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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Limit of integral product

I have an expression of the sort $$\displaystyle \lim_{N \to \infty}\prod_{i=1}^{N}\int_{0}^{1}\mathrm{d}\sigma(p_i)\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^{\infty}\mathrm{d}x \exp{\left [ -\frac{N\...
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There exists such result for any convergence of sequence definiton

Assume that $L(x_n) $ stands for the limit of the sequence $(x_n) $ in some sense, not necessarily the usual limit. For example, could be the almost convergence limit or the statistical limit. I ...
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question on a proof Dirichlet's approximation theorem

We will fix some positive integer $N$ and consider the fractional parts of the numbers $0,α,2α,...,Nα$. Stick them into this collection of $N$ intervals $$[0;\frac{1}{N}),[\frac{1}{N},\frac{2}{N}),...,...
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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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approximation error of $p$-th moment

Suppose that $\delta>0$ is a small quantity and $p\in (1,2)$ is a constant. Suppose that $x_1,\dots,x_n,y_1,\dots,y_m\in (0,\delta)$ and $a_1,\dots,a_n,b_1,\dots,b_m\in (0,1)$ satisfy that $$ \...
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Finding the second derivative using DFT

Suppose you have an even real-valued function f(x), which is periodic with T=2L. Introducing a grid $$x[n]=-L+ndx,\quad f(x[n])\equiv f_n,$$ $$dx=\frac{2L}{N},\quad n=0,\ldots N-1,$$ its DFT is ...
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What is the intuition behind relative error in asymptotics?

I've seen a couple books say that if we approximate a function $f(x)$ with a function $g(x)$ as $x\rightarrow x_0$, then the relative error of the approximation is of order $e(x)$ if $e(x) = o(1)$ and ...
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Bounding sum of independent samples from random distribution

Context Given a discrete variable $X$, and $Y = \sum_{k=0}^N X$, the sum of $N$ independent samples of $X$, I want to an upper bound for $y_b$ so that $p(Y < y_b) \ge 1-l$. So I start with a ...
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Extending Stone–Weierstrass for trigonometric polynomials to more than 1 dimension

I'm trying to prove the extension of Stone–Weierstrass for trigonometric polynomials to more than 1 dimension, but I'm unable to complete it. Here's the exact statement I am trying to prove: Let $f:\...
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Stone–Čech compactification and functions that separate points

Let $\mathfrak X$ be a completely regular space. We can then consider its Stone–Čech compactification $\beta \mathfrak X$. Every continuous bounded function $f:\mathfrak X \to \mathbb C$ admits a ...
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Integral of a continuous 1-form on a sequence of $C^1$ curves

Suppose I am given some continuous $1$- form $\omega$ defined on $\mathbb{R}^2$, and a fixed curve $C$ that is the boundary of a bounded open set $\Omega$. Let $\{C_j\}^\infty_{j=1}$ be some sequence ...
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Error in proof of oscillation theorem in Süli and Mayers' Numerical Analysis

I am reading Süli and Mayers' book "An Introduction to Numerical Analysis" and I think I found an error (page 234, second paragraph for reference). I couldn't find it back in the errata, ...
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Find the polynomial of fixed degree that minimizes the maximal error on a set of equidistant points

I've got a set of $m \in \mathbb{N}$ 2D points which I want to represent as a polynomial of degree $n$: $\left \{ \left( x_i, y_i \right) \in \mathbb{R}^2 : i \in \mathbb{Z}, 0 \leq i \lt m \right \}$,...
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Approximation in Banach spaces using tensor products

I am looking into approximating a trivariate function $u \colon J \times D \to \mathbb{R}$, where $J$ and $D$ are compact domains in $\mathbb{R}$ and $\mathbb{R}^2$, respectively. Think of $u(t,x)$, ...
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Stone-Weierstrass theorem as special case of Bishop theorem?

Referring to the Stone-Weierstrass theorem stated in this question. From Rudin's functional analysis we have the following theorem: 5.7. Bishop Theorem Let $A$ be a closed subalgebra of $C(S)$. ...
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Combining Stone Weirstrass theorem with approximations of $L^p$ spaces, is it possible?

Consider the following version of the Stone-Weierstrass Approximation theorem (from Royden - Real Analysis) The Stone-Weierstrass Approximation Theorem Let $X$ be a compact Hausdorff space. Suppose $\...
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Mollification of a product of two functions in $\mathbb{R}^n$

Consider $f$ and $g$ to be two functions such that $f$ is supported on the unit ball and $g$ is a function that vanishes in the unit ball (for points when $|x|<1$), and is non-zero for $|x|\geq 1$ ...
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Is there an example of universal approximators whose value at the origin is fixed as zero?

I'd like to know whether there is an example of universal approximators whose value at the origin is fixed as zero. Motivation Let's think about $C_{0} := \{f:X \to \mathbb{R}| f \text{is continuous}, ...
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How to show a function with nonzero integral is discriminatory

This is a prove from the artile "G. Cybenko, Approximation by superposition of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)", which uses the Wiener's Tauberian Theorem ...
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Approximation of continuous functions by neural network [closed]

For $f$ as (c) or (d), the answer is yes or no? I wonder that the nozero constant function in $[0,1]$ can not be approximated by such functions.
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Proving that $1, x, x^2,..., f(x)$ is Haar/Chebyshev system

Let's consider function: $$f : [a, b] \ni x \mapsto f(x) \in \mathbb R$$ and $f \in C^n$ such that $\forall_{x \in (a, b)}:f'(x) \neq 0 $. I want to prove that: $$\textrm{lin}\{1,x, x^2,...,x^{n - 1}, ...
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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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Uniform approximation of a continuous function on $[0,1]$ by polynomials with a control on the uniform norm of the polynomials

Let $f \in C([0,1])$. By the Weierstrass approximation theorem, it is possible to uniformly approximate $f$ on $[0,1]$ by a sequence of polynomials $P_i$. (i) Can we also require that $\|P_i\|_\infty \...
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Transfer a nonlieanear function to a linear function

I'm using Java to solve a maximization problem in Cplex. My objective function is quite complex. In a nutshell, there are two parts, A and B. Both of them contain variables. My goal is to maximize A/B,...
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Best-fitting function subspaces in $L^2[-1,1]$

I recently came cross a question related to best-fitting function subspaces as follow. Let $L^2[-1,1]$ be the Hilbert space of real valued square integrable functions on $[-1,1]$ equipped with the ...
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Derivatives of a 2D B-spline with respect to the control points

I'm dealing with an optimal control problem and I want to solve it with the B-splines. In order to compute the gradient of my objective function, I have to derive the B-spline with respect to the ...
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Density of Lipschitz functions on $C^0_b(\mathbb{R},\mathbb{R})$

My problem: Suppose $$V=C^0_b(\mathbb{R},\mathbb{R})= \{ f :\mathbb{R} \to \mathbb{R} | f \mbox{ is continuous and bounded}\}$$ with the usual norm $\|.\|_{\infty}$. I am asking if $$W= \{ f \in V | f ...
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Universal approximation theorem for injections

Let $\sigma$ be a non-polynomial continuous activation function. The classical universal approximation theorem says that functions of the form $f(x) = C\sigma(Ax + b)$ can approximate any continuous ...
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How to represent simple multiplications as additions?

According to Kolmogorov–Arnold representation theorem any multivariate function can be represented as sum of univariate functions. This should hold for multiplications too, so $xy = F(x) + G(y)$ $xyz =...
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Stone–Weierstrass approximation theorem for multivariable case.

Stone–Weierstrass approximation theorem says "Let $A$ be a (complex) unital sub-algebra of $C(K, \mathbb{C})$, such that if $ f\in A$, then $\overline{f} \in A$, and $A$ separates points of $K$. ...
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How to find the best piecewiselinear approximation with fewer intervals

Given a continuous function $f(x)$ on the interval $x\in[a,b]$. How could we find the best piecewise linear approximation of $f(x)$ such that $$J=\|f(x)-\sum_{i=1}^K (m_ix + n_i)\|_{L^2([a,b]} + \...
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Smooth function $C^{\infty}$ that in the limit becomes Coulomb potential

I need to have a scalar function $\psi(\mathbf{r}, \sigma)$ of the coordinate $\mathbf{r}$ in euclidean 3-D space, which depends on a scalar adjustment parameter $\sigma$, such that in the limit $\...
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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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a lower bound for Lebesgue’s constants

How to find a lower bound for Lebesgue’s constants I know that $L_n>\frac{4}{\pi^2}\ log\ n$ Proof. $L_n=\frac{1}{\pi}\int_{0}^{\pi}\frac{\left|sin\left(n+\frac{1}{2}\right)\varphi\right|}{sin\left(...
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Approximate $f(x)= x^2$ with the best approximation in $L_1$ under the given norm.

The given norm is: $||f||= \sqrt{A_0 |f(x_0)|^2 + A_1 |f(x_1)|^2 + A_2 |f(x_2)|^2}$ And from a previous question I have found the following: $A_0 = \frac {-\sqrt{\pi/3}}{2}$ $A_1 = \frac {\sqrt{\pi/3}}...
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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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Non-infinitesimal linear expansion of a differential operator $X$

Consider a differential equation of the form $$ \frac{df}{dt} = X(f) $$ where $X$ is a linear operator. Then, if I am not mistaken, the infinitesimal change of $f$ from time $t$ to $dt$ is given as: $$...
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Error in first derivative of cubic spline interpolant

Let $f: [a,b] \rightarrow \mathbb{R}$ be a $C^{\infty}$ function, and let $a = x_0 < x_1 < \cdots < x_n = b$ be a partition of the interval $[a,b]$. Let $s(x)$ be a piecewise polynomial ...
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Representation of a polynomial in the Chebyshev form

Show that every $ p\in P_n $ has a unique representation in the form $ p\left(x\right)=A_0+A_1T_1\left(x\right)+...+A_nT_n\left(x\right) $ where $ T_i $ are Chebyshev polynomials. [![enter image ...
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Uniform best approximations -- Behavior outside of approximation interval

Consider $f\colon [a, b]\to\mathbb{R}$, with $a,b\in\mathbb{R}\cup \{\pm\infty\}$ and $a<b$, and let $f_n:=\arg\min_{\text{Polynomial p of degree n}} \sup_{x\in (a,b)} |f(x)-p(x)|$. Is there any ...
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Determination of three fitting parameters to approximate $\cos\theta \left( 1-\cos\theta \right) \Theta \left( \frac{\pi}{2} - \theta \right)$

I would like to determine the coefficients $\alpha$, $\beta$, and $\lambda \in \mathbb{R}$ such that $$ f(\theta) = \sin\theta \left( 1+\lambda\cos\theta \right) \left( 1 + \alpha \cos\theta + \beta\...
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