*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 2^{n}th
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
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.