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.
1,142 questions
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Saddle Point / Steepest Descent for Bessel Functions
I am trying to understand how to approximate integrals with Bessel functions. In particular I have something like:
$$I_{\ell} = \int_{0}^{\infty} j_{\ell}(pr) dr = \frac{\sqrt{\pi} \Gamma[(1+\ell)/2] ...
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When does the supremum over a subspace $F\subset C[0,1]$ commute with integration?
Let $F\subset C[0,1]$ be a linear subspace which contains the constant functions and separates points of $[0,1]$.
Assume that for every $f\in C[0,1]$ and every $x\in[0,1]$ we have
$$
f(x)=\sup\{g(x)\...
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Approximate unknown function with Fourier series
Say I have a monotonically increasing function $f : [0,1] \to {\Bbb R}$. I only know the values of $f$ for a finite set of points $x_1, \dots, x_n$. Can I use a Fourier series to approximate the ...
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Is the error of this minmax polynomial approximation strictly increasing?
A paper I am reading [1] makes the claim that for:
$$
D(\tau) := [-1 - \tau, -1 + \tau] \cup [1 - \tau, 1 + \tau]
$$
Let $p(x)$ be the polynomial of a fixed degree $d$ (depending on $\tau$) minimizing ...
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How can I approximate this function to be able to integrate it?
Suppose I have the following real function
$$f(x) = \frac{\left[ (2 + b)^2 - x \right]^{1/2} (b^2 - x)^{3/2} \left[ (2 + b)^2 + 2x \right] (x + 2a^2) (x - 4a^2)^{1/2}}{x^{3/2} (c^2 - x)^2}$$
defined ...
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Clarification about a particular step in a proof of the Kolmogorov-Arnold representation theorem
I am reading this proof of the Kolmogorov-Arnold representation theorem, first this sentences:
By plotting out the entire grid system, one can see that every point in $[0,1]^2$ is contained in $3$ to ...
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Proving universal approximation through ReLU
We were discussing universal approximation theorems for neural networks and showed that the triangular function
$$
h(x) =
\begin{cases}
x+1, & x \in [-1,0] \\
1-x, & x \in [0,1] \\
0, & \...
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Is there a smooth function approximating the minimum of a constant and a variable?
Let $K$ be a constant and $x$ be a variable. What is a smooth, monotonic function that is as close to $\min(K,x)$ as possible, but never exceed $\min(K,x)$?
Also f(x)>=0 for x>=0 and f(0)=0
...
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What is the best algorithm to approximate any given function using modified rational functions over a closed interval domain?
We are given a function $F$ of $x$, with $a_0,a_1,\ldots,a_m$ and $b_1,b_2,\ldots,b_n$ being parameters of the function:
$F(x) = \frac{a_0+a_1 x +\ldots+a_m x^m}{1+|b_1 x + \ldots+b_n x^n|}$
This is a ...
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Error estimation between an analytic function $f$ and its Bernstein polynomial
Let $0<r<1$ be a real number. Suppose $f$ is a continuous function on $[0,1]$, and let
$$B_n(f)=\sum_{k=0}^nf\left(\frac kn\right)\binom{n}{k}x^k(1-x)^{n-k}$$
denote the Bernstein polynomial of ...
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Wavelet Approximation of Piecewise Polynomial
Suppose $f$ is a piecewise degree $\leq N$ polynomial with at most $m$ pieces. Is it true that, using the Daubechies-$N$ wavelet system, we can construct an approximation $f_J$ such that
\begin{align}
...
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Does the Weighted Average of $f$ with $b_{n,K}$ Equal $f(n/K)$?
Let’s suppose that for an affine function
$$
f(u) = au + b,
$$
the Bernstein operator reproduces $f$ exactly; that is, the Bernstein operator $B_K(f)$ satisfies
$$
B_K(f)(x) = \sum_{n=0}^{K} f\left(\...
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Bivariate operators as a 'product' of two univariate operators (Bernstein example)
The Bernstein operator is defined for continous functions $f$ on $[0,1]$ as follows:
$$ B_n (f;x)=\sum_{k=0}^n \begin{pmatrix} n\\ k\end{pmatrix} x^k (1-x)^{n-k} f \left( \frac{k}{n}\right),\quad n\in\...
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Inverse Laplace Transform using Saddle point
I was looking at some slides and got stuck on finding the inverse Laplace transform of the function $ f(s)= \left(\frac{erf(\sqrt{s t})}{\sqrt{s}} \right)^{N}$, where $erf$ is the error function and ...
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Stability Of Approximate Isometries Of $\mathbb{S}^{1}$
Suppose we have a set $X\subset\mathbb{S}^{1}$ such that $d_{H}(X,\mathbb{S}^{1}) < \delta$ and a map $f:X\to\mathbb{S}^{1}$ such that $|d_{\mathbb{S}^{1}}(f(z),f(w)) - d_{\mathbb{S}^{1}}(z,w)| <...
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Error-bounds for linear approximations to arbitrarily sampled functions in R^n
I have an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ evaluated at a set of $k$ points in $\mathbb{R}^n$, $X = \{ x_1,x_2,\cdots,x_k \}$. If I want to find a linear approximation of $f$ at a new ...
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Using Perturbation Theory to estimate solution to 1st order 2nx2n ODE
We have the following expresion (such that $A$ is a $2n\times 2n$ matrix):
$$
\begin{aligned}
{ x}(t_c) &= e^{-t_c\tilde{ A}}{ x}_0+\int_0^{t_c} dt' e^{-(t-t')\tilde{ A}} { c}\\
&=\tilde{ V}e^{...
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When does a set induce a norm? [duplicate]
I have been investigating under which conditions a subset $T$ of a Banach space $X$ induces a gauge norm via
$$ ||f|| := \inf \{ \lambda > 0 \, : \, f \in \lambda T \}. $$
So far I have assumed:
$...
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Theorem concerning Density and Duality in the Banach spaces $L^p(0,\infty)$: yielding $\Rightarrow f^* = \theta^*$
friends :-),
I am reading an old research paper from 1955, and I having problems understanding one part of it: Let
$X$ being one of the Banach spaces of $L^p(0, \infty)$ with $p \geq 1$, and
$S$ ...
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Finding upper bound of degree of approximating polynomial in Stone-Weierstrass
From Stone-Weierstrass Theorem, we know that any subalgebra $C([0,1]^n, \mathbb{R})$ that has constant functions and separates points of $[0,1]^n$ is dense in $C([0,1]^n, \mathbb{R})$ under the ...
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Maximum of real polynomials in terms of its coefficients
Given a closed bounded intervals $[a,b]$, any polynomial
$$p(x)=a_0+a_1 x+\dots+a_n x^n,$$ with $a_0,a_1,\dots,a_n\in \mathbb{R}$, has an absolute maximum on $[a,b]$. That is there exist $c\in [a,b]$ ...
- 3,760
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The equioscillation theorem for Haar manifolds
I am searching for a non-linear generalization of Chebyshev's equioscillation theorem for Haar spaces $V\subseteq\mathcal{C}([a,b])$, where $\mathcal{C}([a,b])$ denotes continuous functions from $[a,b]...
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Asymptotic approximation of Stirling numbers of the second kind
It is well known the asymptotic approximation of Stirling numbers of the second kind to be
$$S(n,k) \sim \frac{k^n}{k!} \ \ \text{as} \ \ \ \ n\rightarrow \infty $$
but Is it possible to comment on ...
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the limit of log-sum-exp under continous condition
$f(x)\in C(\Omega)$ and $\Omega$ is a closed bounded set, now I want to proof $\lim_{\varepsilon\to 0}\varepsilon\log{\left(\int_{\Omega}{e^{\frac{f(y)}{\varepsilon}}dy}\right)}=\max f(y)$. However, I ...
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Higher-order Stirling approximation
I'm trying to obtain higher-order terms of the Stirling series by iteratively substituting $\Gamma(s+1)=s\Gamma(s)$. Here is the procedure:
\begin{eqnarray}
\Gamma(s+1)&=&e^{s\ln s-s}\sqrt{2\...
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