Approximation theorems for double orthogonal series

Abstract

Let { φ ik ( x): i, k = 1, 2,…} be a double orthonormal system on a positive measure space ( X, ƒ, μ) and { a ik } a double sequence of real numbers for which ∑ i = 1 ∞ ∑ k = 1 ∞ a ik 2 < ∞. Then the sum f( x) of the double orthogonal series ∑ i = 1 ∞ ∑ k = 1 ∞ a ik φ ik ( x) exists in the sense of L 2-metric. If, in addition, ∑ i = 1 ∞ ∑ k = 1 ∞ a ik 2 κ 2( 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 s mn ( x) = ∑ i = 1 m ∑ k = 1 n a ik φ ik ( x), by the first arithmetic means of the rectangular partial sums σ mn(x) = ( 1 mn ) ∑ i = 1 m ∑ k = 1 n s ik(x) , by the first arithmetic means of the square partial sums σ r(x) = ( 1 r ) ∑ k = 1 r s kk(x) , etc. The so-called strong approximation to f( x) by s mn ( x) is also studied.