Phase symmetrization and a no-aliasing concentration principle in Paley-Wiener spaces

Abstract

We study a constructive version of the Donoho--Stark concentration problem in a no-aliasing regime. Let $f\in\text{PW}(\mathcal{W})$, where $\mathcal{W}\subset\mathbb R$ is an interval of length $W$, and let $\mathcal{T}\subset\mathbb R$ have measure $T$ and essential diameter $D$. If $WD\leq1$, then the phase-aligned function $g$, defined by $\widehat g=|\widehat f|$, satisfies $$\int_\mathcal{T} |f(t)|^2\,dt \leq \int_{-T/2}^{T/2}|g(t)|^2\,dt.$$ Thus each band-limited function is associated with an explicit comparator having the same Fourier modulus and $L^2$-norm. The proof combines two independent principles: Fourier-phase alignment maximizes concentration on a centered interval for fixed Fourier modulus, while a one-period geometric estimate controls the Fourier transform of the observation set. We show that the latter mechanism is essentially sharp: beyond the one-period threshold, separated components may alias under the exponential map and reinforce coherently. Analogous phase and coefficient rearrangement results are obtained for polynomials on the unit circle, including a constant-one Montgomery-type inequality in the corresponding no-aliasing regime. The results identify both the scope and the limitation of this modewise approach to concentration.