On the strong Cesaro summability of double orthogonal expansions

Abstract

Denoting by σ m, n ( α, β) the ( m, n)th Cesaro mean of the double series ∑ a i, 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=0 m∑ k=0 n|σ i,k (α−1,β−1)—|κ=0(1) . (ii) α or β or both are zero (m+1) −1(n+1) ∑ i=0 m ∑ k=0 n |Δ 11((i+1)(k+1)σ i,k (α,β)—s|κ=0(1) . where Δ 11 ω i, k = ω i, k − ω i − 1, k − ω i, k − 1 + ω - − 1, k − 1 and ω m, n = o(1) means that ω m, n → 0 in Pringsheim's sense and there exists a constant C such that for any pair m, n of natural numbers ¦ω m,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 σ i, k ( o, o) = s i, k the ( i, k)th rectangular partial sum of series. The double orthogonal series {∑ c i,k φ i,k (x, y)} with the coefficient condition ∑ c i, k 2 < ∞ 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 > 1 2 summability of orthogonal series to f( x, y) almost everywhere. Now we extend the investigation for the wider range of parameters and indices.