Codes and Expansions (CodEx) Seminar

Joseph Lakey (New Mexico State University)
New approaches to convergence of Walsh-Fourier series

Almost everywhere convergence of Walsh-Fourier series on L2 of the unit interval was first established in the late 1960s following Carleson’s proof for the trigonometric case. Later approaches established convergence via boundedness of maximal partial sum operators. We outline an approach and preliminary results that aim to provide direct and precise uniform L2 bounds on norms of matrices that correspond to partial sum operators for Walsh-Fourier series in finite dimensions. The special case of dyadic partial sums is analyzed in more detail.  The approach boils down to a method to evaluate and compare norms of specific “truncated Walsh-Hadamard matrices” that represent partial sum operators. A second approach involving a type of matrix dilation is also mentioned as a potential path to establishing sharp bounds for the general (non-dyadic) case.