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Published online by Cambridge University Press: 20 November 2018
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 {ϕ jk (x):i, k = 0, 1, …} an orthonormal system defined on X. We consider the double orthogonal series
(1.1)
where {ϕ ik :i, k = 0, 1, …} is a double sequence of real numbers (coefficients), for which
(1.2)