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The existence of the ‘rare’ sequence of partial sums summable with the method of arithmetical means at each Lebesgue point is proved in the paper. The proof is based on the strategy of random choice. Introduction For a periodic function f ∈ L(T), p ≥ 1, we shall call a point x such that Φx,p(f, h) = ∫ h 0 |f(x+ t)− 2f(x) + f(x− t)|pdt = o(h), h→ 0, an Lp-point. It is well known (see, for example, [Zg]) that Φx,p(f, h) = o(h) almost everywhere. By Sk(f ;x) = ∑ |j|≤k f̂(j)e we denote the partial sum of the Fourier series. It is also well known ([Zg]) that the Fejer means 1 N N ∑ k=1 Sk(f ;x) → f(x), N →∞. uniformly for continuous f , and in each L1-point for summable f . In 1936 Zalcwasser [Zl] asked how ‘rare’ can be the sequence of integers {nk} such that 1 N N ∑ k=1 Snk(f ;x) → f(x), N →∞. This problem was completely solved for continuous functions (uniform convergence) in [Sl], [TZ], [Cl], [Bl]. If {nk} is convex, then the condition supN N−1/2 log nN < ∞ is necessary and sufficient for the uniform convergence for every continuous function. Received by the editors July 30, 1996. 1991 Mathematics Subject Classification. Primary 42A16, 42A24.

Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points