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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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        36 views

        Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$

        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 \...
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        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" ...
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        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 \...
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        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 ...
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        Interpolation Inequality's Proof

        Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$: \begin{equation} \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
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        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 \...
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        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 \\ ...
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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        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{...
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        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}...
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        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 $\phi$ ...
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        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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