The best convergence of multiple trigonometric series

We consider the best convergence of multiple trigonometric series. We indicate essential distinction of the behavior (in this sense) of multiple series from that of simple ones. In particular, the well-known result obtained by S. N. Bernshtein on the best convergence of a series with an odd ratio of frequencies does not hold for a multiple series in the case of the approximation by polynomials with harmonics from rectangles (in the sense of Pringsheim), but it is true for “angular” approximations.     

On the uniform convergence of trigonometric series

In this paper, we obtain sufficient conditions for the uniform convergence of trigonometric series with monotone (in the extended sense) coefficients.     

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.     

Two convergence theorems for Henstock-Kurzweil integrals and their applications to multiple trigonometric series

We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R.P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.     

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

On the uniform convergence of special sine integrals

We study the uniform convergence of sine integrals of measurable functions on the open half-line. We also extend results of F. Móricz and give examples for appropriate functions.     

On the boundedness below of trigonometric polynomials of best approximation

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α _n } _n=0 ^∞ , are uniformly bounded below if the conditions a _n = O(n ^?1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent ?1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L _2π ¹ .     

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.     

On the uniform convergence of samples of lebesgue integrals

Developments and generalizations of a theorem of Halkin, on the limit of the multiple balayage of vector integrals, are deduced via Riemann integration theory.     

On convergence of multiple trigonometric series with monotone coefficients

We study the Pringsheim pointwise convergence of multiple trigonometric series. We obtain a condition on the coefficients of the series that guarantees its Pringsheim convergence and prove the unimprovability of this condition.     

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.     

Accelerating the convergence of trigonometric series

A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.     

The Lebesgue summability of trigonometric integrals

Motivated by the notion of Lebesgue summability of trigonometric series, we define the Lebesgue summability of trigonometric integrals in terms of the symmetric differentiability of the sum of the formally integrated trigonometric integral in question. We extend two theorems of Zygmund from trigonometric series to integrals, and one of them even in a more general form.