On the existence of optimizers for time–frequency concentration problems

We consider the problem of maximum concentration in a fixed measurable subset $$\Omega \subset {\mathbb {R}}^{2d}$$ time-frequency space for functions $$f\in L^2({\mathbb {R}}^{d})$$ . The notion can be made mathematically precise by considering $$L^p$$ -norm on $$ some time–frequency distribution f such as ambiguity function A(f). provide positive answer to an open maximization problem, showing that every finite measure and $$1\le p<\infty , there exists optimizer $$\begin{aligned} \sup \{\Vert A(f)\Vert _{L^p(\Omega )}:\ f\in {R}}^{d}),\ \Vert f\Vert _{L^2}=1 \}. \end{aligned}$$ lack weak upper semicontinuity invariance under shifts make challenging. proof is based compactness with dislocations, certain integral bounds asymptotic decoupling estimates function. also discuss case $$p=\infty related optimization problems time correlation function, cross-ambiguity window, modulation spaces $$M^q({\mathbb $$0<q<2$$ equipped continuous or discrete-type (quasi-)norms.