Besov and Paley-Wiener spaces, Moduli of continuity and Hardy-Steklov operators associated with the group "ax+b"

2024/01/30 by Isaac Z. Pesenson, Pesenson, Isaac
Mathematics · #Mathematical Analysis and Transform Methods #advanced mathematical theories #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2401.16734

Abstract

We introduce and describe relations between Sobolev, Besov and Paley-Wiener spaces associated with three representations of the Lie group of affine transformations of the line. These representations are left and right regular representations and a representation in a space of functions defined on the half-line. The Besov spaces are described as interpolation spaces between respective Sobolev spaces in terms of the Petree's real interpolation method and in terms of a relevant moduli of continuity. By using a Laplace operators associated with these representations a scales of relevant Paley-Wiener spaces are developed and a corresponding L₂-approximation theory is constructed in which our Besov spaces appear as approximation spaces. Another description of our Besov spaces is given in terms of a frequency-localized Hilbert frames. A Jackson-type inequalities are also proven.

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