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I guess this is the theorem : Let $f$ be an infinitely differentiable function defined in $\mathbb R^{n}$, and $u(x,r)$ the mean value of $f$ taken over the sphere with center at $x$ and radius $r$, …

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

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}( …

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 …

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– …

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\ …

One possible reference among many many others... : H. Brezis and F. Browder, Partial differential equations in the 20th century, Advances in Mathematics 135 (1998), 76-144.

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 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 …

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 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, …

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 …

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 …

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 …

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 …