On some properties of double conjugate trigonometric Fourier series

A theorem of Ferenc Lukács [4] states that the partial sums of conjugate Fourier series of periodic Lebesgue integrable functions f diverge at logarithmic rate at the points of discontinuity of first kind of f. F. Móricz [5] proved an analogous theorem for the rectangular partial sums of bivariate functions. The present paper proves analogues of Móricz’s theorem for generalized Cesàro means and for positive linear means.     

AbsoluteN qα -summability of the series conjugate to a Fourier series

The object of the paper is to study the absolute N_qα-summability of the series conjugate to a Fourier series, generalising a known result.     

Weighted integrability of double trigonometric series

We study the double trigonometric series whose coefficients are such that Then its rectangular partial sums converge uniformly to some . We give sufficient conditions for the Lebesgue integrability of , where , or . For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.

     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

On uniformly convergent rearrangements of trigonometric Fourier series

We show that if the module of continuity ω(ƒ, δ) of a 2π-periodic function ƒ ∈ {ie081-01} is o(1/ log log 1/δ) as δ → 0+, then there exists a rearrangement of the trigonometric Fourier series of ƒ converging uniformly to ƒ. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

Sets of absolute convergence of double trigonometric series

We obtain a sufficient condition for a set of plane measure zero to be a set of absolute convergence (an A.C.-set) for a double trigonometric series. Specifically, let y=f(x) (0 x2) be a smooth curve and let inif (x) t <. Then, any set of positive linear measure lying on this curve is an A.C.-set.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 473–480, May, 1972.In conclusion, I am indebted to E. M. Nikishin for suggesting the problem.     

On the convergence in mean of trigonometric Fourier series

We prove the sharpness of Zygmund’s theorem, which asserts that if a 2π-periodic function f belongs to L ln⁺ L, then its Fourier series is convergent in mean.     

Summability of multiple trigonometric Fourier series by linear methods

The paper deals with the problem of estimation of deviations of functions of several variables from linear means of their multiple trigonometric Fourier series. An approach of reducing this problem to the corresponding problem for functions of single variable is developed.     

Convergence of double Fourier series

Summary Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l ₁ , l ₂ , ..., l_k terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l ₁ , l ₂ , ..., l_k, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed. The preparation of this paper was financed, in part, by a Canadian Mathematical Congress Summer Research Grant (1968). Entrata in Redazione il 3 febbraio 1969.     

On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.     

On strong Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 275–279, February, 1992.     

Integrability of double trigonometric series with special coefficients

- , , . , L ¹=L ¹([0, ]×]0, ]). , ; , L ¹ , - . . . . 1976 ., ; 1989 .

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The basic part of this research was done while the author was a visiting professor at the Syracuse University, U.S.A., during the academic year 1986/87.     

On the convergence of greedy approximants of trigonometric Fourier series

Efficient versions of the Cauchy criterion for the convergence in L _p of greedy approximants of trigonometric Fourier series are discussed.     

Concerning the multiplicators of Fourier series in the trigonometric system

In this paper we prove theorems on multiplicators of Fourier series inL _p, where the conditions depend on a parameterp. An example illustrating the importance of these conditions is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 235–247, February, 1998.     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

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In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.