On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm-Liouville Operator

For semicontinuous summation methods generated by I > = {lambda (n) (h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm-Liouville operator of class B, some results on the uniform a.e. behavior of I >-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means lambda (n) (h) = exp(-u(alpha)(n)h) are obtained. An application to a generalized heat equation is given.