On the Rearranged Block-Orthonormal Systems

A condition is established for the sequence {Nk } when there exists a rearrangement of functions of the block-orthonormal system {φn } such that the sequence {log2 n} ({(log log n)2}) is a Weyl multiplier for the convergence (for the summability by the methods (C, α) (α > 0)) almost everywhere of series with respect to the rearranged system {φνn }.

On the Absolute Summability of Series with Respect to Block-Orthonormal Systems

Theorems determining Weyl's multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {ω(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {Nk } such that {ω(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the Δ k -orthonormal systems.