Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Paley-Wiener theorems for slice monogenic functions

Yanshuai Hao, Pei Dang, Weixiong Mai

TL;DR

This work extends Paley-Wiener theory to slice monogenic functions in Clifford analysis by introducing the 1D Clifford Fourier transform and analyzing Hardy and Bergman spaces on slice domains. It proves a non-compact type Paley-Wiener theorem for slice Hardy spaces, linking Fourier support on $(-\infty,0]$ to NTBL representations across slices, and a compact type theorem characterizing exponential type growth via spectrum in $[-B,B]$ with explicit integral representations. A Bergman-space analogue is developed, including norm estimates and explicit reproducing kernels, enabling kernel-based reconstructions. Together, these results unify spectrum-space characterizations for slice monogenic functions and provide practical reproducing-kernel formulas for these function spaces.

Abstract

In this paper, we prove some Paley-Wiener theorems for function spaces consisting of slice monogenic functions such as Paley-Wiener, Hardy and Bergman spaces. As applications, we can compute the reproducing kernel functions for the related function spaces.

Paley-Wiener theorems for slice monogenic functions

TL;DR

This work extends Paley-Wiener theory to slice monogenic functions in Clifford analysis by introducing the 1D Clifford Fourier transform and analyzing Hardy and Bergman spaces on slice domains. It proves a non-compact type Paley-Wiener theorem for slice Hardy spaces, linking Fourier support on to NTBL representations across slices, and a compact type theorem characterizing exponential type growth via spectrum in with explicit integral representations. A Bergman-space analogue is developed, including norm estimates and explicit reproducing kernels, enabling kernel-based reconstructions. Together, these results unify spectrum-space characterizations for slice monogenic functions and provide practical reproducing-kernel formulas for these function spaces.

Abstract

In this paper, we prove some Paley-Wiener theorems for function spaces consisting of slice monogenic functions such as Paley-Wiener, Hardy and Bergman spaces. As applications, we can compute the reproducing kernel functions for the related function spaces.

Paper Structure

This paper contains 6 sections, 25 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Non-compact type Paley-Wiener theorem
  5. Compact type Paley-Wiener theorem
  6. Analogue in Bergman space

Key Result

Proposition 2.5

Let $I=I_1\in\mathbb{S}$. It is possible to choose $I_2,\dots,I_n\in\mathbb{S}$ such that $I_1,\dots,I_n$ form an orthonormal basis for the Clifford algebra $\mathbb{R}_n$ i.e., they satisfy the defining relations $I_rI_s+I_sI_r=-2\delta_{rs}$.

Theorems & Definitions (48)

  • Definition 2.1: Colombo2009Slice
  • Definition 2.2: Colombo2009Slice
  • Definition 2.3: Colombo2009Slice
  • Definition 2.4: MR4752422
  • Proposition 2.5: Colombo2009Slice
  • Proposition 2.6: Colombo2009Slice, Splitting Lemma
  • Proposition 2.7: colombo2009structure, Representation Formula
  • Proposition 2.8: 10.1063/1.3636718
  • Proposition 2.9: 10.1063/1.3636718
  • Proposition 2.10: colombo2010extension, Extension Lemma
  • ...and 38 more