Uniform approximation of sgn(x) by polynomials and entire functions

Uniform approximation of sgn(x) by rational functions with prescribed poles

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function $\sgn(x)$ on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of $2m-1$ at infinity. Then the following limit exists \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{m-{1/2}} (2m-1)^{k+{1/2}}=\frac 2 π(\frac{1-a^2}{2a})^{k+{1/2}} Γ(k+\frac 1 2). \end{equation}

Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$ on $[-1,-a]\cup[a,1]$ by Laurent polynomials.

Weighted uniform convergence of entire Grünwald operators on the real line

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type $τ>0$ at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an accumulation point

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(α)}(z)$ where $α$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].

Lebesgue-type inequalities for de la Vallee Poussin sums on the sets of analytic and entire functions

For the functions from sets $C_β^ψC$ and $C_β^ψL_s, \ 1\leq s\leq\infty$, generated by sequences $ψ(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{ψ(k+1)}{ψ(k)}=q, \ q\in[0,1)$, asymptotically unimprovable estimates for deviations of de la Vallée Poussin sums in the uniform metric, which are represented in terms of values of the best approximations of $(ψ,β)$-differentiable functions of this sort by trigonometric polynomials in the metrics $L_s$ are obtained. Proved that received estimates are unimprovable on some important functional subsets.

一个或两个参数均很大的惠特克函数：涉及贝塞尔函数与艾里函数的简化一致渐近展开式

Whittaker functions with one or both parameters large: simplified uniform asymptotic expansions involving Bessel and Airy functions

该研究针对$\kappa \to \infty$时的惠特克函数$W_{\kappa,\mu}(z)$、$M_{\kappa,\mu}(z)$及相关函数，基于关联微分方程推导了涉及贝塞尔函数与艾里函数的简化一致渐近展开式，可覆盖复$z$平面主分支并可扩展至所有分支及负$\mu$。