Quadratic spectral concentration of characteristic functions

A theorem of Donoho and Stark states that decreasing rearrangement increases the quadratic spectral concentration of a square integrable function supported on a sufficiently small set. Importantly, their condition on the smallness of the support turns out to be necessary. In this talk, we restrict ourselves to considering only characeristic functions and, in this setting, we are able to relax the condition of Donoho and Stark. We also discuss various properties of the sets of fixed measure maximizing the quadratic spectral concentration of their characteristic functions. 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 $\pm 1$.