# 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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### Finding positive roots of quadratic quasi-polynomials

Are there any good exact or approximate closed-form expressions for the strictly positive roots of $x-\ bx^{v}\ +\ c$ that work for any $0 < v < 1$ (when the roots exist)? Using the iterative ...
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### Chebyshev coefficients of Chebyshev interpolants

I am wondering whether there is a known closed form for the $k$th Chebyshev coefficient of a $n$th Chebyshev interpolant, that is, $$\int_{-1}^1T_k(x)L_{n,i}(x)\frac{dx}{\sqrt{1-x^2}}$$ where $T_k$ is ...
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### Analytically tractable solution to system to ODEs?

I have the following system of ODEs \begin{aligned} \dot x (t) & = x(t) \frac{h(t)}{h_0} - \frac{x^2(t)}{x_0} \\ \dot h(t) & = -a h(t) \end{aligned} where $x(0)=x_0$ and $h(0)=h_0$. Using ...
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### generating chebyshev coefficients to replicate Clenshaws table used in zx series computers

I've completely failed to replicate Clenshaw's table for generating a sine approximation function. The results have been reused all over the web, but I cannot find a step by step process to generate ...
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### Exchange Theorem for Haar Condition

I am having some trouble in understanding the meaning of the Exchange Theorem in page 45 of Cheney's "Introduction to Approximation Theory". The Haar condition for a subset $A$ of vectors in ...
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### How can an operator that acts on scalar functions be approximated as a matrix that acts on vectors?

I'm not even sure if this is a valid concern, but I think it stems from my lack of knowledge in operator theory. The point is that I am not able to understand how an operator can be defined to act on ...
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### Interpolation problem (recursive interpolation)

Let $x_0<x_1<...<x_K$ be points on the real line. Let $P$ be the polynomial of degree $K$ such that $P(x_i)=(-1)^i$ for all $i$. Then there exists points $y_0<y_1<...<y_K$ such that ...
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### A version of Ascoli-Arzelà using modulus of continuity and nth entropy numbers

The classical Ascoli-Arzelà theorem could be stated as follows: Let $K$ be a compact metric space and let $\mathcal{H}$ be a bounded subset of $C(K)$ - the space of continuous functions over $K$ with ...
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### Spectrally-Accurate Quadrature of Singular Integrand

I have a set of PDEs governing some function $f(r)$ which I desire to solve via a psuedospectral method (we can consider $f$ to be smooth). It is defined on the interval $r\in[0,\infty)$ with symmetry ...
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### Coefficients of the polynomial approximation of $|x|$

I am looking for a polynomial $p_n$ of degree at most $n$ that will approximate $|x|$ in the interval $[-1,1]$ with $\||x|-p_n\|_\infty=O(\frac{1}{n})$ and will have coefficients whose absolute value ...
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### Mollifiers for a function on $[0,T]\times R$

Do you have any precise and comprehensive reference for how to build a sequence $\phi^\epsilon(t,x)$ of $C^\infty([0,T]\times R)$ functions that converge uniformly on $(0,T)\times R$ to a given ...
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### Bounded approximation property and affine subspaces of codimension $1$

Let $E$ be a Banach space with the bounded approximation property (BAP; definition below) and let $M\subseteq E$ be an affine subset of co-dimension $1$, and let $C\subseteq M$ be a closed and convex ...
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### Chebyshev interpolation of 1-Lipshitz functions

Suppose $f:[-1,1]\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq |x-y|$ for all $x,y\in[-1,1]$. Then Jackson's theorem asserts that $$\min_{\deg p \leq k } \| f - p \|_\infty < C k^{-1}$$ for some ...
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