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As you said, the inequality holds for the orthonormal polynomials $P_{k}$, $k\geq0$, of any positive measure $\mu$, with support $K$, a compact subset of $\mathbb{C}$. It follows from a basic result a …

The (complex) Zernike polynomials $V_{n}^{l}(x,y)$, of total degree $n$, with $|l|\leq n,~n-|l|\text{ even}$, are such that $$ V_{n}^{l}(x,y)=R_{n}^{l}(\rho)e^{il\varphi},\quad \text{with }x=\rho\cos\ …

This is the main result in a paper by J.L. Walsh from 1927 (in german): J.L. Walsh, Über die Entwicklung einer Funktion einer komplexen Veränderlichen nach Polynomen. Math. Ann. 96 (1927), no. 1, 437– …

Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$, $$ I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy. $$ Assuming $f$ is in t …

The inequality and its proof can be found in Chapter 10.18, Eq. (14), p.207 of A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, "Higher Transcendental Functions", vol. 2, Bateman Manuscript P …

E.C. Titchmarsh, The theory of functions, Oxford University Press Section 4.6 Hadamard multiplication theorem, p.158

The set of real numbers for which the property is not satisfied has Lebesgue measure zero : Assume $\alpha>0$ and write $\alpha$ in the form $\alpha=1/\beta^{2}$, $\beta>0$. Then, the approximation p …

This is the two-constants theorem : Let $D$ be a domain of $\mathbb{C}$ with (non-polar) boundary $\partial D$, and let $B$ a Borel subset of $\partial D$. If $u$ is subharmonic on $D$ and satsifies …

The Portmanteau theorem does not seem to be stated in this form in Billingsley or other classical references that I checked. A possible reference for the direct implication is Theorem A.3.12. p.378 of …

So, I write this as an answer rather than a comment to close the question. This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is $$\frac{P_{n}( …

Here, three possible references for the formula: P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Computer Science and Applied Mathematics. Academic Press, New York, 1984 (see …

The paper, by C. Dellacherie himself, C. Dellacherie, Capacities and analytic sets, Cabal seminar 77-79, Proc., Caltech-UCLA logic Semin. 1977-79, Lect. Notes Math. 839, 1-31 (1981). covers, …

4 reference books for the study of PDE with MATLAB: Coleman, Matthew P. An introduction to partial differential equations with MATLAB. Second edition. Chapman & Hall/CRC Applied Mathematics and No …

Here are four references on the subject (the main ones as far as I know) : Baker, George A.; Graves-Morris, Peter, Pad\'e approximants. Second edition. Encyclopedia of Mathematics and its Applicatio …

I edit my post to answer Carlo Beenakker's remark and also because I would like to add a reference, possibly more accurate than the two below. Theorem 7.1 p.13 of A. Adelberg, A finite difference a …