Quadratic spectral concentration of characteristic functions

TL;DR

This paper investigates the spectral concentration inequalities of functions and their rearrangements, extending the range for characteristic functions and analyzing minimizers, with applications to trigonometric polynomials.

Contribution

It extends spectral concentration bounds specifically for characteristic functions and characterizes minimizers of spectral difference functionals.

Findings

Range of spectral concentration inequality extended to WT ≤ 4/3 for characteristic functions.

Identifies properties of sets minimizing the spectral difference functional.

Provides sharp estimates for L2-norms of non-harmonic trigonometric polynomials with alternating coefficients.

Abstract

It is known that the inequality \begin{align*}\int_{-W/2}^{W/2}|\widehat{f}(\xi)|^2d\xi\leq \int_{-W/2}^{W/2}|\widehat{|f|^*}(\xi)|^2d\xi \end{align*} between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function $f\in L^2,\;|\text{supp} f|=T,$ if and only if the product $WT$ does not exceed the critical value $\approx 0.884$. We show that by restricting ourselves to characteristic functions we can enlarge this range up to $WT\leq 4/3$. Besides, we establish various properties of minimizers of the difference $\int_{-W/2}^{W/2}|\widehat{\chi_A^*}(\xi)|^2d\xi-\int_{-W/2}^{W/2}|\widehat{\chi_A}(\xi)|^2d\xi$ over sets $A$ of finite measure and prove that this difference is non-negative for all $W,T>0$ if $A$ is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the $L_2$-norms of non-harmonic trigonometric polynomials with alternating coefficients $\pm 1$.