A pointwise estimate for the Fourier transform and the number of maxima of a function

TL;DR

This paper establishes a pointwise estimate for the Fourier transform based on the function's monotonicity changes, providing tools to bound the number of maxima and roots, with applications to the two weight problem and derivative roots.

Contribution

It introduces a novel pointwise estimate linking Fourier transform behavior to the function's monotonicity changes, enabling new bounds on maxima and roots.

Findings

Derived a pointwise Fourier transform estimate involving monotonicity changes

Provided a lower bound for the number of local maxima of a function

Applied the theorem to the two weight problem and root estimation

Abstract

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.