Topic
Lp space
About: Lp space is a research topic. Over the lifetime, 8797 publications have been published within this topic receiving 212428 citations. The topic is also known as: Lebesgue space.
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Book•
01 Nov 1971
TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.
Abstract: The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.
5,579 citations
Book•
01 Jan 1983
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.
Abstract: How to Measure Smoothness.- Atoms and Pointwise Multipliers.- Wavelets.- Spaces on Lipschitz Domains, Wavelets and Sampling Numbers.- Anisotropic Function Spaces.- Weighted Function Spaces.- Fractal Analysis: Measures, Characteristics, Operators.- Function Spaces on Quasi-metric Spaces.- Function Spaces on Sets.
4,099 citations
Book•
01 Jan 1987
TL;DR: In this article, the classical interpolation theorem is extended to the Banach Function Spaces, and the K-Method is used to find a Banach function space with a constant number of operators.
Abstract: Banach Function Spaces. Rearrangement-Invariant Banach Function Spaces. Interpolation of Operators on Rearrangement-Invariant Spaces. The Classical Interpolation Theorems. The K-Method. Each chapter includes references. Index.
3,388 citations
Book•
01 Jan 1978
TL;DR: In this paper, the spectral theory of linear operators in normed spaces and their spectrum has been studied in the context of bounded self-and-adjoint linear operators and their applications in quantum mechanics.
Abstract: Metric Spaces. Normed Spaces Banach Spaces. Inner Product Spaces Hilbert Spaces. Fundamental Theorems for Normed and Banach Spaces. Further Applications: Banach Fixed Point Theorem. Spectral Theory of Linear Operators in Normed Spaces. Compact Linear Operators on Normed Spaces and Their Spectrum. Spectral Theory of Bounded Self--Adjoint Linear Operators. Unbounded Linear Operators in Hilbert Space. Unbounded Linear Operators in Quantum Mechanics. Appendices. References. Index.
2,781 citations
01 Feb 1978
TL;DR: In this article, the authors give an answer to Ulam's problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist", and prove it for the case n = 1.
Abstract: Let E1, E2 be two Banach spaces, and let f: E1 -* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem. THEOREM. Consider E1, E2 to be two Banach spaces, and let f: E1 -> E2 be a mapping such that f (tx) is continuous in t for each fixed x. Assume that there exists 0 > 0 andp E [0, 1) such that IIf(x + y) f (x) f(A)lI 0. The verification of (3) follows by induction on n. Indeed the case n = 1 is clear because by the hypothesis we can find 0, that is greater or equal to zero, andp such that 0 < p < 1 with 11[f(2x)]/2 -f(x)ll (4) IIxIIp Assume now that (3) holds and we want to prove it for the case (n + 1). However this is true because by (3) we obtain II [f (2n 2x)]/2 n f(2x)llI nI I*2x)2P < m E 2m(P therefore Received by the editors December 1, 1977. AMS (MOS) subject classifications (1970). Primary 47H15; Secondary 39A15.
2,694 citations