Bump Functions With Monotone Fourier Transforms Satisfying Decay Bounds

TL;DR

This paper constructs a smooth, nonnegative, compactly supported function whose Fourier transform is monotone and satisfies specific decay bounds, advancing understanding of Fourier analysis properties.

Contribution

It demonstrates the existence of such functions with monotone Fourier transforms and prescribed decay bounds, a novel contribution in Fourier analysis.

Findings

Existence of smooth, nonnegative, compactly supported functions with monotone Fourier transforms.

Fourier transforms satisfying two-sided decay bounds.

Advancement in understanding Fourier transform decay properties.

Abstract

The existence of a smooth, nonnegative, compactly supported function with monotone (on the half-line) Fourier transform satisfying two-sided decay bounds is demonstrated.