Quadrature formulas for classes with constraints on the Fourier coefficients

Translated from Matematicheskie Zametki, Vol. 49, No. 2, pp. 144–147, February, 1991.     

Estimates for wavelet coefficients on some classes of functions

Let ψ _m ^D be orthogonal Daubechies wavelets that have m zero moments and let {fx1791-01}. We prove that {fx1791-02}. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1594–1600, December, 2007.     

Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, ],f(t)=f(—t), then for odd n while for even n, b₀=0.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 597–604, May, 1968.     

On sequences of Fourier coefficients of functions of Hölder classes

The following theorem is proved. Let { _k(t)} be an arbitrary complete orthonormal system on [0, 1] and let 1/2<<1. Then anf(t) C exists for all< such that _k=1 · |c_k(f)|^p=, p=2/(l+2), where .Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 567–572, November, 1969.The authors wish to thank P. P. Zabreiko and P. L. Ul'yanov for helpful discussions and remarks.     

Remarks on multiple Fourier series

: (1) ( , , ), (2) ( —, , ). , .     

Some classes of functions and Fourier coefficients with respect to general orthonormal systems

The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.     

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by , which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

     

Remarks on Ginzburg's bivariant Chern classes

The convolution product is an important tool in the geometric representation theory. Ginzburg constructed the bivariant Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we give some remarks on the Ginzburg bivariant Chern classes.

     

On some classes of polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots we study polynomials g of lowest degree such that the product gf has positive (nonnegative, respectively) coefficients. We show that for quadratic f with negative linear coefficient every such g must have positive coefficients and exhibit an easy procedure for the determination of g. If f has only integer coefficients we show that g with integer coefficients can be found. Furthermore, for some classes of polynomials f we give upper (lower, respectively) bounds for the degrees of g.     

Some classes of special functions using Fourier transforms of some two-Variable orthogonal polynomials

ABSTRACT

In this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented.     

Remarks on a parametric representation of Nevanlinna-Dzhrbashyan classes

This article presents a parametric representation for classes of functions f that are holomorphic in the unit disk and such that

Remarks on some zero-sum theorems

In the present paper, we give a new proof of a weighted generalization of a result of Gao in a particular case. We also give new methods for determining the weighted Davenport constant and another similar constant for some particular weights.     

Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.