Discrete Fourier restriction theorems in two dimensions

TL;DR

This paper explores Fourier restriction theorems in two dimensions by reformulating a theorem of Yudin within the Fourier algebra of a discrete group, providing visual proofs for boundary restriction properties.

Contribution

It reformulates Yudin's theorem as a restriction property in the Fourier algebra of a discrete plane and offers visual proofs for these restriction statements.

Findings

Reformulation of Yudin's theorem in Fourier algebra context

Visual proofs of restriction properties on convex domain boundaries

Extension of Fourier restriction concepts to discrete groups

Abstract

Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({{\mathbb{R}}_{\rm d}^2})$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({{\mathbb{R}}_{\rm d}^2})$ to the boundary of a strictly convex domain when those functions vanish outside that boundary. We give visual proofs of that statement and a complementary one.