Approximation theorems for double orthogonal series, II

Abstract

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 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 ) 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.