On trigonometric series and primitive functions

It is proved that any measurable, finite function f(x) has a smooth primitive F(x), i.e. there is a function F(x) such that F′(x) = f(x) almost everywhere, and particularly ω(δ; F) = o(δ ln δ). This is an improvement of N. N. Luzin’s theorem which states just the continuity of the primitive F(x).     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

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.     

On the kolmogorov theorems on Fourier series and conjugate functions

We consider Fourier series of summable functions from spaces ??wider?? than L ₁. We describe classes ??(L) which contain conjugate functions, where their conjugate Fourier series converge. The obtained results are more general than A. N. Kolmogorov theorems on the convergence of Fourier series in metrics weaker than that of L ₁.     

Distribution functions and trigonometric series with monotonically decreasing coefficients

The order of the distribution function of the sum of a cosine series with monotonically decreasing coefficients is determined. Theorems concerning integrability and convergence are proved for certain integral classes.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 3–10, July, 1971.The author wishes to thank O. D. Tsereteli for directing this work and P. L. Ul'yanov for the interest he showed in it.     

On a class ofN-dimensional trigonometric series

An analog of Fomin's well-known one-dimensional theorem is proved for trigonometric series of the form

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

On L-convergence of trigonometric series

In the present paper we consider the trigonometric series with (β,r)-general monotone and (β,r)-rest bounded variation coefficients. Necessary and sufficient conditions of L-convergence for such series are obtained in terms of the coefficients. Moreover, we generalize and extend the Tikhonov results [J. Math. Anal. Appl. 347 (2008) 416-427] to the class GM(β,r) or the class RBVS(β,r).     

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 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 and Cesàro summability of trigonometric Fourier series of monotone type functions

The behaviour of the Cesàro means of trigonometric Fourier series of monotone type functions in the space of continuous functions is studied.     

The convergence of trigonometric series to functions of bounded variation

Necessary and sufficient conditions are derived for the convergence of a trigonometric series to a function of bounded variation on the interval (, ) [-, ]. For the case in which the coefficients satisfy certain conditions, the continuity of the sum function is investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 47–58, July, 1970.The author wishes to thank his scientific supervisor S. A. Telyakovskii for suggesting this problem and for his interest in the work.