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

For semicontinuous summation methods generated by $\Lambda=\{\lambda_{n}(h)\}$ ($n=0,1,\dots$; $h>0$) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class $\mathcal{B}$, some results on the uniform a.e. behavior of $\Lambda$-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.