Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A refined variant of Hartley convolution: Algebraic structures and related issues

Trinh Tuan

TL;DR

This work introduces a refined Hartley-type convolution $f \\underset{\\mathscr{H}}{\\ast} g$ tied to the $\\mathscr{H}$-transform, and proves that $(L_1(\\mathbb{R}^n), +, \\underset{\\mathscr{H}}{\\ast})$ is a commutative Banach algebra without a unit. It establishes a key factorization $\\mathscr{H}(f \\underset{\\mathscr{H}}{\\ast} g) = (\\mathscr{H}f)(y)(\\mathscr{H}g)(y)$, a Parseval-type identity in $L_2$, and a sharp Young-type inequality for the convolution, enabling $L_r$ bounds. The Wiener–Lévy invertibility criterion and Gelfand spectral radius are developed for the $\\mathscr{H}$-algebra, linking spectral properties to the $L_\infty$-norm of $\\mathscr{H}f$. Theoretical results are then applied to Fredholm integral equations and 1D heat problems, yielding explicit solvability conditions and a priori estimates via convolution representations that extend classical transform methods.

Abstract

In this work, we propose a novel convolution product associated with the $\mathscr{H}$-transform, denoted by $\underset{\mathscr{H}}{\ast}$, and explore its fundamental properties. Here, the $\mathscr{H}$-transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters $a,b$. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the $\underset{\mathscr{H}}{\ast}$-convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, we prove the Wiener--Lévy invertibility criterion for the $\mathscr H$-algebra and formulate Gelfand's spectral radius theorem. Third, we obtain a sharp form of Young's inequality for the $\underset{\mathscr{H}}{\ast}$-convolution and its direct corollary. Finally, all of these theoretical findings are applied to investigate specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions.

A refined variant of Hartley convolution: Algebraic structures and related issues

TL;DR

This work introduces a refined Hartley-type convolution tied to the -transform, and proves that is a commutative Banach algebra without a unit. It establishes a key factorization , a Parseval-type identity in , and a sharp Young-type inequality for the convolution, enabling bounds. The Wiener–Lévy invertibility criterion and Gelfand spectral radius are developed for the -algebra, linking spectral properties to the -norm of . Theoretical results are then applied to Fredholm integral equations and 1D heat problems, yielding explicit solvability conditions and a priori estimates via convolution representations that extend classical transform methods.

Abstract

In this work, we propose a novel convolution product associated with the -transform, denoted by , and explore its fundamental properties. Here, the -transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters . Our first contribution shows that the space of integrable functions, equipped with multiplication given by the -convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, we prove the Wiener--Lévy invertibility criterion for the -algebra and formulate Gelfand's spectral radius theorem. Third, we obtain a sharp form of Young's inequality for the -convolution and its direct corollary. Finally, all of these theoretical findings are applied to investigate specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions.

Paper Structure

This paper contains 12 sections, 11 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Organization
  4. Two auxiliary lemmas
  5. Algebraic structures
  6. Existence, factorization and Parseval equalities
  7. Main theorems
  8. Gelfand’s spectral radius for $\mathscr H$-algebra
  9. Young-type theorem and inequality
  10. Issues related to applicability
  11. Fredholm integral equation
  12. A Cauchy problem for the heat equation in one dimension

Key Result

Lemma 2.1

Let $f \in L_q(\mathbb{R}^n)$ with $q \geq 1$. Then, for every $v \in \mathbb{R}^n$, the following estimate holds:

Theorems & Definitions (26)

  • Definition 1: Main object
  • Lemma 2.1
  • proof
  • Lemma 2.2: Riemann--Lebesgue type for $\mathscr H$-transform
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2: Banach algebra
  • proof
  • Corollary 3.1
  • ...and 16 more