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Spectral Barron spaces arising from quantum harmonic analysis

Yaogan Mensah

Abstract

In this paper, spectral Barron spaces are defined in the framework of quantum harmonic analysis. Their fundamental properties are studied. These include, among others, their completeness structure and some continuous embedding results. As an application, the existence and the uniqueness of the solution of a Schrödinger-type equation is proved.

Spectral Barron spaces arising from quantum harmonic analysis

Abstract

In this paper, spectral Barron spaces are defined in the framework of quantum harmonic analysis. Their fundamental properties are studied. These include, among others, their completeness structure and some continuous embedding results. As an application, the existence and the uniqueness of the solution of a Schrödinger-type equation is proved.

Paper Structure

This paper contains 9 sections, 13 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact and Schatten $p$-classes operators
  4. Pontrjagin dual of a locally compact abelian group
  5. Quantum Fourier transform
  6. Banach contraction principle
  7. Spectral Barron spaces of compact operators: Properties and embeddings
  8. Application : Schrödinger-type equations for operators
  9. Conclusion

Key Result

Proposition 2.1

Fulsche Assume the representation is integrable. If $f\in L^1(\widehat{G})$, then $\mathcal{F}_U^{-1}(f)\in \mathcal{K}(H)$ and $\|\mathcal{F}_U^{-1}(f)\|_{\hbox{op}}\leq \|f\|_{L^1(\widehat{G})}.$

Theorems & Definitions (23)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Corollary 3.4
  • proof
  • ...and 13 more