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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8 views

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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An example for a seminorm on $\mathbb{R}^n$

Can any one come up with an example of a seminorm that is not a norm on $\mathbb{R}^n$ ? A seminorm on a real vector space $V$ is a function $N:V\rightarrow \mathbb{R}$ that satisfies that 1) $N(x)\...
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Instantaneous smoothing effect of sphere-valued maps

The instantaneous smoothing effect of the heat equation is the property that the solution to $$\begin{cases} \partial_t u= \Delta u, & t>0 \\ u(0, x)=f(x), & x\in \mathbb R^d,\end{cases}$$ ...
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Does every vector space have a Hamel basis ? And is every linear comination representation finite?

If $V$ is a linear space, then a set $B$ of linearly independent vectors in $V$ that span $V$ is called a Hamel basis for $V$. Does every infinite dimensional vector space have a Hamel basis ? My ...
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function parameterization with known sums!

I want to find a function, let's say $y= a x + b$ but I don't have sample $(x,y)$ pairs but what I have is samples of following form $((x_1, x_2, ..., x_n), \sum_{i=1}^n y_i)$ where n is also a known ...
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Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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How to expand $b\int_0^\infty \operatorname{sech}^2\big(b\cdot f(x)\big)\,dx$ for large $b$?

Suppose $f(x)$ has a single zero in $(0,\infty)$ at $x=c$ and has a Taylor expansion about this point with some nonzero radius of convergence $0<R\leq\infty$. For concreteness, I'm working with the ...
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26 views

Wavelet expansions

I'm looking into wavelets to approximate a known square-integrable function $$ f(x) = \sum_{j,k} a_{j,k} \times 2^{j/2}\psi(2^j x - k), \qquad a_{j,k}=2^{j/2}\int f(x) \psi(2^jx-k)dx$$ and I'm happy ...
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35 views

Polynomials can approximate the identity (proof)

From Terence Tao's Analysis II: I'm having problems with 14.8.2.(c), since it seems like the choice of $N$ depends on $c$ (and vice versa). The only proof I have relies on the convergence of the ...
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Approximation theorems and sketches of their proofs

I would like to collect (with support of users here) approximation theorems and sketches of their proofs. Each answer would give one approximation theorem and sketch of a proof. Any other proof of ...
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30 views

Is there a class of functions that have a cyclical or constant derivative?

I'm working on approximating functions in A.I., and I noticed that everyday functions seem to, at some point, have either a cyclical, or constant, derivative. For example, a straight line has a ...
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Approximation subject to derivative constraints, textbook reference?

I'm looking for a reference (preferably a textbook treatment) that can help me answer the following question. Suppose $\mathcal{F}$ is a space of functions on a subset of $\mathbb{R}^d$ with ...
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Approximating function of the integral $\dfrac{\sin(x)^k}{x^k}$

I needed a formula giving the value of the following integral; $$J(k)=\int_0^\pi dx\dfrac{\sin(x)^k}{x^k}$$ I didn't find any analytic expresion for it. I only found this interpolating function: $$J(k)...
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33 views

Stone-Weierstrass Theorem (Lattices)

I am struggling with a portion of a proof concerning the lattice version of the Stone-Weierstrass theorem. In particular, there is a subset $\mathcal{A}$ of the set of all real-valued continuous ...
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20 views

Are metric continuous measures setwise sequentially dense in finite Borel measures?

Let $(\mathcal{X},d)$ be a complete separable metric space. Say that a Borel measure $\sigma$ of $(\mathcal{X},d)$ is a metric continuous measure if for each $x\in\mathcal{X}$ the function $$(0,+\...
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approximation of 'any' bounded continuous function using bounded continuous functions with compact support

Suppose that $\phi$ is a bounded, continuous function with compact support $I$ (i.e. a bounded interval), then given any $\epsilon > 0$, there exists a simple function $\phi_{\epsilon}$ s.t. $$ \...
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Interpolation error of $C^k$ function derivatives

Consider the interval $I=[-1,1]$ and the set of points $A=\{r_1, ..., r_n\}\subset I$. Suppose that there exists a $k$-times continuously differentiable $f\in C^k(I, \mathbb{R})$ of which we know its ...
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Linear Fit: why do we minimize the variance and not the sum of all deviations? [duplicate]

my question is about the linear fit and the least squares method. Why do we decide to minimize the quantity $$ S = \sum_{i=1}^n r_i^2 $$ instead of this one: $$ r_i = y_i - f(x_i, \beta) $$ ? ...
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Choosing basis functions for function approximation

Suppose that $Y$ follows multivariate normal $\mathbb{N}(\mu, \Sigma)$. And we know that $f(Y)$ follows $\mathbb{U}(0,1)$, where $f: \mathbb{R}^n \to \mathbb{R}$ Can we find exact form of $f$ given ...
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60 views

How to prove that $(1+x)^r$ behaves like $1+rx$ for small x without calculus?

Of course the fact that, in the neighborhood of $x=0$, $$ (1+x)^r=1+rx+o(x) $$ can be easily proven for integer $r$. For positive values, it's a trivial consequence of the binomial formula. For ...
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51 views

How small must $x$ be for the error of $\cos(x) \approx 1$ to be below a certain threshold

I might be missing some background knowledge on this subject, but nevertheless I am interested. In some cases like this, the answers talk about finding the taylor series for $\cos(x)$ and then ...
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66 views

Approximating a clamp function using only addition, multiplication, division and subtraction

I'm trying to construct a function which satisfies the following: $$ \begin{align} f(x) = 0 & \qquad x \leq 0\\ f(x) = x & \qquad 0 \lt x \lt 1\\ f(x) = 1 & \qquad x \geq 1\\ \end{align} $...
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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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36 views

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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21 views

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} \...
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38 views

Convergence of piecewise constant/linear approximation to $L^1$ functions

I know that the class of piecewise constant/linear functions with finitely many pieces is dense in $L^1(\mathbb{R})$. I want to get a better understanding of convergence. Let us say that $g \in L^1(\...
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Can we approximate elements in $L^2$ via smooth maps while preserving a pointwise constraint on the derivatives?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open connected* subset $U \subseteq \mathbb{D}^n$ of full measure in $\mathbb{D}^n$, and a ...
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42 views

Can we approximate a smooth function by a continuous and nowhere smooth function uniformly?

From Stone-Weierstrass approximation theorem we know that we can approximate a continuous(no matter differentiable or not) by a polynomial function uniformly within a compact interval domain. ...
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Best approximation of a real number with two functions

There's two functions, called $F(x)$ and $G(x)$, where $F(x)>x,1<G(x) < x$ ,$F'(x)>0, G'(x)>0$ on $(2,\infty)$, and $F(x),G(x)\in(2,\infty)$. Given $x, y\in (2,\infty)$, and now I want ...
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Approximating a Discrete Function with a Polynomial

Consider a discrete function $f : \{x_1,\dots,x_n\} \to [a,b]$ that, for some $\omega>0$, satisfies $$\max_i |f_i-f_{i-1}| \le \omega. \tag1$$ Let $q$ be the polynomial of order $m<n$ that, ...
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$||\phi-\phi_\epsilon||_{L^1(\mu)}<\epsilon$ and $||\phi-\tilde{\phi}_\epsilon ||_{L^1(v\mu)}<\epsilon$, then $\tilde{\phi}_\epsilon= \phi_\epsilon$?

Let $\Omega\subset \mathbb{R}^n$ be an open and bounded set. Let $\mu:\mathcal{B}(\Omega)\to [0,+\infty)$ a bounded Radon measure and let $\varphi, \, v \in L^1(\Omega,\,\mu)$, $v\geq 0$. Then $v\mu$ ...
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Monotonicity of function involving ratios of modified Bessel function of first kind

I am a scientist who is trying to come up with some analytic solutions for a system that I only have approximate answers to and I have run into the problem of proving that the following function is ...
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“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
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modulus of continuity in matlab

How can we apply modulus of continuity and modulus of smoothness in matlab for any function?? These are two rules of modulus of continuity enter image description here
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Convergence speed of discrete approximation

Here I asked the question about approximating the function $g(x) := \mathbb{E}(f(x,Y))$, where $x \in R$ and $Y$ is a random variable. If you follow the link you will see that $g(x)$ can be ...
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187 views

Limits of the wave equation with piecewise constant propagation speed

Consider a wave equation $$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\partial^2 u}{\partial x^2} \tag{1}$$ In frequency domain this becomes an ODE: $$-\omega^2 u = c(x)^2 \frac{\partial^2 u}{\...
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44 views

Polynomial Extrapolation Error Resource

Given $n + 1$ samples of a $n+1$ times continuously differentiable function $f \in C^{k + 1}$: \begin{equation} (x_0, f(x_0)), (x_1, f(x_1)), \dots, (x_n, f(x_n)) \end{equation} Lagrange polynomial is ...
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58 views

Approximation of a continuous function with a particular sequence of smooth functions

Let the interval $[0,1]$ be divided into $n$ subintervals each of length $\frac{1}{n}$. Let $f\in C([0,1], \mathbb R)$ a continuous functions in $[0,1]$ and consider the set $\Omega_n=\big(f\in C^2([0,...
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20 views

Binomial Approximation with Small Exponent

Find an approximation for the expression $ (1 + x)^n $, where $ 0 < x < C $ and $ n $ is positive but small. $ C $ is arbitrarily large (<< 1000) and $ n $ is arbitrarily small ($ n <&...
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13 views

Jaggedness of the boundary of the $\epsilon$ neighborhood of a set

Let $X$ be a simply connected set in $\mathbb{C}$. Of course $X$ can be quite jagged and nasty. Let $\epsilon>0$ be given, and define $C(X;\epsilon)=\{z\in\mathbb{C}:\min(|z-w|:w\in X)=\epsilon\}$...