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 other words, Lr is a space which is intermediate between Lp and Lq.
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).
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.
It depends in an essential way from the specific relative position that X0 and X1 occupy in a larger space Z.
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.
More generally, with continuous injections, so that, under the given condition, Lp(R) is intermediate between Lp0(R) and Lp1(R).
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:
[6] If (R, Σ, μ) is an arbitrary measure space, if 1 ≤ p0, p1 ≤ ∞ and 0 < θ < 1, then with equality of norms.
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 from the dual couple by the J-method; see below.
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 X0 ∩ X1 and such that The norm of x in Jθ,q(X0, X1) is given by the formula The two real interpolation methods are equivalent when 0 < θ < 1.
[10] 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.
This follows from an inequality of Hardy and from the value given above of the K-functional for this compatible couple.
An intermediate space X of the compatible couple (X0, X1) is said to be of class θ if [14] with continuous injections.
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.
For the complex interpolation method, the following duality result holds: In general, the dual of the space (X0, X1)θ is equal[17] to
[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: Since the function t → K(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 x → K(x, 2n; X0, X1) defines an equivalent norm on X1.
Therefore, when q is finite, the dual of (X0, X1)θ,q is a quotient of the ℓ 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.
Lions and Peetre have proved that: Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result: The space ℓ q used for the discrete definition can be replaced by an arbitrary sequence space Y with unconditional basis, and the weights an = 2−θn, 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.
For the rest of the section, the following setting and notation will be used: Complex interpolation works well on the class of Sobolev spaces