Abstract. It is well known that for a non-negative sequence $\{a_n\}_{n=1}^\infty $ the continuity of the sum $\sum ^\infty_{n=1}a_n\cos nx$ is equivalent to the convergence of the series $\sum ^\infty_{n=1}a_n$. We prove that for generalized monotone $\{a_n\}_{n=1}^\infty $ the last condition implies the so-called $p$-absolute continuity in the sense of L. C. Young and E. R. Love, where $1< p< \infty $. In this case we give estimates for the $p$-variation moduli of continuity and best approximations in terms of Fourier coefficients of a function. As a corollary of the above results some Konyushkov-type theorems on the equivalence of $O$- and $\asymp $-relations are established.

DOI: 10.14232/actasm-014-574-4

AMS Subject Classification (1991): 42A32, 42A10, 42A16, 41A25

Keyword(s): $p$-variation, $L^p$, best approximation, fractional moduli of continuity, Fourier coefficients, equivalence of $O$- and $\asymp $-relations

Received November 12, 2014, and in revised form August 15, 2015. (Registered under 74/2014.)