Trigonometric series with general monotone coefficients

Abstract

We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin.

A concept of general monotonicity and applications

“…Then, if p m ≤ Cq m with C independent of m and n, the sequence {c n } is a QM S (and, consequently, a GM S, see [18]). With assumption p m ≤ Cq m in hand, we observe that if…”

“…Recently certain efforts were made in extending the class of monotone sequences (in [8], [18]) and functions (in [11]) in such a way that known results hold true for wider classes of sequences/functions. The property that characterized those new classes was called general monotonicity.…”

Trigonometric series of Nikol’skii classes

“…We note (see [17]) that a = {a n } ∞ n=1 ∈ GM if and only if {a n } ∞ n=1 satisfies the following two conditions:…”

“…where M is the class of monotone sequences, QM is the class of quasi monotone sequences, ORV is the class of O-regularly quasi monotone sequences, and RBV S is the class of sequences of rest bounded variation (see references in [17]). …”

On the uniform convergence of double sine series

“…Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [2], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences.…”