We investigate the asymptotic behavior of Jacobi-Pi\~neiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.
We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $\beta_K=-K$ and $\alpha_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(\alpha,\beta)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^...
For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Via...
We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential...
We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\frac{\pi}{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function $\varphi_d$, and determine their limiting d...
We study uniform approximation of $\exp(tz)$, $t>0$, on $(-\infty,0]$ by rational functions $P_m(z)/(q_m-z)^m$ with a prescribed repeated pole $q_m>0$. A M\"obius transformation reduces the problem to polynomial approximation of $F_\lambda(x)=\exp\!\left(-\lambda\frac{1-x}{1+x}\right)$ on $[-1,1]$, where $\lambda=tq_m$...
For the Jacobi family with parameters $(0,\beta)$, $\beta\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi--Radau functions. When two consecutive functions are parametrised by the same Pr\"ufer phase, the function of higher degree attains that phase closer to $\theta=0$ and has stri...
K. Castillo, Peng-Cheng Hang· 0 citations
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