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The Extended Paley-Wiener Theorem over the Hardy-Sobolev Spaces

Detian Liu, Haichou Li, Kit Ian Kou

Abstract

We examine how the square-integrable function subspaces are transformed using the holomorphic Fourier transform. On account of this, the extended Paley-Wiener theorem over the Hardy-Sobolev spaces is produced. The theorem also asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found. We discuss the relationship between the disc and the upper half-plane.

The Extended Paley-Wiener Theorem over the Hardy-Sobolev Spaces

Abstract

We examine how the square-integrable function subspaces are transformed using the holomorphic Fourier transform. On account of this, the extended Paley-Wiener theorem over the Hardy-Sobolev spaces is produced. The theorem also asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found. We discuss the relationship between the disc and the upper half-plane.
Paper Structure (8 sections, 8 theorems, 60 equations, 1 table)

This paper contains 8 sections, 8 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related works
  3. Notations
  4. Paper contributions
  5. Paper outlines
  6. Preliminary
  7. Main results
  8. Conclusion

Key Result

Proposition 2.2

(4) For $n\in\mathbb{N}$ and $\varphi\in L^2 (t^n)$. Then for $1\le k\le n$, we have Moreover, $W^{-k}\varphi$ is $(k-1)$-times differentiable with $(W^{-k}\varphi)^{(l)}=(-1)^l W^{-(k-l)}\varphi$ for every $1\le l\le k-1$, and $(-1)^k(W^{-k}\varphi)^{(k)}=\varphi$.

Theorems & Definitions (16)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • ...and 6 more