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

Isaac Pesenson

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 $L_{2}$-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.

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

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 -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.
Paper Structure (21 sections, 20 theorems, 193 equations)

This paper contains 21 sections, 20 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. The group $ax+b$ and its representations
  3. The group $"ax+b"$
  4. Representations
  5. Left-regular representation of the group $"ax+b "$
  6. Right-regular representation of the group $"ax+b "$
  7. Representation of the $"ax+b"$ group in spaces ${\bf X}^{p}$
  8. General framework
  9. $K$-functional and modulus of continuity
  10. Besov spaces
  11. The case of a unitary representation
  12. Paley-Wiener vectors in ${\bf H}$
  13. A Jackson-type inequality
  14. Comparison with the previous examples
  15. Proofs of Theorems \ref{['Main-Ineq-000']} and \ref{['Main-000']}
  16. ...and 6 more sections

Key Result

Theorem 2.1

The space $\mathbf{W}^{m}_{2}(\mathbb{D}_{1}^{L}, \mathbb{D}_{2}^{L})$ coincides with the domain $\mathcal{D}(\Delta_{L}^{m/2})$ and the norm (SobL) is equivalent to the graph norm $\|f\|_{{\bf L}^{2}(G, d\mu_{l})}+\|\Delta_{L}^{m/2}f\|_{{\bf L}^{2}(G, d\mu_{l})}$.

Theorems & Definitions (38)

  • Definition 1
  • Theorem 2.1
  • Definition 2
  • Theorem 2.2
  • Definition 3
  • Definition 4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 28 more