On the pointwise convergence of Cesàro means of two-variable functions with respect to unbounded Vilenkin systems Original Research Article

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C,1)) means of functions on unbounded Vilenkin groups. There was no known positive result before the authorʹs paper appeared in 1999 (J. Approx. Theory 101(1) (1999) 1) with respect to the a.e. convergence of the one-dimensional (C,1) means of Lp (p>1) functions. This paper is concerned with the almost everywhere convergence of a subsequence of the two-dimensional Fejér means of functions in L log+ L. Namely, we prove the a.e. relation limn,k→∞ σMn,M̃kf=f (for the indices the condition |n−k|<α is provided, where α>0 is some constant).