# 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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### Multi-dimensional Chebyshev system?

I’m wondering what is the definition of multi-dimensional (not multi-variate) Chebyshev system? Specifically, a Chebyshev system is a set of linearly independent functions $\{f_0,\ldots, f_n\}$ ...
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### Orthogonal Polynomials approximation and $L^2(\mathbb(R))$

I have another basic question, this time about approximation of functions. Given $(p)_{i\in \mathbb{N}}\in \mathcal{A}=\{$all the families of orthogonal polynomials $\}$ and $f\in L^2(\mathbb{R})$, ...
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### How to apply the Stone-Weierstrass-Theorem to nonlinear systems? How to integrate Volterra kernels for TI systems?

I'm currently studying Volterra series as an input-ouput-representation of nonlinear systems. For this, I found a variety of interesting papers. For instance, Lesiak & Krener (1978), Brockett (...
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### Multi-dimensional uniform approximation results

Recently I've been investigating results from approximation theory, especially the uniform approximation by polynomials. I find most of the interesting results are for one-dimensional, uni-variate ...
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### Taylor series to approximate derivative with difference quotient function

Consider the following example on Taylor series. Let's consider the difference quotient function of center $x_0$: $$f'_h(x_0)=\frac{f(x_0+h)-f(x_0)}{h}$$ For $h$ sufficiently small, the difference ...
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### Checking the density of a set of polynomials on $C([-1, 1])$

Suppose $P = \{\sum_{j = 0}^{n_i} a_{ij} x^j | a_{ij} \in \mathbb{R}\}$ is a set of polynomials defined on $[-1, 1]$. Is there any way to test whether $\text{span}\: (P)$ is dense in $C([-1, 1])$ with ...
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### Necessary condition for the linear span of a set of functions to be dense in $C([-1,1])$

Let $P=\{x^{k_i}\}_i$ be a set of monomials defined on $[-1, 1]$. By Stone-Weierstrass theorem, if $k_i= i$, then $\text{span}\: (P)$ is dense in $C[-1, 1]$ under sup-norm topology. However the ...
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### On the invertibility of basis function matrix for interpolating functions

Problem: Let $f: \mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R}^2 \to \mathbb{R}$ satisfies $$f(y(x)) = g(x) \ \forall x \in \mathbb{R}^2$$ where $y : \mathbb{R}^2 \to \mathbb{R}^2$ is a map that ...
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### even and odd cubic polynomials

Book says approximating even function $P_3$ must be of form $ax^2+b\$so they neglected $x,x^3\$. This function approximate even function $|x|^3$ very nicely in the book after some tweaking. I am ...
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### Approximation formula for a simple counting problem

Let $a,b,c$ be positive integers with $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1$ and let $n =$ the number of positive integers $\leq N$ not divided by $a,b,c$. Set, $$m = N\cdot (1-1/a)(1-1/b)(1-1/c).$$ I ...
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### Is an infinite sequence of orthogonal functions in $H$ closed in $H$?

Consider some countably infinite sequence of elements $f_n$, each belonging to an infinite dimensional Hilbert space $H$, that are all orthogonal to every other member of the sequence. Is this set ...
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### Generalizing the Runge approximation theorem for Riemann Surfaces: approximating by nonvanishing functions

The Runge approximation theorem for open Riemann Surfaces says the following. If $X$ is an open Riemann surface and $Y \subset X$ is a Runge subset, i.e., $X \setminus Y$ has no relatively compact ...
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### Problem on othogonal polynomials with Lebesgue mesure

This is an over my level assignmet i got: Let $\mu$ denote the Lebesgue (of arc-length) measure on the unit circle $\mathbb{T}:=\{|z|=1\}$. Prove that for the measere $dv(z)=|1-z|^{2}d\mu$ the ...
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### Approximating the gradient of a Finite Element solution on nodes

I have been working on a Finite Element implementation that approximates the solution to the following PDE in 3D using tetrahedral elements and piecewise linear basis functions. \begin{equation} \...