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# Questions tagged [approximation-theory]

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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I am trying to prove equations (3) given in this paper http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf. The authors use taylor series to approximate function $f(q) = \left(1 + q \... 2answers 249 views ### Low-degree polynomial approximation of the piecewise-linear function$x \mapsto \max(x, 0)$on an interval$x \in [-R,R]$For$R > 0$, consider the piecewise-linear function$\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by$\sigma_R(x) := \max(x,0)$. Question Given$\epsilon> 0$, find a "low-degree" ... 0answers 22 views ### Convergence rate of cardinal series (Whittaker-Shannon interpolant) Given$f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series $$s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \... 2answers 133 views ### Multiple series calculation Let n be a positive integer. I would like to find a numerical evaluation of the convergent (!) series$$ S_{n,s}=\sum_{k\in \mathbb Z^{n}}\frac{1}{(1+\vert k\vert^{2})^{s/2}},\quad s> n, $$where ... 1answer 95 views ### Interpolation Inequality's Proof Let \Omega \subseteq R^{n} bounded domain. I need to prove that for u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega): \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\... 0answers 203 views ### Interlacing sequences by polynomials? Given t=2^\ell where \ell\in\mathbb N_{>0} and M\in\mathbb Z and two sets of integers \{a_1,\dots,a_t\} and \{b_1,\dots,b_t\} with 0<a_1\leq \dots\leq a_t<M and 0<b_1\leq \... 0answers 87 views ### Closed form expression for Tr\left[ (\mathbf{DW})^k \right] Given the N \times N diagonal matrices \mathbf{D} and \mathbf{W} as defined below \begin{split} \mathbf{DW} &= \left[ \begin{array}{cccc} \beta_{1} & 0 & \cdots & 0 \\ ... 2answers 419 views ### An expansion from Ramanujan related to birthday problem A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ... 0answers 124 views ### Closed subvariety that is unique in its small analytic neighborhood Let Y be some smooth projective variety over \mathbb C with \dim Y \geq 2. For a closed sub-variety X \hookrightarrow Y, consider the following property: There is some small open neighborhood ... 2answers 243 views ### Reference request: Extensions of Wiener's Tauberian Theorem Wiener's Tauberian Theorem says that linear combinations of translations of a function f are dense in L^1(\mathbb{R}) if and only if the zero set of the Fourier transform of f is empty. This is ... 0answers 24 views ### Estimating quality of projection I asked this question on math.se, but didn't receive an answer Suppose we are given a vector v and vectors \mu_i: v = \mu_1+\mu_2+...+\mu_m, where \mu_i \in R^n, all \mu_i are of unit ... 0answers 52 views ### Can we approximate this matrix field with an invertible matrix field? Let \mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\} be the closed unit disk, and set$$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{... 0answers 114 views ### Padé Approximants of Power Series with Natural Boundaries Consider a power series$\sum_{n=0}^{\infty}c_{n}z^{n}$for which$c_{n}\in\left\{ 0,1\right\}$for all$n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}... 1answer 97 views ### Proving that \lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i If I have the following integral equation$$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$An approximate solution of$\phi$... 0answers 51 views ### Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally? Let$\mathbb{D}^n$be the closed unit ball, and let$f:\mathbb{D}^n \to \mathbb{R}^n$be harmonic; More precisely, I assume that$f$is real-analytic and harmonic on the interior$(\mathbb{D}^n)^o\$ ...

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