Growth of Cesàro means of double Vilenkin-Fourier series of unbounded type

Abstract

Let N represent the set of natural numbers, and Q := [0, 1) x [0, 1) represent the unit cube. Let p ₀, p ₁, p ₂, … and q ₀, q ₁,q ₂, … be two sequences of natural numbers with p _n≥ 2 and q _n≥ 2. For each n ∈ N set P _n := p ₀ p ₁ … p _n-1 and Q _n := q ₀ q ₁…q _n-1, where the empty product is by definition 1. The double Vilenkin system associated with these generators is the system (w _n,m ;n,m ∈ N) defined on Q as follows:

$${{w}_{{n,m}}}(x,y): = {{w}_{n}}(x){{w}_{m}}(y): = \prod\limits_{{k = 0}}^{\infty } {\exp } \left( {\frac{{2\pi i{{n}_{k}}{{x}_{k}}}}{{{{p}_{k}}}}} \right)\prod\limits_{{k = 0}}^{\infty } {\exp } \left( {\frac{{2\pi i{{m}_{k}}{{y}_{k}}}}{{{{q}_{k}}}}} \right) $$

, where the coefficients n _k , m _k , x _k , y _k all are integers which satisfy

$$0 \leqslant {{n}_{k}} \leqslant {{p}_{k}}, 0 \leqslant {{m}_{k}} < {{q}_{k}}, 0 \leqslant {{x}_{k}} < {{p}_{k}},0 \leqslant {{y}_{k}} < {{q}_{k}}, $$

(see Vilenkin [5] for more details). When p _k ≡q _k ≡2 for all k, the system w _n,m is the double Walsh system. When p _k = O(1) and q _k =O(1), the system w _n,m is called a double Vilenkin system of bounded type.