Strong approximation by rectangular partial sums of double orthogonal series (Q1113407)

Two theorems connected with strong approximation by rectangular partial sums of double orthogonal series of the form \(\sum^{\infty}_{i=1}\sum^{\infty}_{k=1}a_{ik}\phi_{ik}(x),\) where \(\{a_{ik}:\) \(i,k=1,2,...\}\) is a double sequence of real numbers for which \(\sum^{\infty}_{i=1}\sum^{\infty}_{k=1}a^ 2_{ik}<\infty\), have been established. The proofs are based on some well-expressed lemmas.