Properties of functions representable by sine trigonometric series with multiple-monotone coefficients

Functions being sums of sine series with multiple-monotone coefficients are considered. Upper and lower estimates for norms of such functions via their coefficients are presented in the spaces L _p (0 < p < ∞).     

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.     

Convergence of Riemann sums for stochastic integrals

Summary Let b be a Brownian motion and f a function in L ²[0,1]. If is a partition of [0,1], denote by f the step function obtained by replacing f by its mean values in each subinterval. As becomes fine, the martingale f db converges to fdb in L ² but not necessarily almost surely. We determine precisely which Lipschitz conditions on f imply a.s. convergence. A similar thing is done for non-anticipating random functions.     

An example of the sequence of coefficients of a double trigonometric series

We construct an example of a double sequence a of nonnegative numbers that are monotone decreasing to zero in the first index for any fixed value of the second index and two Hadamard lacunary sequences of natural numbers such that the double trigonometric lacunary monotone series with the coefficients a constructed from the first lacunary sequence is squaredivergent almost everywhere and the one constructed from the second lacunary sequence is squareconvergent almost everywhere.     

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997     

Estimates of norms of functions represented as double series over cosines with multiple-monotone coefficients

Lower and upper bounds are proved for the norms of functions being sums of double series in cosines.     

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum

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.     

Asymptotics of multiple trigonometric series with monotone coefficients

The paper describes a general method of determination of the asymptotic behavior near zero of multiple trigonometric series whose coefficients possess certain monotonicity properties.     

Some upper bounds of partial sums of a trigonometric series cannot be sharpened via lower bounds

Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 447–451, September, 1995.     

Functions represented as sums of two quasicontinuous functions with a closed graph

Let X=(X,d) be a metric space. We prove that if the set D(f) of discontinuity points of a function is “sufficiently small,” then f can be decomposed into a sum of two quasicontinuous functions with a closed graph.     

On a class of entire functions represented by Dirichlet series

The present paper deals with the study on a class of entire functions represented by Dirichlet series whose coefficients belong to a commutative Banach algebra with identity. We consider a class of such series which satisfy certain conditions and establish some results.     

The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ( $ \tfrac{1} {2} Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ (, 1). Namely, the following assertion is true. Let α ∈ (, 1), < p < 2, a sequence a ∈ M _α, and . Then the series cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π). Original Russian Text ? M.I. D’yachenko, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2008, No. 5, pp. 38–47.