Fourier transforms on an amalgam type space

TL;DR

This paper introduces a new amalgam space within L^1(R+), explores Fourier transform integrability for functions with derivatives in this space, and applies these results to estimate the integrability of trigonometric series.

Contribution

It defines a novel amalgam space and establishes integrability results for Fourier transforms of functions with derivatives in this space, with applications to trigonometric series.

Findings

Fourier transforms of functions with derivatives in the amalgam space are integrable.

Provides estimates for the integrability of trigonometric series.

Introduces a new subspace of L^1(R+).

Abstract

We introduce an amalgam type space, a subspace of $L^1(\mathbb R_+).$ Integrability results for the Fourier transform of a function with the derivative from such an amalgam space are proved. As an application we obtain estimates for the integrability of trigonometric series.