# Decay of Fourier transforms and generalized Besov spaces

## TL;DR

This paper characterizes generalized Lipschitz and Besov spaces through Fourier transform decay, providing necessary and sufficient Titchmarsh-type conditions using approximation estimates of multiplier operators.

## Contribution

It introduces a new characterization of Besov spaces based on Fourier decay and develops Titchmarsh-type conditions for these spaces.

## Key findings

- Established decay conditions for Fourier transforms characterizing Besov spaces
- Derived necessary and sufficient Titchmarsh-type conditions
- Provided estimates for approximation rates of multiplier operators

## Abstract

A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given. In particular, necessary and sufficient conditions of Titchmarsh type are obtained. The method is based on two-sided estimate for the rate of approximation of a $\beta$-admissible family of multipliers operators in terms of decay properties of Fourier transform.