Approximation of analytic functions by polynomials with integer coefficients

For some classes of analytic functions we estimate the rate of approximation by polynomials with integer coefficients.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 182–188, February, 1996.     

Calculation of fourier coefficients of discrete functions using cubic splines

Formulas are given for the calculation of the finite Fourier transform of a B-spline. These formulas are useful for the computation of Fourier coefficients of a function which is given at a discrete set of arbitrary points.     

On the factorization of some polynomial analogues of binomial coefficients

Archiv der Mathematik -     

Constructive almost periodic factorization of some triangular matrix functions

A factorability criterion is obtained constructively, and the respective factorization obtained explicitly, for 2×2 triangular almost periodic matrix functions of the form . Here f=c−1e−α−c₀+c₁eβ, eμ(x):=eiμx, cj are non-zero constants and 0<α,β, α+β<λ?α+β+max{α,β} with α/β being irrational. Note that the factorization problem, even for triangular matrix functions as above with an arbitrary trinomial f, is open. The result obtained is yet another step towards its solution.     

Winding numbers and fourier series

This is an expository paper on a topic of classical analysis arising from the BMO-theory of topological degree (Brezis-Nirenberg, 1995). We sketch the history of the subject and some of its recent developments.     

On the fourier coefficients of functions concentrated on a subset of the circle

We consider ,mE > 0,G(E) is a certain subspace of L ¹ (E) consisting of functions concentrated on E and integrable, and {d_k}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=L^p(E), p≥2, then there exists f∈G(E) such that |f(n)|≥d_n, (one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function classes G(E). We study the question of partial majorization. Bibliography: 2 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 42–48.     

Stirling numbers and integer partitions

In this paper, we prove that the Stirling numbers of both kinds can be written as sums over integer partitions. As corollaries, we rewrite some identities with Stirling numbers of both kinds without Stirling numbers.     

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.     

整数分解式的指数函数及其均值的计算

给出了整数标准分解式的数论函数β(n，p)和它的均值∑n≤Nβ(n，p)的定义，推出了整数n标准分解式的指数函数计算公式和均值∑n≤Nβ(n,p)的精确计算公式．