Proof of two conjectures of Móricz on double trigonometric series

We use a unified approach to obtain several integrability theorems of Boas [R.P. Boas, Integrability of trigonometric series, I, Duke Math. J. 18 (1951) 787-793]. In particular, we settle two conjectures of Móricz [F. Móricz, On the integrability of double cosine and sine series, II, J. Math. Anal. Appl. 154 (1991) 466-483] concerning double trigonometric series.     

Completeness of perturbed systems of trigonometric functions

The paper addresses the completeness of perturbed systems of sines and cosines with non-integer indices and, moreover, complex, in the space of Lebesgue-integrable functions. The criteria for the variation from the integer value have been found such that sine and cosine systems are complete.     

Estimates of trigonometric sums by the third derivative

The well-known Van der Corput estimate of a trigonometric sum by the third derivative is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 383–389, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-00002.     

Positivity of certain sums over Jacobi kernel polynomials

We present a computer-assisted proof of positivity of sums over kernel polynomials for ultraspherical Jacobi polynomials.     

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.     

A hierarchy of thin sets related to the boundedness of trigonometric series

We study the family  of the sets on which some series of the form is uniformly bounded. We show that the families  of all sets admitting the boundary  form a hierarchy which is incontinuous with respect to the operations of intersection and union.

     

Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums

If at each point of a set of positive Lebesgue measure every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier series of a function to which it converges uniformly.

     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas)

Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ _n ^(α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w ^(a,b) σ _n ^(α,β) ‖_∞ ≦ C‖w ^(a,b) f‖_∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.      

Positivity of the generalized translation associated with the q-Hankel transform and applications

A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel- positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q^?2-Lommel polynomials.     

Moment inequalities for the partial sums of random variables

This paper discusses the conditions under which Rosenthal type inequality is obtained from M-Z-B type inequality. And M-Z-B type inequality is proved for a wide class of random variables. Hence Rosenthal type inequalities for some classes of random variables are obtained.     

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 .     

Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the jumps are approximated by solution of a system of linear equations. The accuracy of the jump approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation of nonnegative functions by means of exponentiated trigonometric polynomials

We consider the problem of approximating a nonnegative function from the knowledge of its first Fourier coefficients. Here, we analyze a method introduced heuristically in a paper by Borwein and Huang (SIAM J. Opt. 5 (1995) 68–99), where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function. In this note, we prove that approximations given by Borwein and Huang's method, in the trigonometric case, can be related to a nonlinear constrained optimization problem, and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of Toeplitz matrices. Moreover, they allow to obtain an improved convergence theorem for best entropy approximations.     

Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

WZ方法与一个二项式级数的部分和公式

本文基于WZ理论给出了Peter Paule与Carsten Schneider的一篇文章中的一个二项式级数的部分和公式的新证明,并且发现他们的文章中所给的另一个二项式级数的部分和公式实际上是错误的,我们给出了其相应的一个正确公式.     

Alternative proofs for inequalities of some trigonometric functions

By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established.     

Multidimensional generalization of the Korous inequality

This paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.