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В области математического анализа , интерполяция пространство является пространством , которое лежит «между» двумя другими банаховыми пространствами . Основные области применения находится в пространствах Соболева , где пространство функций , которые имеют нецелое число производных интерполируются из пространств функций с целым числом производных.

История [ править ]

Теория интерполяции векторных пространств началась с наблюдения Юзефа Марцинкевича , позже обобщенного и теперь известного как теорема Рисса-Торина . Проще говоря, если линейная функция непрерывна на некотором пространстве L p, а также на некотором пространстве L q , то она также непрерывна на пространстве L r для любого промежуточного r между p и q . Другими словами, L r - это пространство, которое занимает промежуточное положение между L p и L q .

При разработке пространств Соболева стало ясно, что следовые пространства не являются обычными функциональными пространствами (с целым числом производных), и Жак-Луи Лионс обнаружил, что действительно эти следовые пространства составлены из функций, имеющих нецелую степень дифференцируемости.

Многие методы были разработаны для создания таких пространств функций, включая преобразование Фурье , комплексную интерполяцию, [1] действительную интерполяцию [2], а также другие инструменты (см., Например, дробную производную ).

Настройка интерполяции [ править ]

Банахово пространство X называется непрерывно вложено в хаусдорфовом топологических векторном пространстве Z , когда Х представляет собой линейное подпространство Z таких , что отображение вложения из X в Z непрерывно. Совместимая пара ( X 0 , X 1 ) банаховых пространств состоит из двух банаховых пространств Х 0 и Х 1 , которые непрерывно внедренных в том же хаусдорфовом топологическое векторное пространство Z . [3] Вложение в линейное пространство.Z позволяет рассматривать два линейных подпространства

а также

Интерполяция не зависит только от классов изоморфной (или изометрической) эквивалентности X 0 и X 1 . Это зависит существенным образом от конкретного относительного положения , что X 0 и X 1 занимают в большем пространстве Z .

Можно определить нормы на X 0X 1 и X 0 + X 1 следующим образом:

С этими нормами пересечение и сумма являются банаховыми пространствами. Следующие включения являются непрерывными:

Interpolation studies the family of spaces X that are intermediate spaces between X0 and X1 in the sense that

where the two inclusions maps are continuous.

Примером такой ситуации является пара ( L 1 ( R ), L ( R )) , где два банаховых пространства непрерывно вложены в пространство Z измеримых функций на вещественной прямой, снабженное топологией сходимости по мере . В этой ситуации пространства L p ( R ) при 1 ≤ p ≤ ∞ занимают промежуточное положение между L 1 ( R ) и L ( R ) . В более общем смысле,

с непрерывными впрысками, так что при данном условии L p ( R ) занимает промежуточное положение между L p 0 ( R ) и L p 1 ( R ) .

Определение. Учитывая две совместимые пары ( X 0 , X 1 ) и ( Y 0 , Y 1 ) , пара интерполяции - это пара ( X , Y ) банаховых пространств с двумя следующими свойствами:
  • Пространство X занимает промежуточное положение между X 0 и X 1 , а Y занимает промежуточное положение между Y 0 и Y 1 .
  • Если L является любой линейный оператор из X 0 + Х 1 до Y 0 + Y 1 , который отображает непрерывно X 0 до Y 0 и Х 1 до Y 1 , то она также непрерывно отображает X на Y .

Пара интерполяции ( X , Y ) называется экспонентой θ (при 0 < θ <1 ), если существует константа C такая, что

для всех операторов L, как указано выше. Обозначение || L || X , Y для нормы L как отображение из X в Y . Если C  = 1 , мы говорим, что ( X , Y ) - пара точных интерполяций показателя θ .

Сложная интерполяция [ править ]

If the scalars are complex numbers, properties of complex analytic functions are used to define an interpolation space. Given a compatible couple (X0, X1) of Banach spaces, the linear space consists of all functions f  : CX0 + X1, that are analytic on S = {z : 0 < Re(z) < 1}, continuous on S = {z : 0 ≤ Re(z) ≤ 1}, and for which all the following subsets are bounded:

{ f (z) : zS} ⊂ X0 + X1,
{ f (it) : tR} ⊂ X0,
{ f (1 + it) : tR} ⊂ X1.

is a Banach space under the norm

Definition.[4] For 0 < θ < 1, the complex interpolation space (X0, X1)θ is the linear subspace of X0 + X1 consisting of all values f(θ) when f varies in the preceding space of functions,

The norm on the complex interpolation space (X0, X1)θ is defined by

Equipped with this norm, the complex interpolation space (X0, X1)θ is a Banach space.

Theorem.[5] Given two compatible couples of Banach spaces (X0, X1) and (Y0, Y1), the pair ((X0, X1)θ, (Y0, Y1)θ) is an exact interpolation pair of exponent θ, i.e., if T : X0 + X1Y0 + Y1, is a linear operator bounded from Xj to Yj, j = 0, 1, then T is bounded from (X0, X1)θ to (Y0, Y1)θ and

The family of Lp spaces (consisting of complex valued functions) behaves well under complex interpolation.[6] If (R, Σ, μ) is an arbitrary measure space, if 1 ≤ p0, p1 ≤ ∞ and 0 < θ < 1, then

with equality of norms. This fact is closely related to the Riesz–Thorin theorem.

Real interpolation[edit]

There are two ways for introducing the real interpolation method. The first and most commonly used when actually identifying examples of interpolation spaces is the K-method. The second method, the J-method, gives the same interpolation spaces as the K-method when the parameter θ is in (0, 1). That the J- and K-methods agree is important for the study of duals of interpolation spaces: basically, the dual of an interpolation space constructed by the K-method appears to be a space constructed form the dual couple by the J-method; see below.

K-method[edit]

The K-method of real interpolation[7] can be used for Banach spaces over the field R of real numbers.

Definition. Let (X0, X1) be a compatible couple of Banach spaces. For t > 0 and every xX0 + X1, let

Changing the order of the two spaces results in:[8]

Let

The K-method of real interpolation consists in taking Kθ,q(X0, X1) to be the linear subspace of X0 + X1 consisting of all x such that ||x||θ,q;K < ∞.

Example[edit]

An important example is that of the couple (L1(R, Σ, μ), L(R, Σ, μ)), where the functional K(t, f ; L1, L) can be computed explicitly. The measure μ is supposed σ-finite. In this context, the best way of cutting the function f  ∈ L1 + L as sum of two functions f0L1 and f1L is, for some s > 0 to be chosen as function of t, to let f1(x) be given for all xR by

The optimal choice of s leads to the formula[9]

where f ∗ is the decreasing rearrangement of f.

J-method[edit]

As with the K-method, the J-method can be used for real Banach spaces.

Definition. Let (X0, X1) be a compatible couple of Banach spaces. For t > 0 and for every vector xX0X1, let

A vector x in X0 + X1 belongs to the interpolation space Jθ,q(X0, X1) if and only if it can be written as

where v(t) is measurable with values in X0X1 and such that

The norm of x in Jθ,q(X0, X1) is given by the formula

Relations between the interpolation methods[edit]

The two real interpolation methods are equivalent when 0 < θ < 1.[10]

Theorem. Let (X0, X1) be a compatible couple of Banach spaces. If 0 < θ < 1 and 1 ≤ q ≤ ∞, then
with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if X0 and X1 form a direct sum, then the intersection and the J-spaces are the null space, and a simple computation shows that the K-spaces are also null.

When 0 < θ < 1, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters θ and q. The notation for this real interpolation space is (X0, X1)θ,q. One has that

For a given value of θ, the real interpolation spaces increase with q:[11] if 0 < θ < 1 and 1 ≤ qr ≤ ∞, the following continuous inclusion holds true:

Theorem. Given 0 < θ < 1, 1 ≤ q ≤ ∞ and two compatible couples (X0, X1) and (Y0, Y1), the pair ((X0, X1)θ,q, (Y0, Y1)θ,q) is an exact interpolation pair of exponent θ.[12]

A complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method. However, there is a general relationship.

Theorem. Let (X0, X1) be a compatible couple of Banach spaces. If 0 < θ < 1, then

Examples[edit]

When X0 = C([0, 1]) and X1 = C1([0, 1]), the space of continuously differentiable functions on [0, 1], the (θ, ∞) interpolation method, for 0 < θ < 1, gives the Hölder space C0,θ of exponent θ. This is because the K-functional K(f, t; X0, X1) of this couple is equivalent to

Only values 0 < t < 1 are interesting here.

Real interpolation between Lp spaces gives[13] the family of Lorentz spaces. Assuming 0 < θ < 1 and 1 ≤ q ≤ ∞, one has:

with equivalent norms. This follows from an inequality of Hardy and from the value given above of the K-functional for this compatible couple. When q = p, the Lorentz space Lp,p is equal to Lp, up to renorming. When q = ∞, the Lorentz space Lp,∞ is equal to weak-Lp.

The reiteration theorem[edit]

An intermediate space X of the compatible couple (X0, X1) is said to be of class θ if [14]

with continuous injections. Beside all real interpolation spaces (X0, X1)θ,q with parameter θ and 1 ≤ q ≤ ∞, the complex interpolation space (X0, X1)θ is an intermediate space of class θ of the compatible couple (X0, X1).

The reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a convex combination a = (1 − θ)x0 + θx1: taking a further convex combination of two convex combinations gives another convex combination.

Theorem.[15] Let A0, A1 be intermediate spaces of the compatible couple (X0, X1), of class θ0 and θ1 respectively, with 0 < θ0θ1 < 1. When 0 < θ < 1 and 1 ≤ q ≤ ∞, one has

It is notable that when interpolating with the real method between A0 = (X0, X1)θ0,q0 and A1 = (X0, X1)θ1,q1, only the values of θ0 and θ1 matter. Also, A0 and A1 can be complex interpolation spaces between X0 and X1, with parameters θ0 and θ1 respectively.

There is also a reiteration theorem for the complex method.

Theorem.[16] Let (X0, X1) be a compatible couple of complex Banach spaces, and assume that X0X1 is dense in X0 and in X1. Let A0 = (X0, X1)θ0 and A1 = (X0, X1)θ1, where 0 ≤ θ0θ1 ≤ 1. Assume further that X0X1 is dense in A0A1. Then, for every 0 ≤ θ ≤ 1,

The density condition is always satisfied when X0X1 or X1X0.

Duality[edit]

Let (X0, X1) be a compatible couple, and assume that X0X1 is dense in X0 and in X1. In this case, the restriction map from the (continuous) dual of Xj, j = 0, 1, to the dual of X0X1 is one-to-one. It follows that the pair of duals is a compatible couple continuously embedded in the dual (X0X1)′.

For the complex interpolation method, the following duality result holds:

Theorem.[17] Let (X0, X1) be a compatible couple of complex Banach spaces, and assume that X0X1 is dense in X0 and in X1. If X0 and X1 are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals,

In general, the dual of the space (X0, X1)θ is equal[17] to a space defined by a variant of the complex method.[18] The upper-θ and lower-θ methods do not coincide in general, but they do if at least one of X0, X1 is a reflexive space.[19]

For the real interpolation method, the duality holds provided that the parameter q is finite:

Theorem.[20] Let 0 < θ < 1, 1 ≤ q < ∞ and (X0, X1) a compatible couple of real Banach spaces. Assume that X0X1 is dense in X0 and in X1. Then
where

Discrete definitions[edit]

Since the function tK(x, t) varies regularly (it is increasing, but 1/tK(x, t) is decreasing), the definition of the Kθ,q-norm of a vector n, previously given by an integral, is equivalent to a definition given by a series.[21] This series is obtained by breaking (0, ∞) into pieces (2n, 2n+1) of equal mass for the measure dt/t,

In the special case where X0 is continuously embedded in X1, one can omit the part of the series with negative indices n. In this case, each of the functions xK(x, 2n; X0, X1) defines an equivalent norm on X1.

The interpolation space (X0, X1)θ,q is a "diagonal subspace" of an  q-sum of a sequence of Banach spaces (each one being isomorphic to X0 + X1). Therefore, when q is finite, the dual of (X0, X1)θ,q is a quotient of the  p-sum of the duals, 1/p + 1/q = 1, which leads to the following formula for the discrete Jθ,p-norm of a functional x' in the dual of (X0, X1)θ,q:

The usual formula for the discrete Jθ,p-norm is obtained by changing n to n.

The discrete definition makes several questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators. Lions and Peetre have proved that:

Theorem.[22] If the linear operator T is compact from X0 to a Banach space Y and bounded from X1 to Y, then T is compact from (X0, X1)θ,q to Y when 0 < θ < 1, 1 ≤ q ≤ ∞.

Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:

Theorem.[23] A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.

A general interpolation method[edit]

The space  q used for the discrete definition can be replaced by an arbitrary sequence space Y with unconditional basis, and the weights an = 2θn, bn = 2(1−θ)n, that are used for the Kθ,q-norm, can be replaced by general weights

The interpolation space K(X0, X1, Y, {an}, {bn}) consists of the vectors x in X0 + X1 such that[24]

where {yn} is the unconditional basis of Y. This abstract method can be used, for example, for the proof of the following result:

Theorem.[25] A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with symmetric basis.

Interpolation of Sobolev and Besov spaces[edit]

Several interpolation results are available for Sobolev spaces and Besov spaces on Rn,[26]

These spaces are spaces of measurable functions on Rn when s ≥ 0, and of tempered distributions on Rn when s < 0. For the rest of the section, the following setting and notation will be used:

Complex interpolation works well on the class of Sobolev spaces (the Bessel potential spaces) as well as Besov spaces:

Real interpolation between Sobolev spaces may give Besov spaces, except when s0 = s1,

When s0s1 but p0 = p1, real interpolation between Sobolev spaces gives a Besov space:

Also,

See also[edit]

  • Riesz–Thorin theorem
  • Marcinkiewicz interpolation theorem

Notes[edit]

  1. ^ The seminal papers in this direction are Lions, Jacques-Louis (1960), "Une construction d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 251: 1853–1855 and Calderón (1964).
  2. ^ first defined in Lions, Jacques-Louis; Peetre, Jaak (1961), "Propriétés d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 253: 1747–1749, developed in Lions & Peetre (1964), with notation slightly different (and more complicated, with four parameters instead of two) from today's notation. It was put later in today's form in Peetre, Jaak (1963), "Nouvelles propriétés d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 256: 1424–1426, andPeetre, Jaak (1968), A theory of interpolation of normed spaces, Notas de Matemática, 39, Rio de Janeiro: Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, pp. iii+86.
  3. ^ see Bennett & Sharpley (1988), pp. 96–105.
  4. ^ see p. 88 in Bergh & Löfström (1976).
  5. ^ see Theorem 4.1.2, p. 88 in Bergh & Löfström (1976).
  6. ^ see Chapter 5, p. 106 in Bergh & Löfström (1976).
  7. ^ see pp. 293–302 in Bennett & Sharpley (1988).
  8. ^ see Proposition 1.2, p. 294 in Bennett & Sharpley (1988).
  9. ^ see p. 298 in Bennett & Sharpley (1988).
  10. ^ see Theorem 2.8, p. 314 in Bennett & Sharpley (1988).
  11. ^ see Proposition 1.10, p. 301 in Bennett & Sharpley (1988)
  12. ^ see Theorem 1.12, pp. 301–302 in Bennett & Sharpley (1988).
  13. ^ see Theorem 1.9, p. 300 in Bennett & Sharpley (1988).
  14. ^ see Definition 2.2, pp. 309–310 in Bennett & Sharpley (1988)
  15. ^ see Theorem 2.4, p. 311 in Bennett & Sharpley (1988)
  16. ^ see 12.3, p. 121 in Calderón (1964).
  17. ^ a b see 12.1 and 12.2, p. 121 in Calderón (1964).
  18. ^ Theorem 4.1.4, p. 89 in Bergh & Löfström (1976).
  19. ^ Theorem 4.3.1, p. 93 in Bergh & Löfström (1976).
  20. ^ see Théorème 3.1, p. 23 in Lions & Peetre (1964), or Theorem 3.7.1, p. 54 in Bergh & Löfström (1976).
  21. ^ see chap. II in Lions & Peetre (1964).
  22. ^ see chap. 5, Théorème 2.2, p. 37 in Lions & Peetre (1964).
  23. ^ Davis, William J.; Figiel, Tadeusz; Johnson, William B.; Pełczyński, Aleksander (1974), "Factoring weakly compact operators", Journal of Functional Analysis, 17 (3): 311–327, doi:10.1016/0022-1236(74)90044-5, see also Theorem 2.g.11, p. 224 in Lindenstrauss & Tzafriri (1979).
  24. ^ Johnson, William B.; Lindenstrauss, Joram (2001), "Basic concepts in the geometry of Banach spaces", Handbook of the geometry of Banach spaces, Vol. I, Amsterdam: North-Holland, pp. 1–84, and section 2.g in Lindenstrauss & Tzafriri (1979).
  25. ^ see Theorem 3.b.1, p. 123 in Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Berlin: Springer-Verlag, pp. xiii+188, ISBN 978-3-540-08072-5.
  26. ^ Theorem 6.4.5, p. 152 in Bergh & Löfström (1976).

References[edit]

  • Calderón, Alberto P. (1964), "Intermediate spaces and interpolation, the complex method", Studia Math., 24 (2): 113–190, doi:10.4064/sm-24-2-113-190.
  • Lions, Jacques-Louis.; Peetre, Jaak (1964), "Sur une classe d'espaces d'interpolation", Inst. Hautes Études Sci. Publ. Math. (in French), 19: 5–68, doi:10.1007/bf02684796.
  • Bennett, Colin; Sharpley, Robert (1988), Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, pp. xiv+469, ISBN 978-0-12-088730-9.
  • Bergh, Jöran; Löfström, Jörgen (1976), Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Berlin-New York: Springer-Verlag, pp. x+207, ISBN 978-3-540-07875-3.
  • Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8.
  • Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 978-3-540-08888-2.
  • Tartar, Luc (2007), An Introduction to Sobolev Spaces and Interpolation, Springer, ISBN 978-3-540-71482-8.