LAUSR

In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular partial sums S2A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums SnAMA f ,nA ∈ {1, 2, ...,mA − 1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ∈ L2. Let P denote the set of positive integers,N := P∪ {0}. Let m := (m0,m1, ...) denote a sequence of positive integers not less than 2. Denote by Zmk := {0, 1, ...,mk − 1} the additive group of integers modulo mk. Define the group Gm as the complete direct product of the groups Zm j , with the product of the discrete topologies of Zm j ’s. The direct product μ of the measures μk({ j}) := 1 mk ( j ∈ Zmk ) is the Haar measure on Gm with μ(Gm) = 1. The elements of Gm can be represented by sequences x := (x0, x1, ..., x j, ...), (x j ∈ Zm j ). The group operation + in Gm is given by x + y = ( x0 + y0 (modm0) , ..., xk + yk (modmk) , ... ) , where x = (x0, ..., xk, ...) and y = ( y0, ..., yk, ... ) ∈ G. The inverse of + will be denoted by −. It is easy to give a base for the neighborhoods of Gm : I0(x) := Gm, In(x) := {y ∈ Gm|y0 = x0, ..., yn−1 = xn−1} for x ∈ Gm, n ∈N. Define In := In(0) for n ∈N. The sets In(x) are called (m-adic) intervals. Define AA,B the σ-algebra generated by rectangles IA(x) × IB(x) as x = (x1, x2) rolls over Gm × Gm. Let EA,B be the conditional expectation operator with respect to σ-algebraAA,B. That is, EA,B f (x1, x2) = MAMB ∫ IA(x1)×IB(x2) f (y1, y2)dμ(y1, y2). 2010 Mathematics Subject Classification. Primary 42C10