An Eigenvector Problem Arising in the Study of Convergence of Walsh–Fourier Series

This work establishes bounds for certain matrices that arise in the study of the convergence of expansions in Walsh functions, or Walsh–Fourier series. The matrices in question arise by “truncating” certain orthogonal matrices corresponding to expansions of dyadic step functions on the unit interval in the basis of Walsh functions. Here, truncation means, for each column, replacing all entries in that column below a column-dependent row by zero. The truncations correspond to partial sum operators in the Walsh basis. We study here a specific family of these truncated matrices that are shown elsewhere to have optimal norms among certain families of truncations. The main result here provides an approximate eigenvalue bound from which one can conclude that the norm of the truncation approaches a fixed value as the dimension of the truncation matrix approaches infinity. Its proof relies on the interplay between continuous and discrete sets. In particular, it is shown that integer samples of certain sinusoidal functions form approximate eigenvectors of a compressed version of the truncation. This bound plays an important role in a bigger new approach to the convergence of Walsh–Fourier series that this work is part of.