A vectorial Ingham-Beurling theorem

TL;DR

This paper extends the Ingham-Beurling theorem, originally about scalar sums, to vector coefficient sums, providing a broader understanding of the theorem's applicability in harmonic analysis.

Contribution

It generalizes the classical Ingham-Beurling theorem to the case of vector coefficients, building on previous work that used divided differences.

Findings

Extension of the theorem to vector coefficients.

Proof of optimality of assumptions in the scalar case.

Broader applicability in harmonic analysis.

Abstract

Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend these results to vector coefficient sums.