On series by general Franklin system

We prove that if the admissible sequence $\mathcal{T}$ is strongly regular or is quasi-dyadic and weakly regular, then the Paley operator by the corresponding general Franklin system is of weak type (1, 1). Examples of general Franklin system with some properties are also discussed.     

On absolute convergence of multiple fourier series

We obtain sufficient conditions for β-absolute convergence (0 < β ≤ 1) of multiple Fourier series of functions of the classes $L^2 ([0,2\pi ]^N ),(\Lambda ^1 ,\Lambda ^2 ,...,\Lambda ^N )BV^{(p)} ([0,2\pi ]^N ),r - BV([0,2\pi ]^N )$ .     

On the absolute convergence of power series

We obtain a two-dimensional analog of the Hardy-Littlewood result on the absolute convergence of power series in the case of multiple series on the boundary of a unit polydisk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 594–602, May, 1999.     

On the absolute convergence of multiple fourier series

Let f: R ^N → C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series. This research was started while the first author was a visiting professor at the Department of Mathematics, Texas A&M University, College Station during the fall semester in 2005; and it was also supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

On sets of absolute convergence for multiple trigonometric series

We obtain a sufficient condition for a set of measure zero in N-dimensional space to be a set of absolute convergence (A.C. set) for an N-tuple trigonometric series. We also show that, in a certain subclass of sets of measure zero (namely in the subclass of monotonic curves), this condition cannot be sharpened.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 625–635, May, 1973.The author wishes to thank E. M. Nikishin for some valuable remarks.     

On unconditional and absolute convergence of series in wavelet systems

The paper gives some conditions under which the absolute and the unconditional convergence almost everywhere on a set of positive measure are equivalent for series in wavelet systems.     

The absolute convergence of orthogonal series

We obtain sufficient conditions for the absolute convergence of Fourier series for functions of L _d ² depending on the properties of the function being expanded and the rate of growth of the sums of the system of functions {_k(t)} orthonormalized in [a, b] with respect to d(t). We show that if at some point x [a, b] the function (t) has a discontinuity, at that point the Fourier series of any functionf(t) L _d ² , converges absolutely.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 511–516, November, 1972.     

The absolute convergence of lacunary series

A theorem is proved from which it follows that there exists a complete U-set E and a number p such that: a) if the p-lacunary trigonometric series $$\sum\nolimits_{k = 1}^\infty {a_k \sin (n_k x + \varepsilon _k ),} \frac{{\lim }}{{k \to \infty }}n_{k + 1} /n_k > p,$$ converges on E, the series of the moduli of its coefficients converges; b) if the sum of the p-lacunary trigonometric series is differentiable on E, it is continuously differentiable everywhere.     

On the unconditional almost-everywhere convergence of general orthogonal series

The Orlicz and Tandori theorems on the unconditional almost-everywhere convergence, with respect to Lebesgue measure, of real orthogonal series defined on the interval (0; 1) are extended to general complex orthogonal series defined on a space with arbitrary measure.