Acta Math. Acad. Paed. Nyíregyháziensis
15 (1999), 27-34
www.bgytf.hu/~amapn

On the a.e. convergence of Fourier series on unbounded Vilenkin groups

G. Gát

Abstract:

It is well known that the 2ⁿth partial sums of the Walsh-Fourier series of an integrable function converges a.e. to the function. This result has been proved [Sto] by techniques known in the martingale theory. The author gave "purely dyadic harmonic analysis'' proof of this in the former volume of this journal [Gát]. The Vilenkin groups are generalizations of the Walsh group. We prove the a.e. convergence $S_{M_n}f \to f \, (n\to\infty), \, f\in L^1(G_m)$ even in the case when G_m is an unbounded Vilenkin group. The nowelty of this proof is that we use techniques, which are elementary in dyadic harmonic analysis. We do not use any technique in martingale theory used in the former proof [Sto].

 

[Sto] Stout, W. F., Almost sure convergence, Academic Press, 1974.

 

Mathematics Subject Classification. 42C10, 43A75.