология
Напомним, что обозначим все эти функции в которые имеют компактную опору в где отметить, что это союз всех поскольку K переходитКроме того, для каждого к , плотное подмножество Частный случай, когда дает нам пространство тестовых функций.
называется пространством тестовых функций на и его также можно обозначить как
В этом разделе каноническая топология LF определяется как прямой предел . Также возможно определить эту топологию в терминах ее окрестностей начала координат, что будет описано позже.
- Топология определяется прямыми ограничениями
Для любых двух множеств K и L мы заявляем, что если и только если что, в частности, делает коллекцию компактных подмножеств U в направленное множество (мы говорим, что такой набор направлен включением подмножеств ). Для всех компактных удовлетворение есть карты включения
Напомним, что карта является топологическим вложением . Коллекция карт
образует прямую систему в категории из локально выпуклых топологических векторных пространств , что направленная на(при включении подмножества). Прямым пределом этой системы (в категории локально выпуклых ТВП) является пара где природные включения и где теперь наделен (уникальной) сильнейшей локально выпуклой топологией, делающей все отображения включения непрерывный.
Каноническая топология LF наявляется тончайшей локально выпуклой топологией на создание всех карт включения непрерывный (где K пробегает ).
Предположение : как это принято в математической литературе, в этой статье впредь предполагается, что наделен своей канонической LF-топологией (если явно не указано иное).
- Топология, определяемая окрестностями начала координат
Если U - выпуклое подмножеството U является окрестностью начала координат в канонической ЛФ топологии тогда и только тогда, когда она удовлетворяет следующему условию:
- Для всех является окрестностью начала координат в
( CN )
Заметим, что любое выпуклое множество, удовлетворяющее этому условию, обязательно поглощает вПоскольку топология любого топологического векторного пространства трансляционно-инвариантна, любая TVS-топология полностью определяется множеством окрестностей начала координат. Это означает, что можно фактически определить каноническую топологию LF, объявив, что выпуклое сбалансированное подмножество U является окрестностью начала координат тогда и только тогда, когда оно удовлетворяет условию CN .
- Топология, определяемая с помощью дифференциальных операторов
Линейный дифференциальный оператор в U с гладкими коэффициентами является суммой
где и почти все, кроме конечного множества тождественно 0 . Целое числоназывается порядком дифференциального оператора Если является линейным дифференциальным оператором порядка k, то он индуцирует каноническое линейное отображение определяется где мы будем использовать обозначения повторно, а также обозначим это отображение через [8]
Для любой каноническая LF топология на является самой слабой локально выпуклой топологией TVS, делающей все линейные дифференциальные операторы в U порядка в непрерывные карты из в [8]
Свойства канонической LF топологии
- Независимость канонической LF топологии от
Одно из преимуществ определения канонической LF-топологии как прямого предела прямой системы состоит в том, что мы можем сразу использовать универсальное свойство прямых пределов. Еще одно преимущество состоит в том, что мы можем использовать хорошо известные результаты теории категорий, чтобы сделать вывод, что каноническая топология LF фактически не зависит от конкретного выбора направленного набора.компактов. И рассматривая разные коллекции (в частности, те упомянутые в начале статьи), мы можем вывести различные свойства этой топологии. В частности, мы можем вывести, что каноническая топология LF делаетв хаусдорфово локально выпуклое строгое LF-пространство (а также строгое LB-пространство, если), что, конечно же, является причиной того, что эта топология называется «канонической топологией LF» (см. эту сноску для более подробной информации). [примечание 6]
- Универсальная собственность
Из универсального свойства прямых пределов мы знаем, что еслиесть линейное отображение в локально выпуклое пространство Y (не обязательно хаусдорфово), то у непрерывно тогда и только тогда , когда у будет ограничена , если и только если для каждогоограничение u нанепрерывно (или ограничено). [9] [10]
- Зависимость канонической топологии СФ от U
Предположим, что V - открытое подмножество содержащий Позволять обозначают карту, которая отправляет функцию в к его тривиальному продолжению на V (которое было определено выше). Эта карта представляет собой непрерывную линейную карту. [11] Если (и только если) тогда это не плотное подмножество а также is not a topological embedding.[11] Consequently, if then the transpose of is neither one-to-one nor onto.[11]
- Bounded subsets
A subset B of is bounded in if and only if there exists some such that and B is a bounded subset of [10] Moreover, if is compact and then S is bounded in if and only if it is bounded in For any any bounded subset of (resp. ) is a relatively compact subset of (resp. ), where [10]
- Non-metrizability
For all compact the interior of in is empty so that is of the first category in itself. It follows from Baire's theorem that is not metrizable and thus also not normable (see this footnote[note 7] for an explanation of how the non-metrizable space can be complete even though it does not admit a metric). The fact that is a nuclear Montel space makes up for the non-metrizability of (see this footnote for a more detailed explanation).[note 8]
- Relationships between spaces
Using the universal property of direct limits and the fact that the natural inclusions are all topological embedding, one may show that all of the maps are also topological embeddings. Said differently, the topology on is identical to the subspace topology that it inherits from where recall that 's topology was defined to be the subspace topology induced on it by In particular, both and induces the same subspace topology on However, this does not imply that the canonical LF topology on is equal to the subspace topology induced on by ; these two topologies on are in fact never equal to each other since the canonical LF topology is never metrizable while the subspace topology induced on it by is metrizable (since recall that is metrizable). The canonical LF topology on is actually strictly finer than the subspace topology that it inherits from (thus the natural inclusion is continuous but not a topological embedding).[7]
Indeed, the canonical LF topology is so fine that if denotes some linear map that is a "natural inclusion" (such as or or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on the fine nature of the canonical LF topology means that more linear functionals on end up being continuous ("more" means as compared to a coarser topology that we could have placed on such as for instance, the subspace topology induced by some which although it would have made metrizable, it would have also resulted in fewer linear functionals on being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making into a complete TVS[12]).
- Other properties
- The differentiation map is a surjective continuous linear operator.[13]
- The bilinear multiplication map given by is not continuous; it is however, hypocontinuous.[14]
Distributions
As discussed earlier, continuous linear functionals on a are known as distributions on U. Thus the set of all distributions on U is the continuous dual space of which when endowed with the strong dual topology is denoted by
By definition, a distribution on U is defined to be a continuous linear functional on Said differently, a distribution on U is an element of the continuous dual space of when is endowed with its canonical LF topology.
We have the canonical duality pairing between a distribution T on U and a test function which is denoted using angle brackets by
One interprets this notation as the distribution T acting on the test function to give a scalar, or symmetrically as the test function acting on the distribution T.
- Characterizations of distributions
Proposition. If T is a linear functional on then the following are equivalent:
- T is a distribution;
- Definition: T is continuous;
- T is continuous at the origin;
- T is uniformly continuous;
- T is a bounded operator;
- T is sequentially continuous;
- explicitly, for every sequence in that converges in to some [note 9]
- T is sequentially continuous at the origin; in other words, T maps null sequences[note 10] to null sequences;
- explicitly, for every sequence in that converges in to the origin (such a sequence is called a null sequence),
- a null sequence is by definition a sequence that converges to the origin;
- T maps null sequences to bounded subsets;
- explicitly, for every sequence in that converges in to the origin, the sequence is bounded;
- T maps Mackey convergence null sequences[note 11] to bounded subsets;
- explicitly, for every Mackey convergent null sequence in the sequence is bounded;
- a sequence is said to be Mackey convergent to 0 if there exists a divergent sequence of positive real number such that the sequence is bounded; every sequence that is Mackey convergent to 0 necessarily converges to the origin (in the usual sense);
- The kernel of T is a closed subspace of
- The graph of T is closed;
- There exists a continuous seminorm g on such that
- There exists a constant a collection of continuous seminorms, that defines the canonical LF topology of and a finite subset such that [note 12]
- For every compact subset there exist constants and such that for all [1]
- For every compact subset there exist constants and such that for all with support contained in [15]
- For any compact subset and any sequence in if converges uniformly to zero for all multi-indices p, then
- Any of the three statements immediately above (i.e. statements 14, 15, and 16) but with the additional requirement that compact set K belongs to
Topology on the space of distributions
Definition and notation: The space of distributions on U, denoted by is the continuous dual space of endowed with the topology of uniform convergence on bounded subsets of [7] More succinctly, the space of distributions on U is
The topology of uniform convergence on bounded subsets is also called the strong dual topology.[note 13] This topology is chosen because it is with this topology that becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.[16] No matter what dual topology is placed on [note 14] a sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, will be a non-metrizable, locally convex topological vector space. The space is separable[17] and has the strong Pytkeev property[18] but it is neither a k-space[18] nor a sequential space,[17] which in particular implies that it is not metrizable and also that its topology can not be defined using only sequences.
Topological properties
- Topological vector space categories
The canonical LF topology makes into a complete distinguished strict LF-space (and a strict LB-space if and only if [19]), which implies that is a meager subset of itself.[20] Furthermore, as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of is a Fréchet space if and only if so in particular, the strong dual of which is the space of distributions on U, is not metrizable (note that the weak-* topology on also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives ).
The three spaces and the Schwartz space as well as the strong duals of each of these three spaces, are complete nuclear[21] Montel[22] bornological spaces, which implies that all six of these locally convex spaces are also paracompact[23] reflexive barrelled Mackey spaces. The spaces and are both distinguished Fréchet spaces. Moreover, both and are Schwartz TVSs.
Convergent sequences
- Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of and are sequential spaces but not Fréchet-Urysohn spaces.[17] Moreover, neither the space of test functions nor its strong dual is a sequential space (not even an Ascoli space),[17][24] which in particular implies that their topologies can not be defined entirely in terms of convergent sequences.
A sequence in converges in if and only if there exists some such that contains this sequence and this sequence converges in ; equivalently, it converges if and only if the following two conditions hold:[25]
- There is a compact set containing the supports of all
- For each multi-index the sequence of partial derivatives tends uniformly to
Neither the space nor its strong dual is a sequential space,[17][24] and consequently, their topologies can not be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is not enough to define the canonical LF topology on The same can be said of the strong dual topology on
- What sequences do characterize
Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,[26] which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually define the convergence of a sequence of distributions; this is fine for sequences but it does not extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology).
Sequences characterize continuity of linear maps valued in locally convex space. Suppose X is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map into a locally convex space Y is continuous if and only if it maps null sequences[note 10] in X to bounded subsets of Y.[note 15] More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequences[note 11] to bounded subsets of So in particular, if a linear map into a locally convex space is sequentially continuous at the origin then it is continuous.[27] However, this does not necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs.
For every is sequentially dense in [28] Furthermore, is a sequentially dense subset of (with its strong dual topology)[29] and also a sequentially dense subset of the strong dual space of [29]
- Sequences of distributions
A sequence of distributions converges with respect to the weak-* topology on to a distribution T if and only if
for every test function For example, if is the function
and is the distribution corresponding to then
as so δ in Thus, for large the function can be regarded as an approximation of the Dirac delta distribution.
- Other properties
- The strong dual space of is TVS isomorphic to via the canonical TVS-isomorphism defined by sending to value at (that is, to the linear functional on defined by sending to );
- On any bounded subset of the weak and strong subspace topologies coincide; the same is true for ;
- Every weakly convergent sequence in is strongly convergent (although this does not extend to nets).
Localization of distributions
There is no way to define the value of a distribution in at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.
Restrictions to an open subset
Let U and V be open subsets of with V ⊆ U. Let be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U. The transpose of is called the restriction mapping and is denoted by
The map is a continuous injection where if V ⊆ U then it is not a topological embedding and its range is not dense in which implies that this map's transpose is neither injective nor surjective and that the topology that transfers from onto its image is strictly finer than the subspace topology that induces on this same set.[11] A distribution is said to be extendible to U if it belongs to the range of the transpose of and it is called extendible if it is extendable to [11]
For any distribution the restriction ρVU(T) is a distribution in defined by:
Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = ℝ and V = (0, 2), then the distribution
is in but admits no extension to
Gluing and distributions that vanish in a set
Theorem[30] — Let be a collection of open subsets of For each let and suppose that for all the restriction of to is equal to the restriction of to (note that both restrictions are elements of ). Then there exists a unique such that for all the restriction of T to is equal to
Let V be an open subset of U. is said to vanish in V if for all such that we have T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map ρVU.
- Corollary.[30] Let be a collection of open subsets of and let T = 0 if and only if for each the restriction of T to is equal to 0.
- Corollary.[30] The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.
Support of a distribution
This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and does not vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called the support of T.[30] Thus
If is a locally integrable function on U and if is its associated distribution, then the support of is the smallest closed subset of U in the complement of which is almost everywhere equal to 0.[30] If is continuous, then the support of is equal to the closure of the set of points in U at which does not vanish.[30] The support of the distribution associated with the Dirac measure at a point is the set [30] If the support of a test function does not intersect the support of a distribution T then Tf = 0. A distribution T is 0 if and only if its support is empty. If is identically 1 on some open set containing the support of a distribution T then fT = T. If the support of a distribution T is compact then it has finite order and furthermore, there is a constant C and a non-negative integer N such that:[6]
If T has compact support then it has a unique extension to a continuous linear functional on ; this functional can be defined by where is any function that is identically 1 on an open set containing the support of T.[6]
If and then and Thus, distributions with support in a given subset form a vector subspace of ; such a subspace is weakly closed in if and only if A is closed in U.[31] Furthermore, if is a differential operator in U, then for all distributions T on U and all we have and [31]
Distributions with compact support
- Support in a point set and Dirac measures
For any let denote the distribution induced by the Dirac measure at x. For any and distribution the support of T is contained in if and only if T is a finite linear combination of derivatives of the Dirac measure at [32] If in addition the order of T is then there exist constants such that:[33]
Said differently, if T has support at a single point then T is in fact a finite linear combination of distributional derivatives of the δ function at P. That is, there exists an integer m and complex constants aα such that
where is the translation operator.
- Distribution with compact support
Theorem[6] — Suppose T is a distribution on U with compact support K. There exists a continuous function defined on U and a multi-index p such that
where the derivatives are understood in the sense of distributions. That is, for all test functions on U,
- Distributions of finite order with support in an open subset
Theorem[6] — Suppose T is a distribution on U with compact support K and let V be an open subset of U containing K. Since every distribution with compact support has finite order, take N to be the order of T and define There exists a family of continuous functions defined on U with support in V such that
where the derivatives are understood in the sense of distributions. That is, for all test functions on U,
Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of (or the Schwartz space for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.
- Distributions as sheafs
Theorem[34] — Let T be a distribution on U. There exists a sequence in such that each Ti has compact support and every compact subset K ⊆ U intersects the support of only finitely many Ti, and the sequence of partial sums defined by converges in to T; in other words we have:
Recall that a sequence converges in (with its strong dual topology) if and only if it converges pointwise.
Decomposition of distributions as sums of derivatives of continuous functions
By combining the above results, one may express any distribution on U as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on U. In other words for arbitrary we can write:
where are finite sets of multi-indices and the functions are continuous.
Theorem[35] — Let T be a distribution on U. For every multi-index p there exists a continuous function gp on U such that
- any compact subset K of U intersects the support of only finitely many gp, and
Moreover, if T has finite order, then one can choose gp in such a way that only finitely many of them are non-zero.
Note that the infinite sum above is well-defined as a distribution. The value of T for a given can be computed using the finitely many gα that intersect the support of
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if is a linear map which is continuous with respect to the weak topology, then it is possible to extend A to a map by passing to the limit.[note 16][citation needed][clarification needed]
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator because it provides a unified approach that the many definitions in the theory of distributions and because of its many well-known topological properties.[36] In general the transpose of a continuous linear map is the linear map defined by or equivalently, it is the unique map satisfying for all and all Since A is continuous, the transpose is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).
In the context of distributions, the characterization of the transpose can be refined slightly. Let be a continuous linear map. Then by definition, the transpose of A is the unique linear operator that satisfies:
- for all and all
However, since the image of is dense in it is sufficient that the above equality hold for all distributions of the form where Explicitly, this means that the above condition holds if and only if the condition below holds:
- for all
Differential operators
Differentiation of distributions
Let is the partial derivative operator In order to extend we compute its transpose:
Therefore Therefore the partial derivative of with respect to the coordinate is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction is a linear operator on
More generally, if is an arbitrary multi-index, then the partial derivative of the distribution is defined by
Differentiation of distributions is a continuous operator on this is an important and desirable property that is not shared by most other notions of differentiation.
If T is a distribution in ℝ then
where is the derivative of T and τx is translation by x; thus the derivative of T may be viewed as a limit of quotients.[37]
Differential operators acting on smooth functions
A linear differential operator in U with smooth coefficients acts on the space of smooth functions on Given we would like to define a continuous linear map, that extends the action of on to distributions on In other words we would like to define such that the following diagram commutes:
Where the vertical maps are given by assigning its canonical distribution which is defined by: for all With this notation the diagram commuting is equivalent to:
In order to find we consider the transpose of the continuous induced map defined by As discussed above, for any the transpose may be calculated by:
For the last line we used integration by parts combined with the fact that and therefore all the functions have compact support.[note 17] Continuing the calculation above we have for all
Define the formal transpose of which will be denoted by to avoid confusion with the transpose map, to be the following differential operator on U:
The computations above have shown that:
- Lemma. Let be a linear differential operator with smooth coefficients in Then for all we have
- which is equivalent to:
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, i.e. [8] enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator defined by We claim that the transpose of this map, can be taken as To see this, for every , compute its action on a distribution of the form with :
We call the continuous linear operator the differential operator on distributions extending P.[8] Its action on an arbitrary distribution is defined via:
If converges to then for every multi-index converges to
Multiplication of distributions by smooth functions
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if is a smooth function then is a differential operator of order 0, whose formal transpose is itself (i.e. ). The induced differential operator maps a distribution T to a distribution denoted by We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of multiplication by a smooth function. If is a smooth function and T is a distribution on U, then the product mT is defined by
This definition coincides with the transpose definition since if is the operator of multiplication by the function m (i.e., ), then
so that
Under multiplication by smooth functions, is a module over the ring With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if δ′ is the Dirac delta distribution on ℝ, then mδ = m(0)δ, and if δ′ is the derivative of the delta distribution, then
The bilinear multiplication map given by is not continuous; it is however, hypocontinuous.[14]
Example. For any distribution T, the product of T with the function that is identically 1 on U is equal to T.
Example. Suppose is a sequence of test functions on U that converges to the constant function For any distribution T on U, the sequence converges to [38]
If converges to and converges to then converges to
Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. 1/x is the distribution obtained by the Cauchy principal value
If δ is the Dirac delta distribution then
but
so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions.
Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non linear, like for example the Navier–Stokes equations of fluid dynamics.
Several not entirely satisfactory[citation needed] theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.
Inspired by Lyons' rough path theory,[39] Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures[40]), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.
Composition with a smooth function
Let T be a distribution on Let V be an open set in and F : V → U. If F is a submersion, it is possible to define
This is the composition of the distribution T with F, and is also called the pullback of T along F, sometimes written
The pullback is often denoted F*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative dF(x) of F is a surjective linear map for every x ∈ V. A necessary (but not sufficient) condition for extending F# to distributions is that F be an open mapping.[41] The inverse function theorem ensures that a submersion satisfies this condition.
If F is a submersion, then F# is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F# is a continuous linear operator on Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.[42]
In the special case when F is a diffeomorphism from an open subset V of onto an open subset U of change of variables under the integral gives
In this particular case, then, F# is defined by the transpose formula:
Convolution
Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions. Recall that if and g are functions on then we denote by the convolution of and g, defined at to be the integral
provided that the integral exists. If are such that 1/r = (1/p) + (1/q) - 1 then for any functions and we have and [43] If and g are continuous functions on at least one of which has compact support, then and if then the value of on A do not depend on the values of outside of the Minkowski sum [43]
Importantly, if has compact support then for any the convolution map is continuous when considered as the map or as the map [43]
- Translation and symmetry
Given the translation operator τa sends to defined by This can be extended by the transpose to distributions in the following way: given a distribution T, the translation of by is the distribution defined by [44][45]
Given define the function by Given a distribution T, let be the distribution defined by The operator is called the symmetry with respect to the origin.[44]
Convolution of a test function with a distribution
Convolution with defines a linear map:
which is continuous with respect to the canonical LF space topology on
Convolution of with a distribution can be defined by taking the transpose of Cf relative to the duality pairing of with the space of distributions.[46] If then by Fubini's theorem
Extending by continuity, the convolution of with a distribution T is defined by
for all
An alternative way to define the convolution of a test function and a distribution T is to use the translation operator τa. The convolution of the compactly supported function and the distribution T is then the function defined for each by
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support then if is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on to the restriction of an entire function of exponential type in to ) then the same is true of [44] If the distribution T has compact support as well, then is a compactly supported function, and the Titchmarsh convolution theorem Hörmander (1983, Theorem 4.3.3) implies that
where ch denotes the convex hull and supp denotes the support.
Convolution of a smooth function with a distribution
Let and and assume that at least one of and T has compact support. The convolution of and T, denoted by or by is the smooth function:[44]
satisfying for all :
If T is a distribution then the map is continuous as a map where if in addition T has compact support then it is also continuous as the map and continuous as the map [44]
If is a continuous linear map such that for all and all then there exists a distribution such that for all [6]
Example.[6] Let H be the Heaviside function on ℝ. For any
Let be the Dirac measure at 0 and its derivative as a distribution. Then and Importantly, the associative law fails to hold:
Convolution of distributions
It is also possible to define the convolution of two distributions S and T on provided one of them has compact support. Informally, in order to define S ∗ T where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the associativity formula
continues to hold for all test functions [47]
It is also possible to provide a more explicit characterization of the convolution of distributions.[46] Suppose that S and T are distributions and that S has compact support. Then the linear maps
are continuous. The transposes of these maps,
are consequently continuous and one may show that
- [44]
This common value is called the convolution of S and T and it is a distribution that is denoted by or It satisfies [44] If S and T are two distributions, at least one of which has compact support, then for any [44] If T is a distribution in and if is a Dirac measure then [44]
Suppose that it is T that has compact support. For consider the function
It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index α,
The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.[44]
This definition of convolution remains valid under less restrictive assumptions about S and T.[48]
The convolution of distributions with compact support induces a continuous bilinear map defined by where denotes the space of distributions with compact support.[14] However, the convolution map as a function is not continuous[14] although it is separately continuous.[49] The convolution maps and given by both fail to be continuous.[14] Each of these non-continuous maps is, however, separately continuous and hypocontinuous.[14]
Convolution versus multiplication
In general, regularity is required for multiplication products and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let be a rapidly decreasing tempered distribution or, equivalently, be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let be the normalized (unitary, ordinary frequency) Fourier transform [50] then, according to Schwartz (1951),
hold within the space of tempered distributions.[51][52][53] In particular, these equations become the Poisson Summation Formula if is the Dirac Comb.[54] The space of all rapidly decreasing tempered distributions is also called the space of convolution operators and the space of all ordinary functions within the space of tempered distributions is also called the space of multiplication operators More generally, and [55][56] A particular case is the Paley-Wiener-Schwartz Theorem which states that and This is because and In other words, compactly supported tempered distributions belong to the space of convolution operators and Paley-Wiener functions better known as bandlimited functions, belong to the space of multiplication operators [57]
For example, let be the Dirac comb and be the Dirac delta then is the function that is constantly one and both equations yield the Dirac comb identity. Another example is to let be the Dirac comb and be the rectangular function then is the sinc function and both equations yield the Classical Sampling Theorem for suitable functions. More generally, if is the Dirac comb and is a smooth window function (Schwartz function), e.g. the Gaussian, then is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.
Tensor product of distributions
Let and be open sets. Assume all vector spaces to be over the field where or For we define the following family of functions:
Given and we define the following functions:
Note that and Now we define the following continuous linear maps associated to and :
Moreover if either (resp. ) has compact support then it also induces a continuous linear map of (resp. ).[58]
Fubini's theorem for distributions[58] — Let and For every we have:
Definition. The tensor product of and denoted by or is a distribution in and is defined by:[58]
Schwartz kernel theorem
The tensor product defines a bilinear map
the span of the range of this map is a dense subspace of its codomain. Furthermore, [58] Moreover induces continuous bilinear maps:
where denotes the space of distributions with compact support and is the Schwartz space of rapidly decreasing functions.[14]
Schwartz kernel theorem[59] — We have canonical TVS isomorphisms:
Here represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product, since these spaces are nuclear) and has the topology of uniform convergence on bounded subsets.
This result does not hold for Hilbert spaces such as and its dual space.[60] Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space ? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because is a nuclear space that the Schwartz kernel theorem holds.
Spaces of distributions
For all 0 < k < ∞ and all 1 < p < ∞, all of the following canonical injections are continuous and have a range that is dense in their codomain:
where the topologies on () are defined as direct limits of the spaces in a manner analogous to how the topologies on were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, is even sequentially dense in every [28] All of the canonical injections () are continuous and the range of this injection is dense in the codomain if and only if (here has its usual norm topology).[61]
Suppose that is one of the spaces () or () or (). Since the canonical injection is a continuous injection whose image is dense in the codomain, the transpose is a continuous injection. This transpose thus allows us to identify with a certain vector subspace of the space of distributions. This transpose map is not necessarily a TVS-embedding so that topology that this map transfers to the image is finer than the subspace topology that this space inherits from A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced by is called a space of distributions.[61] Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order some integer, distributions induced by a positive Radon measure, distributions induced by an -function, etc.) and any representation theorem about the dual space of X may, through the transpose be transferred directly to elements of the space
Radon measures
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Note that the continuous dual space can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is,
- if then there exists a Radon measure on U such that for all and
- if is a Radon measure on U then the linear functional on defined by is continuous.
Through the injection every Radon measure becomes a distribution on U. If is a locally integrable function on U then the distribution is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally functions in U:
- Theorem.[34] Suppose is a Radon measure, V ⊆ U is a neighborhood of the support of T, and There exists is a family of locally functions in U such that
- and for very
- Positive Radon measures
A linear function T on a space of functions is called positive if whenever a function that belongs to the domain of T is non-negative (i.e. is real-valued and ) then One may show that every positive linear functional on is necessarily continuous (i.e. necessarily a Radon measure).[62]Note that Lebesgue measure is an example of a positive Radon measure.
Locally integrable functions as distributions
One particularly important class of Radon measures are those that are induced locally integrable functions. The function is called locally integrable if it is Lebesgue integrable over every compact subset K of U.[note 18] This is a large class of functions which includes all continuous functions and all Lp functions. The topology on is defined in such a fashion that any locally integrable function yields a continuous linear functional on – that is, an element of – denoted here by Tf, whose value on the test function is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying Tf with provided no confusion can arise, and thus the pairing between Tf and is often written
If and g are two locally integrable functions, then the associated distributions Tf and Tg are equal to the same element of if and only if and g are equal almost everywhere (see, for instance, Hörmander (1983, Theorem 1.2.5)). In a similar manner, every Radon measure on U defines an element of whose value on the test function is As above, it is conventional to abuse notation and write the pairing between a Radon measure and a test function as Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.
- Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on [29] This means that for any there is a sequence of test functions, that converges to (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
Furthermore, is also sequentially dense in the strong dual space of [29]
Distributions with compact support
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).[31]
The elements of can be identified as the space of distributions with compact support.[31] Explicitly, if T is a distribution on U then the following are equivalent,
- ;
- the support of T is compact;
- the restriction of to when that space is equipped with the subspace topology inherited from (a coarser topology than the canonical LF topology), is continuous;[31]
- there is a compact subset K of U such that for every test function whose support is completely outside of K, we have
Compactly supported distributions define continuous linear functionals on the space ; recall that the topology on is defined such that a sequence of test functions converges to 0 if and only if all derivatives of converge uniformly to 0 on every compact subset of U. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from to
Distributions of finite order
Let The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Consequently, the image of denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so 's topology is finer than the subspace topology that this set inherits from ). The elements of are the distributions of order ≤ k.[34] The distributions of order ≤ 0, which are also called distributions of order 0, are exactly the distributions that are Radon measures (described above).
For a distribution of order k is a distribution of order ≤ k that is not a distribution of order ≤ k − 1.[34]
A distribution is said to be of finite order if there is some integer k such that it is a distribution of order ≤ k, and the set of distributions of finite order is denoted by Note that if k ≤ l then so that is a vector subspace of and furthermore, if and only if [34]
- Structure of distributions of finite order
Every distribution with compact support in U is a distribution of finite order.[34] Indeed, every distribution in U is locally a distribution of finite order, in the following sense:[34] If V is an open and relatively compact subset of U and if is the restriction mapping from U to V, then the image of under is contained in
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:
- Theorem.[34] Suppose has finite order ≤ k and Given any open subset V of U containing the support of T, there is a family of Radon measures in U, such that for very and
Example. (Distributions of infinite order) Let U := (0, ∞) and for every test function let
Then S is a distribution of infinite order on U. Moreover, S can not be extended to a distribution on ℝ; that is, there exists no distribution T on ℝ such that the restriction of T to U is equal to T.[63]
Tempered distributions and Fourier transform
Defined below are the tempered distributions, which form a subspace of the space of distributions on This is a proper subspace: while every tempered distribution is a distribution and an element of the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in
- Schwartz space
The Schwartz space, is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of multiplied with any power of |x|, converges to 0 as |x| → ∞. These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices and define:
Then is in the Schwartz space if all the values satisfy:
The family of seminorms pα, β defines a locally convex topology on the Schwartz space. For n = 1, the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:[64]
Otherwise, one can define a norm on via
The Schwartz space is a Fréchet space (i.e. a complete metrizable locally convex space). Because the Fourier transform changes into multiplication by and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of which implies that such a sequence must converge to zero in [64]
is dense in The subset of all analytic Schwartz functions is dense in as well.[65]
The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
where represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).[59]
- Tempered distributions
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus, the image of the transpose map, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).
The space is called the space of >tempered distributions is it is the continuous dual of the Schwartz space. Equivalently, a distribution T is a tempered distribution if and only if
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of L p ( R n ) {\displaystyle L^{p}(\mathbb {R} ^{n})} for p ≥ 1 are tempered distributions.
The tempered distributions can also be characterized as slowly growing, meaning that each derivative of T grows at most as fast as some polynomial. This characterization is dual to the rapidly falling behaviour of the derivatives of a function in the Schwartz space, where each derivative of decays faster than every inverse power of |x|. An example of a rapidly falling function is for any positive n, λ, β.
- Fourier transform
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform is a TVS-automorphism of the Schwartz space, and the Fourier transform is defined to be its transpose which (abusing notation) will again be denoted by F. So the Fourier transform of the tempered distribution T is defined by (FT)(ψ) = T(Fψ) for every Schwartz function ψ. FT is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
and also with convolution: if T is a tempered distribution and ψ is a slowly increasing smooth function on ψT is again a tempered distribution and
is the convolution of FT and Fψ. In particular, the Fourier transform of the constant function equal to 1 is the δ distribution.
- Expressing tempered distributions as sums of derivatives
If is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions
This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function F and a multi-index α such that
Restriction of distributions to compact sets
If then for any compact set there exists a continuous function F compactly supported in (possibly on a larger set than K itself) and a multi-index α such that on
Using holomorphic functions as test functions
The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.
See also
- Colombeau algebra
- Current (mathematics)
- Distribution (number theory)
- Distribution on a linear algebraic group
- Gelfand triple
- Generalized function
- Homogeneous distribution
- Hyperfunction
- Laplacian of the indicator
- Limit of a distribution
- Linear form
- Malgrange–Ehrenpreis theorem
- Pseudodifferential operator
- Riesz representation theorem
- Vague topology
- Weak solution
Notes
- ^ turns out to also be linear and continuous when the space of test functions is given a certain topology called the canonical LF topology.
- ^ The Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin.
- ^ Except for the trivial (i.e. identically 0) map, which of course is always analytic.
- ^ Note that i being an integer implies This is sometimes expressed as Since the inequality "" means: if while if then it means
- ^ The image of the compact set under a continuous -valued map (e.g. for ) is itself a compact, and thus bounded, subset of If then this implies that each of the functions defined above is -valued (i.e. none of the supremums above are ever equal to ).
- ^ If we take to be the set of all compact subsets of U then we can use the universal property of direct limits to conclude that the inclusion is a continuous and even that they are topological embedding for every compact subset If however, we take to be the set of closures of some countable increasing sequence of relatively compact open subsets of U having all of the properties mentioned earlier in this in this article then we immediately deduce that is a Hausdorff locally convex strict LF-space (and even a strict LB-space when ). All of these facts can also be proved directly without using direct systems (although with more work).
- ^ For any TVS X (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using only the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not require the existence of any metric. However, if the TVS X is metrizable and if d is any translation-invariant metric on X that defines its topology, then X is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if math>(X, d) is a complete metric space. So if a TVS X happens to have a topology that can be defined by such a metric d then d may be used to deduce the completeness of X but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on ).
- ^ One reason for giving the canonical LF topology is because it is with this topology that and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. infinite differentiability, such as and ) end up being nuclear TVSs while TVSs associated with finite continuous differentiability (such as with K compact and ) often end up being non-nuclear spaces, such as Banach spaces.
- ^ Even though the topology of is not metrizable, a linear functional on is continuous if and only if it is sequentially continuous.
- ^ a b A null sequence is a sequence that converges to the origin.
- ^ a b A sequence is said to be Mackey convergent to 0 in if there exists a divergent sequence of positive real number such that is a bounded set in
- ^ If is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
- ^ In functional analysis, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space where if X is a normed space then this strong dual topology is the same as the usual norm-induced topology on
- ^ Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology.
- ^ Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences.
- ^ This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.
- ^ For example let and take to be the ordinary derivative for functions of one real variable and assume the support of to be contained in the finite interval then since
- ^ For more information on such class of functions, see the entry on locally integrable functions.
References
- ^ a b Trèves 2006, pp. 222-223.
- ^ See for example Grubb 2009, p. 14.
- ^ a b Trèves 2006, pp. 85-89.
- ^ a b Trèves 2006, pp. 142-149.
- ^ Trèves 2006, pp. 356-358.
- ^ a b c d e f g h Rudin 1991, pp. 149-181.
- ^ a b c d Trèves 2006, pp. 131-134.
- ^ a b c d Trèves 2006, pp. 247-252.
- ^ Trèves 2006, pp. 126-134.
- ^ a b c Trèves 2006, pp. 136-148.
- ^ a b c d e Trèves 2006, pp. 245-247.
- ^ Rudin 1991, pp. 149-155.
- ^ Narici & Beckenstein 2011, pp. 446-447.
- ^ a b c d e f g Trèves 2006, p. 423.
- ^ See for example Grubb 2009, p. 14.
- ^ See for example Schaefer & Wolff 1999, p. 173.
- ^ a b c d e Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
- ^ a b Gabriyelyan, S.S. Kakol J., and·Leiderman, A. "The strong Pitkeev property for topological groups and topological vector spaces"
- ^ Trèves 2006, pp. 195-201.
- ^ Narici & Beckenstein 2011, p. 435.
- ^ Trèves 2006, pp. 526-534.
- ^ Trèves 2006, p. 357.
- ^ "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020.
It is a Montel space, hence paracompact, and so normal.
- ^ a b T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
- ^ According to Gel'fand & Shilov 1966–1968, v. 1, §1.2
- ^ Trèves 2006, pp. 351-359.
- ^ Narici & Beckenstein 2011, pp. 441-457.
- ^ a b Trèves 2006, pp. 150-160.
- ^ a b c d Trèves 2006, pp. 300-304.
- ^ a b c d e f g Trèves 2006, pp. 253-255.
- ^ a b c d e Trèves 2006, pp. 255-257.
- ^ Trèves 2006, pp. 264-266.
- ^ Rudin 1991, p. 165.
- ^ a b c d e f g h Trèves 2006, pp. 258-264.
- ^ Rudin 1991, pp. 169-170.
- ^ Strichartz 1994, §2.3; Trèves 2006.
- ^ Rudin 1991, p. 180.
- ^ Trèves 2006, p. 261.
- ^ Lyons, T. (1998). "Differential equations driven by rough signals". Revista Matemática Iberoamericana: 215–310. doi:10.4171/RMI/240.
- ^ Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4.
- ^ See for example Hörmander 1983, Theorem 6.1.1.
- ^ See Hörmander 1983, Theorem 6.1.2.
- ^ a b c Trèves 2006, pp. 278-283.
- ^ a b c d e f g h i j Trèves 2006, pp. 284-297.
- ^ See for example Rudin 1991, §6.29.
- ^ a b Trèves 2006, Chapter 27.
- ^ Hörmander 1983, §IV.2 proves the uniqueness of such an extension.
- ^ See for instance Gel'fand & Shilov 1966–1968, v. 1, pp. 103–104 and Benedetto 1997, Definition 2.5.8.
- ^ Trèves 2006, p. 294.
- ^ Folland, G.B. (1989). Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press.
- ^ Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
- ^ Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
- ^ Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
- ^ Woodward, P.M. (1953). Probability and Information Theory with Applications to Radar. Oxford, UK: Pergamon Press.
- ^ Trèves 2006, pp. 318-319.
- ^ Friedlander, F.G.; Joshi, M.S. (1998). Introduction to the Theory of Distributions. Cambridge, UK: Cambridge University Press.
- ^ Schwartz 1951.
- ^ a b c d Trèves 2006, pp. 416-419.
- ^ a b Trèves 2006, p. 531.
- ^ Trèves 2006, pp. 509-510.
- ^ a b Trèves 2006, pp. 240-252.
- ^ Trèves 2006, p. 218.
- ^ Rudin 1991, pp. 177-181.
- ^ a b Trèves 2006, pp. 92-94.
- ^ Trèves 2006, pp. 160.
Bibliography
- Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
- Benedetto, J.J. (1997), Harmonic Analysis and Applications, CRC Press.
- Folland, G.B. (1989). Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press.
- Friedlander, F.G.; Joshi, M.S. (1998). Introduction to the Theory of Distributions. Cambridge, UK: Cambridge University Press..
- Gårding, L. (1997), Some Points of Analysis and their History, American Mathematical Society.
- Gel'fand, I.M.; Shilov, G.E. (1966–1968), Generalized functions, 1–5, Academic Press.
- Grubb, G. (2009), Distributions and Operators, Springer.
- Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.
- Horváth, John (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. 1. Reading, MA: Addison-Wesley Publishing Company. ISBN 978-0201029857.
- Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing..
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schwartz, Laurent (1954), "Sur l'impossibilité de la multiplications des distributions", C. R. Acad. Sci. Paris, 239: 847–848.
- Schwartz, Laurent (1951), Théorie des distributions, 1–2, Hermann.
- Sobolev, S.L. (1936), "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales", Mat. Sbornik, 1: 39–72.
- Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 0-691-08078-X.
- Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, ISBN 0-8493-8273-4.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Woodward, P.M. (1953). Probability and Information Theory with Applications to Radar. Oxford, UK: Pergamon Press.
Further reading
- M. J. Lighthill (1959). Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. ISBN 0-521-09128-4 (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
- V.S. Vladimirov (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0
- Vladimirov, V.S. (2001) [1994], "Generalized function", Encyclopedia of Mathematics, EMS Press.
- Vladimirov, V.S. (2001) [1994], "Generalized functions, space of", Encyclopedia of Mathematics, EMS Press.
- Vladimirov, V.S. (2001) [1994], "Generalized function, derivative of a", Encyclopedia of Mathematics, EMS Press.
- Vladimirov, V.S. (2001) [1994], "Generalized functions, product of", Encyclopedia of Mathematics, EMS Press.
- Oberguggenberger, Michael (2001) [1994], "Generalized function algebras", Encyclopedia of Mathematics, EMS Press.