Chaundry and Jolliffe  proved that if ak is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence
of the series Σk=1∞ak sin kx is limk→∞kak = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu  generalized this classical result. In this paper we propose new classes
of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov 
on the L1-convergence of Fourier series.
Our aim is to find the source why the logarithm sequences play the crucial role in the L1-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences,
the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms
of these classes, extended analogues of Zhou’s theorems are proved.
A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions
on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin
and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means
of these Fourier series is bounded from Hp to Lp (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin,
Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.