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 summability of trigonometric series

{_n} a _{n n} , s _n n- , _n -n- (, 1)- . . . -, . , «|s₂n+1–s ₂ n¦< . .» «¦ _n – _n–1¦< . .» . s. , {a _n } 0, a _n/n<, {a _n } , {it|_n¦<. , , -.     

On the series of absolute values of blocks of trigonometric series

Conditions of integrability with a power weight are obtained for the sums of series of absolute values of blocks of sine series and cosine series under certain special conditions imposed on the coefficients of these series.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

On the absolute Riesz summability of orthogonal series

- , [4]. , .     

On the absolute summability of factored Fourier series

Summary In this paper the author has considered the harmonic summability of the factored Fourier series when ϕα(t), α > 0, is of bounded variation.     

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.