Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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

A description of approximation theory can also be found in this question: Approximation theory.

Some important topics and results in approximation theory:

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Why is the polynomial best approximation to an even function itself even?

I have seen this stated and it seems intuitively obvious but I cannot prove it. I have a feeling it may be because a non-even best approximant would not satisfy the equioscillation property of the best approximation (that if |f-p|=d, where p is the…
Monty123
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Selecting degrees for a given number of terms in Pade approximants

Let suppose that I have a function $f[x]$ I want to approximate using a Pade expansion and that I decide what would be the maximum number of terms to be used. Is there any methodology to find what have to be the degrees of numerator and denominator…
Claude Leibovici
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Uniform approximation on $[0,\infty)$

Let $f:[0,\infty) \to \mathbb R$ be a continuous function such that $f(x) \to 0$ as $x \to \infty$. If $\epsilon >0$ then there exists a polynomial $p$ such that $|f(x)-e^{-x}p(x)|<\epsilon$ for all $x \in [0,\infty)$. This is an exercise by Serge…
Kavi Rama Murthy
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Approximating a discontinuous by a smooth function

The discontinuous function I want to approximate is defined as $x \mapsto g(x) : \mathbb{R}_+\mapsto \mathbb{R}_-$ with $g(x) := -x\mathbb{1}_{x < b}$ where $b > 0$ is some given real number. I want to approximate this function on the whole…
Calculon
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Need some good problems on Weierstrass Approximation Theorem

I know the Weierstrass Approximation Theorem, and I know its proof. I however till now have not really found any good application of the theorem except in one problem where it is given that if $f$ is continuous on $[0,1]$ and $\int_0^1x^nf(x)dx=0$…
Landon Carter
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Prove that $\left\{x^{\lambda_i}\right\}_{i=0}^n$ is a Chebyshev system on $(0,\infty)$

The definition of Chebyshev system is as follows. A linearly independent system of $n$ basis functions $\varphi_1,\ldots,\varphi_n$ on $[a,b]$ is a Chebyshev system on $[a,b]$ if every non-trivial linear combination $\displaystyle\sum_{k=1}^n…
synack
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Error Analysis of Gaussian Interpolation

Is there any theory available on how well I can approximate a function using a basis of Gaussians $$ \phi_k(x) = \exp(-\tfrac{x - x_k}{\lambda_k}) ? $$ I've heard a bit about Gaussian process regression/Kriging, which seems related, but the…
gTcV
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Error term when Lagrange interpolating continuous non-differentiable functions

Suppose I know the values of a continuous function on $[a,b]$ in some finite number of points $x_0,x_1 \ldots x_n$. I can form the Lagrange interpolating polynomial, $p$. I am curious if there is any interesting estimate of the expression…
Johan
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approximating with a class of indicator functions: any theorems?

Let $G$ be a class of $\it{indicator}$ functions where $g\in G$ implies that $g : X \rightarrow \{0,1\}$, where the domain $X$ is a compact subset of $\mathbf{R}^n$. For example: $$ g(x) = 0 \text{ if } w_1x_1+\cdots+w_dx_d < \eta $$ $$ g(x) = 1…
zorank
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Is there Stone-Weierstrass theorem for piecewise continuous functions

There is a version of the theorem for continuous mappings $f$ from a compact metric space $X$ to real numbers $R$. Namely, if an algebra $A$ of continuous functions separates points and is non-vanishing, then any $f$ can be approximated arbitrarily…
zorank
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How to interpolate a function with Gaussian functions?

A function $f:[a,b]\to \mathbb{R}$ is given (we can assume it is continuous or differentiable) and we want to interpolate it with Gaussian functions $$\phi_i(x) = e^{-\alpha(x-x_i)^2}\text{,}$$ where $x_i$'s, $x_i\in [a, b]$, and $\alpha > 0$ are…
Antoine
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How to show that $e^{a_0t},\dots,e^{a_nt}$ form a Chebyshev system

How to show that $e^{a_0t},\dots,e^{a_nt}$ form a Chebyshev system on any interval? I have a hint saying that I should use Rolle's theorem, but I don't see how Rolle's theorem can give an upper bound on the number of zeroes. A system of real…
Nick Sun
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Padé approximant for a function $ f(x) $ always valid?

Is there any function $f(x)$ which can not be approximated by a Padé rational approximant? $$f(x) \approx \frac{a_0+a_1 x+ \ldots +a_nx^n}{b_0+b_1 x+ \ldots +b_m x^m} $$ What happens with $f(x)= \tan(x)$ or $f(x)=\log^{a}(x)$
Jose Garcia
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How can one measure how well is a function approximated over an interval?

I am currently studying Taylor polynomials. I was wondering if the fact that a (continuous) function approximates another over an interval can be quantified? At a point it is easy, you just compute $|f(a)-g(a)|$ and this tells you how the functions…
Adam
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Hash Collision Probability Approximation

If an item is chosen at random $k$ times from a set of $n$ items, the probability the chosen items are all different is exactly $\dfrac{n^\underline{k}}{n^k}=\dfrac{n!}{(n-k)!n^k}$. For large $n$, the expression is said to be approximately equal to…
Roman Odaisky
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