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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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What is the complete Partial Derivatives approximation formula to calculate the value of a function?

We know the approximation formula using partial derivatives to calculate the value of a function with some variables $x$ and $y$, which can be extended to any number of variables to be \begin{...
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22 views

A question on the existence and uniqueness of a cubic Hermite interpolant

I have been trying to solve a particular problem that establishes both the existence and uniqueness of a cubic hermite interpolant on some generic interval $[a,b]$. Briefly, for a function $f$ we ...
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24 views

How could I obtain this approximation of the May-Wigner theorem?

I'm trying to understand the complete proof of the May-Wigner theorem. We have a real random $n\times n$ matrix $B$ with its non-zero elements $B_{ij}$ are chosen independiently from a fixed ...
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50 views

Can we approximate continuous functions arbitrarily well with polynomials? (beyond Weierstrass )

Let $f:(0,1) \to \mathbb{R}$ be continuous, and let $\delta:(0,1) \to \mathbb{R}$ be continuous and positive. Does there always exist a polynomial $p(x)$ satisfying $|f(x)-p(x)| < \delta(x)$ for ...
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Do all convergent sub-sequences of a sequence in a compact set $\mathcal K$ converge to the same element in $\mathcal K$?

I came upon this when trying to understand the proof to "Theorem on Existence of Best Approximations in a Metric Space" as given by Cheney (1981). Let K denote a compact set in a metric space. To ...
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35 views

Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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27 views

Smooth approximation (under supremum norm) of distance to algebraic set in $\mathbb{R}^n$.

Given a set $S$ which is the zeroes of a finite number of homogenous polynomials in $x\in\mathbb{R}^n$, I want a constant $\alpha$ and a $C^2$ approximation, denoted $d$, to the function $d(x,S)=\inf_{...
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11 views

Approximating the lateral derivatives

Let $f : [0,2] \to \mathbb{R}$ be a continuous function with continuous derivatives of all orders in every point except at $t = 1,$ where the lateral derivatives exist. We know that one can ...
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16 views

Optimize a fixed size susbset

So I'm trying to solve this problem: There are many people who apply for jobs at a company. Each applicant has some technical skills required for jobs. The skills possessed by different ...
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172 views

Why do deep neural networks work well?

The universal approximation theorem, as I understand it, states that for any continuous bounded function $f: X \rightarrow \mathbb{R}$ with compact domain $X$ and any threshold $\varepsilon$ there is ...
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Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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21 views

Advanced Methods for Approximating Surfaces based only on partial derivative estimates

I'm looking for information on interpolating a surface function p(x,y) based only on estimates of the partial derivatives at points on a grid. Obviously, any such approximation is subject to a ...
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1answer
17 views

Generalization of Power Series

Given a smooth function $f(x):$ Does there always exist an expansion of x around a point $x_{0}$ of the form $$\sum_{n=0}^\infty \frac{h(f^{(n)}(x_{0}))}{n!}g(x-x0,n)$$ for some functions $h(c),g(a,b)...
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24 views

Regression through linear Fourier coefficient fitting?

Basically suppose on was given an unknown function/data and expected to write a function so that $Y=f(X)$, this can be done by linear regression in simple cases very easily. However, suppose that the ...
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53 views

Smoothing of a step function using smoothstep. (Curve fitting)

I was trying to smoothen the step function (zero when $x$ is less than $2/3$ and equal to $1$ when $x$ is greater then $5/6$) as in the picture below. Trying to fit $f$ in between $2/3$ and $5/6$ ...
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15 views

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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24 views

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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28 views

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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19 views

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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24 views

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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33 views

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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19 views

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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40 views

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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1answer
42 views

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
121 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
48 views

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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21 views

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
42 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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17 views

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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1answer
37 views

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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2answers
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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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29 views

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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43 views

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
57 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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58 views

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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22 views

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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32 views

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
101 views

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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49 views

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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16 views

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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39 views

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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26 views

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
63 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 ...