On the existence of optimizers for time–frequency concentration problems

Abstract

We consider the problem of the maximum concentration in a fixed measurable subset $\Omega \subset {\mathbb {R}}^{2d}$ of the time-frequency space for functions $f\in L^2({\mathbb {R}}^{d})$. The notion of concentration can be made mathematically precise by considering the $L^p$-norm on $\Omega$ of some time–frequency distribution of f such as the ambiguity function A(f). We provide a positive answer to an open maximization problem, by showing that for every subset $\Omega \subset {\mathbb {R}}^{2d}$ of finite measure and every $1\le p<\infty$, there exists an optimizer for

$$\begin{aligned} \sup \{\Vert A(f)\Vert _{L^p(\Omega )}:\ f\in L^2({\mathbb {R}}^{d}),\ \Vert f\Vert _{L^2}=1 \}. \end{aligned}$$

The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time–frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case $p=\infty$ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces $M^q({\mathbb {R}}^{d})$, $0<q<2$, equipped with continuous or discrete-type (quasi-)norms.