In simple terms, if a linear function is continuous on a certain space Lp and also on a certain space Lq, then it is also continuous on the space Lr, for any intermediate r between p and q. In other words, Lr is a space which is intermediate between Lp and Lq. 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,  real interpolation,  as well as other tools see e. The setting of interpolation[ edit ] 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.
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Some Classical Theorems. The Riesz-Thorin Theorem. Applications of the Riesz-Thorin Theorem. The Marcinkiewicz Theorem. An Application of the Marcinkiewicz Theorem. Two Classical Approximation Results. Notes and Comment. General Properties of Interpolation Spaces.
Categories and Functors. Normed Vector Spaces. Couples of Spaces. Definition of Interpolation Spaces. The Aronszajn-Gagliardo Theorem. A Necessary Condition for Interpolation. A Duality Theorem. The Real Interpolation Method. The K-Method. The J-Method.
The Equivalence Theorem. Simple Properties of?? The Reiteration Theorem. A Formula for the K-Functional. The Duality Theorem.
A Compactness Theorem. An Extremal Property of the Real Method. Quasi-Normed Abelian Groups. The Complex Interpolation Method.
Definition of the Complex Method. Simple Properties of? Multilinear Interpolation. On the Connection with the Real Method. Interpolation of Lp-Spaces.
Interpolation of Lp-Spaces: the Complex Method. Interpolation of Lp-Spaces: the Real Method. Interpolation of Lorentz Spaces. Interpolation of Lp-Spaces with Change of Measure: p0? Interpolation of Sobolev and Besov Spaces. Fourier Multipliers. Definition of the Sobolev and Besov Spaces. The Homogeneous Sobolev and Besov Spaces.
An Embedding Theorem. A Trace Theorem. Interpolation of Semi-Groups of Operators. Applications to Approximation Theory. Approximation Spaces. Approximation of Functions. Approximation of Operators.
Approximation by Difference Operators.
INTERPOLATION SPACES BERGH LOFSTROM PDF
Shakajinn To learn more about Copies Direct watch this short online video. However, there is a general relationship. Lions and Peetre have proved that:. Interpolation does not depend only upon the isomorphic nor isometric equivalence classes of X 0 lofstron X 1. From Wikipedia, the free encyclopedia. Details Collect From To learn more about how to request items watch this short online video. This is because the K-functional K ft ; X 0X 1 of this couple is equivalent to.