A bound for Mean values of Fourier transforms

TL;DR

This paper establishes a new bound on the mean values of Fourier transforms, demonstrating a specific growth condition for sequences and relating integrals of the Fourier transform to weighted measures.

Contribution

It introduces a novel bound for Fourier transform mean values involving sequences with geometric growth and non-negative measures with non-negative Fourier transforms.

Findings

Existence of a sequence with at least geometric growth rate.

A bound relating Fourier transform integrals to weighted measure integrals.

The bound involves exponential and sine kernel terms.

Abstract

We show that there exists a sequence $\{n_k, k\ge 1\}$ growing at least geometrically such that for any finite non-negative measure $\nu$ such that $\hat \nu\ge 0$, any $T>0$, $$ \int_{-2^{n_k} T}^{2^{n_k} T} \hat \nu(x) \dd x \ll_\e T\,2^{2^{(1+\e)n_k}} \int_\R \Big|{\sin {xT} \over xT} \Big|^{n_k^2} \nu(\dd x). $$