Approximation theorems for double orthogonal series

AbstractLet {φik(x): i, k = 1, 2,…} be a double orthonormal system on a positive measure space (X, ƒ, μ) and {aik} a double sequence of real numbers for which ∑i = 1∞ ∑k = 1∞ aik2 < ∞. Then the sum f(x) of the double orthogonal series ∑i = 1∞ ∑k = 1∞ aikφik(x) exists in the sense of L2-metric. If, in addition, ∑i = 1∞ ∑k = 1∞ aik2κ2(i, k) < ∞ with an appropriate double sequence {κ(i,k)} of positive numbers, then a rate of approximation to f(x) can be concluded by the rectangular partial sums smn(x) = ∑i = 1m ∑k = 1n aikφik(x), by the first arithmetic means of the rectangular partial sums σmn(x) = (1mn) ∑i = 1m ∑k = 1n sik(x), by the first arithmetic means of the square partial sums σr(x) = (1r) ∑k = 1r skk(x), etc. The so-called strong approximation to f(x) by smn(x) is also studied