In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives.

History

The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem. In simple terms, if a linear function is continuous on a certain space L^p and also on a certain space L^q, then it is also continuous on the space L^r, for any intermediate r between p and q. In other words, L^r is a space which is intermediate between L^p and L^q.

In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability.

Many methods were designed to generate such spaces of functions, including the Fourier transform, complex interpolation,^[1] real interpolation,^[2] as well as other tools (see e.g. fractional derivative).

The setting of interpolation

A Banach space X is said to be continuously embedded in a Hausdorff topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous. A compatible couple (X₀, X₁) of Banach spaces consists of two Banach spaces X₀ and X₁ that are continuously embedded in the same Hausdorff topological vector space Z.^[3] The embedding in a linear space Z allows to consider the two linear subspaces

${\displaystyle X_{0}\cap X_{1}}$

and

${\displaystyle X_{0}+X_{1}=\left\{z\in Z:z=x_{0}+x_{1},\ x_{0}\in X_{0},\,x_{1}\in X_{1}\right\}.}$

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of X₀ and X₁. It depends in an essential way from the specific relative position that X₀ and X₁ occupy in a larger space Z.

One can define norms on X₀ ∩ X₁ and X₀ + X₁ by

${\displaystyle \|x\|_{X_{0}\cap X_{1}}:=\max \left(\left\|x\right\|_{X_{0}},\left\|x\right\|_{X_{1}}\right),}$

${\displaystyle \|x\|_{X_{0}+X_{1}}:=\inf \left\{\left\|x_{0}\right\|_{X_{0}}+\left\|x_{1}\right\|_{X_{1}}\ :\ x=x_{0}+x_{1},\;x_{0}\in X_{0},\;x_{1}\in X_{1}\right\}.}$

Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:

${\displaystyle X_{0}\cap X_{1}\subset X_{0},\ X_{1}\subset X_{0}+X_{1}.}$

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

${\displaystyle X_{0}\cap X_{1}\subset X\subset X_{0}+X_{1},}$

where the two inclusions maps are continuous.

An example of this situation is the pair (L¹(R), L^∞(R)), where the two Banach spaces are continuously embedded in the space Z of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces L^p(R), for 1 ≤ p ≤ ∞ are intermediate between L¹(R) and L^∞(R). More generally,

${\displaystyle L^{p_{0}}(\mathbf {R} )\cap L^{p_{1}}(\mathbf {R} )\subset L^{p}(\mathbf {R} )\subset L^{p_{0}}(\mathbf {R} )+L^{p_{1}}(\mathbf {R} ),\ \ {\text{when}}\ \ 1\leq p_{0}\leq p\leq p_{1}\leq \infty ,}$

with continuous injections, so that, under the given condition, L^p(R) is intermediate between L^p₀(R) and L^p₁(R).

Definition. Given two compatible couples (X₀, X₁) and (Y₀, Y₁), an interpolation pair is a couple (X, Y) of Banach spaces with the two following properties:

• The space X is intermediate between X₀ and X₁, and Y is intermediate between Y₀ and Y₁.
• If L is any linear operator from X₀ + X₁ to Y₀ + Y₁, which maps continuously X₀ to Y₀ and X₁ to Y₁, then it also maps continuously X to Y.

The interpolation pair (X, Y) is said to be of exponent θ (with 0 < θ < 1) if there exists a constant C such that

${\displaystyle \|L\|_{X,Y}\leq C\|L\|_{X_{0},Y_{0}}^{1-\theta }\;\|L\|_{X_{1},Y_{1}}^{\theta }}$

for all operators L as above. The notation ||L||_X,Y is for the norm of L as a map from X to Y. If C = 1, we say that (X, Y) is an exact interpolation pair of exponent θ.

Complex interpolation

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 ${\displaystyle {\mathcal {F}}(X_{0},X_{1})}$ consists of all functions  f  : C → X₀ + 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) : z ∈ S} ⊂ X₀ + X₁,

{ f (it) : t ∈ R} ⊂ X₀,

{ f (1 + it) : t ∈ R} ⊂ X₁.

${\displaystyle \mathcal{F}(X_0,X_{}}$Zdroj:https://en.wikipedia.org?pojem=Complex_interpolation
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