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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 optimal knots for function approximations

I would like to approximate a continuous (complex) function $f(x)$ in the interval $[a,b]$ $ (x\in\mathbb{R})$ by local polynomial functions of order $3$ (cubic Hermite spline or cubic C2 spline). Is ...
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Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family? Details: Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\...
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Reference request: Bounded function can be approximated by continuous functions in $L_1$ with bounded $L_\infty$-norm

I think that it is well-known that a real valued function $f\in L_\infty[a,b]$ can be approximated by continuous functions $f_n$ with respect to the $L_1$-norm, i.e. $||f_n-f||_{L^1}\to0$, where the $...
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References for a proof of a Jackson's inequality?

Let $g:[0,2\pi]\to \mathbb{C}$ which is $\mathcal{C}^k([0,2\pi],\mathbb{R})$ and periodic. If $\mid f^{(k)}(x) \mid\le 1$ then for each $n\in \mathbb{N}^*$, there exists a trigonometric polynomial $T_{...
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Approximating a bounded measurable function from below by a sequence of smooth functions

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a bounded measurable function. Is it possible to find a sequence of functions $\{f_n \}_n: \mathbb{R} \to \mathbb{R}$ in $C^{\infty}_c( \mathbb{R})$ ...
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Metric projection from space of bounded functions to finite-dimensional linear space

Apologies if the answer is obvious or should be easy to find, but so far I've had no luck. Let $X$ be a subspace of $\mathbb{R^k}$ for a finite $k$ and let $\mathcal{B}(X)$ be the Banach space of ...
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Does the Sobolev space $W^{1,p}(\Omega), p>2$ has a monotone basis?

A Shauder basis in a Banach space is monotone if $\|P_{n}f\|\leq\|f\|,$ where $P_{n}$ is the projection to the sum of the first n elements of the basis. For Hilbert spaces this is always the case if ...
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Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
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approximating a decreasing function with hyperbolic functions

Let $y = f(x)$, where $x, y \in \mathbb{R}_1$ and $f \in \mathcal{C}^1$ with $f'(x) \le 0$. Partition the range using $t$ points ${y_1, \ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - ...
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Mitigating the Runge Phenomenon with Constrained Norm Minimization

I am interested in polynomial interpolation of a set of points in $\{(x_1, y_1), \ldots, (x_n, y_n)\} \subset \mathbb{R}^2$. On the wikipedia page for Runge's phenomenon, the Constrained ...
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Do two close functions share some local minima?

Let $f,g:\mathbb{R}^n \to \mathbb{R}$ be two differentiable functions. Assume that $\| f -g \|_{\infty} \leq \epsilon$. On what conditions on $f$, for every local minima $x$ of $g$, there is a local ...
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Error analysis of approximating Fourier transforms

Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ \hat{f}(k) = \int_{-\infty}^{\infty} dx~ f(x)~ e^{i k x} .$$ Suppose I want to approximate this transform by a ...
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2answers
106 views

Approximation of conditional expectation of unknown function

I am given a multidimensional markovian stochastic process $X_1,X_2,...X_n$ with continuous state space and unknown to me function $V$. I want to approximate expectation $E(V(X_k)|X_{k-1} = x)$ ...
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1answer
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Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...
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Approximation in probability

To prove a CLT, I need first to prove that the following approximation holds $$ \sqrt{n}\sum_{j=1}^n\left(\left|\int_{(j-1)/n}^{j/n}\nu_s\,dW_s\right|\,\left|\int_{j/n}^{(j+1)/n}\nu_s\,dW_s\right|-\...
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1answer
40 views

A strengthening of Stone–Weierstrass which applies to arbitrary closed intervals

Note in this question, we concern ourselves only with the space $C([a,b])$ of continuous real-valued functions on compact intervals $[a,b]$. The Müntz–Szász theorem is a well-known result related to ...
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Designing a polynomial whose coefficients are somehow small

Consider vectors $v\in\mathbb{R}^n$ where: The polynomial $p_v(x)$, taking its coefficients from $v$, has $p_v(x)\geq 1$ along $[0,a]$ and is nonnegative. Take $B_\mu$ a box: $B_\mu:=\{b: b_1=0, \mu^...
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Taylor Series Expansion on error propagation.

I am reading through Stoer and Bulirsch's Introduction to Numerical Analysis. In their section on error propagation they are describing a derivation of the Jacobian as it related to a problems ...
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Clarification of assumptions made in deriving error of implicit midpoint rule

In my derivation for $y^\prime = f(t,y)$, I begin by writing the method as an expression which should simplify to the error, by substitution of the exact solution \begin{equation} y(t_{n+1}) - y(t_n) -...
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Computing the solution of an ODE using power series

I have a system of ODEs defined on $\mathbb{R}\times\mathbb{S}$, $$\begin{aligned}\dot{x}={ }&y\\\dot{y}={ }&-0.2y+\frac{300\cos(2)\sin(x)}{1.8(1.3+\cos(x-2))-2\sin(x)\sin(2)},\end{aligned}$$ ...
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Polynomial approximation of a function in a chosen interval

I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section. I would like to approximate nonlinear functions typically used in ...
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Chebyshev approximation for large interval

In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose. It seems that ...
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1answer
32 views

Orthonormal basis for L2 (0,1) by using Laplacian's eigenfunctions.

A standard orthonormal basis for L2 (0,1) is given by the Fourier expansion, as described here, for example (Orthonormal Basis of $L^2$). On the other hand, it seems a standard result that the ...
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Approximate a solution for $\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx 0$

Is it possible to approximate (or even find) a solution for the following equation: $$\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx0,$$ where $x\ge 0$ and integer, and the ...
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Methods of approximating sine waves (given other sine waves)

I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by: $$f(n) = 2^{\frac{n-49}{12}}\times440$$ $$\sin(2\pi f(n)x), \sin(2\pi f(n+...
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From wavelets to curvelets

I've read that the curvelet is a generalisation of wavelet. I am now looking for references such as books, lecture note or research papers introducing the mathematical theoretic aspect of curvelet to ...
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3answers
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Approximation of a sum with an integral…

Let $G$ a continuous function in $C([0,1], \mathbb R)$. I think that $$\frac{1}{N}\sum_{x \;\text{odd}\in \{1,\ldots, N\}}G\Big (\frac{x}{N}\Big )\xrightarrow{N\to +\infty}\frac{1}{2}\int_0^1G(r)dr,$$ ...
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Is there any example demonstrating nonlinearity of best polynomial approximation operator?

For any $f\in C[0,1]$, it is well known that there exists an unique $p^{*}\in P_n[0,1]$ such that $||f-p^{*}||_{\infty}=\inf\limits_{p\in P_n[0,1]}||f-p||_{\infty}$. In this fashion, one can define an ...
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How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
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Single hidden layer with finite #neurons limitations

I need to prove that MNN with one hidden layer, and finite number of neurons does not have compact support, i.e. the integral of the normal of f (network function) upon all R^d equal to infinity. It ...
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Approximation of Cantor function by piecewise constant function in $L^1$

Let $c(x)$ be Cantor function. How can we prove that constant $\frac{1}{2}$ gives the best approximation in $L^1$ metric? Let $ h(x)= \begin{cases} \frac14 \quad\text{for}\quad x\in[0,\frac13]\\ \...
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Problem on Riemann-Stieltjes Integration and function approximation

Let $\alpha$ be continuous and increasing function on [a,b]. Given $f$ $\in$ ${R_\alpha }$[a,b] and $\epsilon$> 0; Then, Prove that there exist (i) a step function $h$ on [a,b] with ${||h||_\infty}$$...
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an approximation to the generalized hypergeometric function

Relating to the article An approximation to the generalized hypergeometric function, I would like to calculated example of the Poisson distribution $Po(10)$. If you have the possibility, please see ...
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1answer
56 views

Extended Global Approximation Theorem

In Evans, $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
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Family of graphs that have approximation ratio = 2

My question today is about the approximation algorithms. Well, for Approx-Vertex-Cover problems , we know we can get ratio of 2 just by picking an edge and taking 2 endpoints of the same and ...
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how to estimate maximum of Lebesgue function of arbitrary nodes?

Denote $S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0}<x_{n1}<\cdots<x_{nn}\leq b\right\}$ with $-\infty<a<b<+\infty$ and $n\geq 1$. For any $x_n\in S_n$, define the ...
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38 views

Interpolation error with Legendre/Chebyshev polynomials

I remember seeing somewhere that the Lagrange interpolation over Chebyshev nodes has least possible deviation in the sense of $\|\cdot\|_\infty$-norm, while Legendre nodes are optimal in the sense of $...
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Find an L_{∞} distance between g=|x| and f=x{erf}_q(\frac{x}{\theta}).

Let $g=|x|$ and $f=x\cdot\textrm{erf}_q(\frac{x}{\theta})$, where $\textrm{erf}_q(\frac{x}{\theta})=\frac{q}{\Gamma(1/q)}\int_{0}^{\frac{x}{\theta}}\exp(-t^{q})dt$ is a generalized error function. ...
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1answer
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Can we express the integral of the nth derivative of this function analytically?

I am currently working on an assignment with Legendre Polynomials. The integral I get stuck with is in fact the integral of the Legendre Polynomial itself i.e. $$\int \frac{1}{2^n n!} \frac{d^n}{dt^n} ...
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1answer
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Why is overhat notation used both for a unit vector, $\hat{x}$, and for the closest vector, $\hat{x}$, in the best approximation theorem?

vector notation - why is overhat notation used both for a unit vector, $\hat{\mathbf x} = {\mathbf x \over || \mathbf x ||}$, and for the closest vector in a subspace $\hat{\mathbf x}$ to a vector $\...
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If graphical observations of a limit are feasible for totient approxinmation\extension onto $\mathbb R$

Essentially I want to know if the following can be considered true, despite the fact that the Euler totient is not actually a continuous function on $\mathbb R$ for which all the implications of an ...
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Does any continuous function on $[0,1]$ have a best $n$th degree polynomial approximation in the supremum norm?

Recently I am stuck in a problem in approximation theory which actually is problem in functional analysis. $C[0,1]$ is a normed vector space with $||\cdot ||_{\infty}$. $\Pi_n$ is a subspace which ...
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1answer
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A density result in $W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$

is the following result valid?: If $ u \in W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$, then $\forall \epsilon > 0 ~ \exists f \in C_{c}^{\infty}$ s.t. $\|u-f\|_{W^{1,p}(\mathbb{R}^n)} < ...
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1answer
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Showing the existence of a polynomial $p$ to approximate $f : [2,7] \rightarrow \Bbb{R}$

Let $f:[2,7] \rightarrow \Bbb{R}$ be a continuous function and for given $\epsilon >0$,we have to prove that there exists a polynomial $p$ such that $f(2)=p(2)$, $p'(2) = 0$ and $\sup\{|p(x) - f(x)|...
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What is the rational function that deviates least from $0$?

It is a well known result that among the set of polynomials of degree $n$ with leading coefficient $1$, the $n^{th}$ order Chebyshev polynomial $T_n (x)$ minimizes the (point-wise maximum) deviation ...
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How to improve derivative approximation errors along the boundary using radial basis functions

I am using radial basis functions to approximate the derivatives of a function. The test function I am using is: $g=y\cos(x)+x\sin(y)$ on the interval from 0 to $2\pi$ in both x and y directions. The ...
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1answer
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Best approximation and orthogonality

Let $\mathscr{B} := \mathbb{R}^n$ equipped with the euclidian norm, let $M \in \operatorname{Mat}_{n,m}(\mathbb{R})$ a Matrix with $\operatorname{rank} M = n \le m$ and $\mathscr{A} := \{ Mx \...
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Approximation to combinations

I have to show that for very large $N$ and $n$: $$ ^{N+n_1-1}C_{n_1} \cdot ^{N+n-n_1-1}C_{n-n_1} \propto \exp{-\frac{\left(n_1-n/2 \right)^2}{\sigma ^2}}$$ where $$ \sigma ^2 = \frac{n \left(2N + ...
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1answer
60 views

How does Weierstrass' theorem follow from Mergelyan's theorem?

According to Theorems 1 and 3 in this review article we have Weierstrass: Suppose $f$ is a continuous function on a closed bounded interval $[a,b] \subset\mathbb{R}$. For every $\epsilon > 0$ ...
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1answer
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Applying Bishop’s Theorem to $\langle xy;x^2y\rangle$

I am using Bishop’s Theorem in the version given by Wikipedia¹: Let $\mathfrak{A}$ be a closed subalgebra of the Banach space $C(X,ℂ)$ of continuous complex-valued functions on a compact Hausdorff ...