On the Restricted Cesàro Summability of Multiple Orthogonal Series

We actually treat double orthogonal series in detail, simply for the sake of brevity in notations. Multiple orthogonal series will be shortly indicated in the concluding Section 8.Let (X, , μ) be an arbitrary positive measure space and {ϕjk(x):i, k = 0, 1, …} an orthonormal system defined on X. We consider the double orthogonal series(1.1)where {ϕik:i, k = 0, 1, …} is a double sequence of real numbers (coefficients), for which(1.2)