On the strong Cesaro summability of double orthogonal expansions

AbstractDenoting by σm, n(α, β) the (m, n)th Cesaro mean of the double series ∑ai,k, we say that the double series is strongly Cesaro summable with parameters α, β ⩾ 0 and index λ >0—or summable [C, (α, β)]λ—to the sum s if in the cases (i) α,β>0, (m+1)−1(n+1)−1∑i=0m∑k=0n|σi,k(α−1,β−1)—|κ=0(1). (ii) α or β or both are zero (m+1)−1(n+1)∑i=0m∑k=0n|Δ11((i+1)(k+1)σi,k(α,β)—s|κ=0(1). where Δ11ωi,k = ωi,k − ωi − 1,k − ωi,k − 1 + ω- − 1,k − 1 and ωm,n = o(1) means that ωm,n→ 0 in Pringsheim's sense and there exists a constant C such that for any pair m, n of natural numbers ¦ωm,n¦ ⩽ C holds. This definition for single series was introduced in the case (i) by J. M. Hyslop (Proc. Glasgow Math. Assoc. 1 (1952), 16–20) and in the case (ii) by N. Tanovic-Miller, (Acta Math. Hungar. 42 (1983), 35–43). The case α = β = 0 is the strong convergence, where σi,k(o,o) = si,k the (i, k)th rectangular partial sum of series. The double orthogonal series {∑ ci,k φi,k (x, y)} with the coefficient condition ∑ ci,k2 < ∞ is an orthogonal expansion of a square-integrable function f(x, y). Exchanging the coefficient condition by a stronger one F. Móricz (Stud. Math. 81 (1985), 79–94) proved a theorem for the [C, (α, β)]2; α, β > 12 summability of orthogonal series to f(x, y) almost everywhere. Now we extend the investigation for the wider range of parameters and indices