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Concerning the other coefficients $c_k$, Branan [1] showed that $(k+1)c_{k+1}=(n-k)\bar c_{n-k}$ is a necessary condition for univalence in $|z|<1$ if $c_1=1$ and $c_n=1/n$. For univalent polynomials …

There has been recent advances in the study of orthogonal polynomials with respect to logarithmic weights of the form $$w(x)=\log\frac{2k}{(1-x)}~\text{on}~(-1,1),\qquad k>1,$$ in particular the asymp …

Uniform approximation by polynomials on a finite set of points is studied in Section 1.3 of T.J. Rivlin, An introduction to the approximation of functions, Dover, 2003. An explicit solution to …

Let $P_{n}(z)=\gamma_{n}z^{n}+\cdots$ be a sequence of orthonormal polynomials with respect to some weight $w$ on $\mathbb{R}$. Given an $n\geq0$, consider the following Riemann-Hilbert problem for th …

Appendix A4 of the book P. Borwein, T. Erdelyi, Polynomials and Polynomial inequalities, Graduate Texts in Mathematics 161, Springer should be a good source for your question. In particular, (A …

Yes, indeed, the maximum principle for subharmonic functions (hence for harmonic functions) is valid for domains, independently of the smoothness of their boundaries : Let $u$ be a subharmonic functi …

It is a conjecture of Stieltjes, apparently still open, see T.J. Stieltjes, Letter No. 275 of Oct. 2, 1890, in Correspondance d'Hermite et de Stieltjes, vol 2, Gauthier-Villars, Paris, 1905. that Le …

Results for the $L^p$ norms of general orthogonal polynomials have been given in Ref. [1]. In the case of the orthonormal Hermite polynomials $H_n$, it was given in [1] for $L^{2p}$ norms, $0<p<4/3$ …

The Funk-Hecke formula, in its simplest form, says that for $f$, a bounded measurable function on $[-1,1]$, and $y\in S_{n}$, one has $$ \int_{S_{n}}f( \langle x,y \rangle)d\sigma_{n}(x)=\frac{2\pi^{n …

A nice and classical reference for splines (at third-year undergraduate or graduate level) is Powell, M. J. D. Approximation theory and methods. Cambridge University Press, Cambridge-New York, 1 …

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set, and let $(x_{n})_{n\geq1}$ be a sequence of all points of $\Omega$ with rational coordinates. Consider the discrete measure of finite mass, $$\ …

A reference in english for Bochner's theorem is section 20.1, p.508, of the book by Mourad E.H.Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and it …

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, endowed w …