Reconstruction of Eigenfunctions of q-ary n-dimensional Hypercube

TL;DR

This paper develops methods to reconstruct eigenfunctions of q-ary n-dimensional hypercubes from partial data, establishing conditions for unique determination of the functions based on their values in spheres and balls.

Contribution

It introduces new reconstruction techniques and conditions for eigenfunctions of hypercubes, enhancing understanding of their structure from limited information.

Findings

Values in the sphere determine the function in the ball under certain conditions.

When the radius equals the eigenvalue number, the eigenfunction is uniquely determined.

Reconstruction methods depend on supplementary conditions for uniqueness.

Abstract

Under study are eigenfunctions of $q$-ary $n$-dimensional hypercube. Given all values of an eigenfunction in the sphere we develop methods to reconstruct the function in full or in part. First, we obtain that all values of the function in the corresponding ball are uniquely determined under some supplementary conditions. Secondly, if the radius is equal to the eigenvalue number we obtain that all values of the eigenfunction are uniquiely determined under some supplementary conditions.