Constructing Infinite Sets of Orthogonal Exponentials for Convex Polytopes

TL;DR

This paper demonstrates the existence and explicit construction of infinite orthogonal exponential sets for specific convex polytopes, including simple-rational and certain non-simple polytopes, and explores positive density sets via affine transformations.

Contribution

It provides new explicit constructions of infinite orthogonal exponential sets for convex polytopes, expanding understanding beyond previous limited cases.

Findings

Infinite orthogonal exponential sets exist for certain convex polytopes.

Explicit constructions are provided for these sets.

Orthogonal projections of affine transformations of hypercubes can produce positive density sets.

Abstract

The aim of this article is to show the existence, and also give an explicit construction, of infinite sets of orthogonal exponentials for certain families of convex polytopes which include simple-rational polytopes and also non simple polytopes which satisfy other nontrivial conditions. We also show that by considering weight functions one can construct infinite sets of orthogonal exponentials with a positive density by considering orthogonal projections of affine transformations of hypercubes (i.e., zonotopes).