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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 collection of functions that a given finite neural network can approximate with ease?

To my understanding, one version of the universal approximation theorem runs as follows: Let $\Phi$ be the family of (trained) feedforward neural networks of bounded width, arbitrary depth, and mild ...
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46 views

Integral divergence implies summation divergence

Assume $f:(0,1)\times (0,1) \to \mathbb{R}$ is a nonnegative continiuos bivariate function such that $$\int_0^1 \int_0^1 f^2(x,y) dx dy = \infty,\quad \int_0^1 \int_0^1 f(x,y) dx dy = 1.$$ Can we ...
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Proving the high precision of the series correct to at least half a billion digits

I recently learned about this high-precision series. It is claimed that it is correct to at least half a billion digits. I am curious to know how it works. $$ \sum_{n=1}^\infty\frac{\left\lfloor n e^{\...
Pustam Raut's user avatar
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2 votes
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Problem in A Course in Approximation Theory by Cheney and Light

In chapter 2 of "A Course in Approximation Theory" by Cheney and Light, problem 11 states: How large can the coefficients be in a polynomial $p$ of degree at most $n$, if $p$ satisfies the ...
tox123's user avatar
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Is the best degree $n$ polynomial approximation an interpolation on $L^2[0,1]$?

The Question: Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous. Let $p:[0,1] \rightarrow \mathbb{R}$ be the $L^2$-closest degree $n$ polynomial to $f$. That is, $p$ minimizes $\int_0^1|f(x) - p(x)|^...
Joe's user avatar
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4 votes
2 answers
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Is Kolmogorov-Arnold (representation) neural network dense?

The Kolmogorov-Arnold neural networks (KAN), Ziming Liu et al, KAN: Kolmogorov-Arnold Networks draws inspiration from the Kolmogorov-Arnold representation theorem(KA theorem). However, the former, as ...
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Polynomial Approximation of Piecewise Continuous Functions

I'm looking for results about constructing polynomial approximations of piecewise continuous functions. Specifically, I'm wondering about whether there is a straightforward approach to the following ...
coult099's user avatar
1 vote
1 answer
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Approximating integral of product of gaussian and cosines

I am trying to evaluate the following integral \begin{equation} I=\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\prod_i^N\cos{\left(a_ix\right)}dx, \end{equation} where $b>0$ and $a_i$ are ...
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$f$ convex $\Rightarrow$ $L(f)\geq f$

Assume that $L:C[a,b]\to C[a,b]$ is a sequence of linear operator acting on the set $C[a,b]$ of continuous functions over [a,b]. Besides, $L(1)=1$ and $L(x)=x$. I recently found that from this is ...
Senna's user avatar
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Approximating the function 1 in Sobolev energy

I would like to find a sequence of smooth compactly supported functions $ u_n \in C^\infty_c([0,+\infty))$ with the following two properties: $u_n \to 1$ a.e. on $[0,+\infty)$ as $n \to + \infty$; $\...
Bremen000's user avatar
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Good approximation of $\sin(x)^5$ to use for ODE?

I need to find an approximate solution of $x'(t)= - \sin(x)^5$ with $x \in [0, \pi]$. I know there's no explicit solution, but I wonder if there are good approximations (whatever that means, let's say ...
tommy1996q's user avatar
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Approximating self-maps of $[0,1]$

Let $\mathrm{Inc}([0,1])$ denote the space of continuous, increasing functions $f:[0,1]\rightarrow [0,1]$ such that $f(0)=0$ and $f(1)=1$. I want to find a countable family of functions $f_n\in \...
Alvaro Martinez's user avatar
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How to approximate any line segment within a circular region using the minimum number of connected rotating axes

This problem arises from my personal experience in developing a game mod. At that time, I wanted to create a drone system for vehicles, but due to the limitations of the game itself, I could only ...
S PLATEX's user avatar
1 vote
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172 views

Explain the proof of Kolmogorov Arnold representation theorem

Can someone explain the outline of proof strategy of Kolmogorov Arnold representation theorem? Any proof of any variant (eg. George Lorentz's variant) would suffice. I would be grateful if you could ...
HIH's user avatar
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Closed subspaces of $L^1(\mathbb{R})$ that is not isomorphic to a subspace of a space with an unconditional basis

It is known that $L^1([0,1])$ does not admit any unconditional basis. Even more, $L^1([0,1])$ is not isomorphic to a subspace of a space with an unconditional basis. My question is the following ``...
Roddick Yu's user avatar
1 vote
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Hilbert spaces that include algebraic polynomials

This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
FDK's user avatar
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4 votes
1 answer
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Approximating exponential function using piecewise constant function

I want to construct a piecewise function to approximate the function $f:[0,1]^d \to \mathbb{R}, ~f(x) = \exp(\|x\|^2)$. My approach is to partition the space $[0,1]^d$ into non-overlapping cubes $B_1,\...
WeakLearner's user avatar
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Approximation by smooth functions - do the derivatives converge locally uniformly?

Say I have a function $f:\mathbb{R} \to \mathbb{R}$ which is continously differentiable and has a bounded derivative. Then I know I can approximate $f$ with smooth functions $\phi_n$ by mollifications ...
Snildt's user avatar
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1 answer
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Linearization that holds for any fixed parameter but not with limit

Consider this problem for $x\ll1$, and assume initially $a\neq 1$ $$ \frac{1}{1+(a-1)\frac{1}{x}} = \frac{x}{a-1} \frac{1}{1+\frac{x}{a-1}} = \frac{x}{a-1}\left (1 - \frac{x}{a-1} +\cdots\right)\...
atapaka's user avatar
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3 votes
1 answer
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Rate of convergence of Fourier modes of a mollifier on the torus in a negative regilarity Besov norm

Consider a standard mollifier $\rho_\delta$ on $\mathbb{T}^2$ (the 2 dimensional torus) and let $$ \hat{\rho_\delta}(m): = \int_{\mathbb{T}^2} \rho_\delta(z) e^{2 \pi i m \cdot z} dz $$ for any $m \...
Marco's user avatar
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0 answers
35 views

Best uniform approximation of complex exponential function $e^z$ over unit disc in complex plane

It is known that the best uniform approximation for a real function defined in interval $[1,-1]$ is via the Chebyshev polynomials. ([see optimal polynomials])1. Such polynomials are also called min-...
Manish Kumar's user avatar
1 vote
0 answers
41 views

Lipschitz approximation of the identity in metric spaces

Let $(X,d)$ be a complete and separable metric space and let $Y \subset X$ be a sigma compact subset. Does there exist a sequence of functions $(T_\epsilon)_{\epsilon>0}$ such that $T_\epsilon: Y \...
Bremen000's user avatar
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1 vote
0 answers
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$f$ smooth non-analytic, $f(0) = 0$. Is $f(x)/x$ smooth at $0$? [duplicate]

I'd like to know if the following statement is true: For any $f \in \mathcal{C}^\infty(\mathbb{R})$ such that $f(0) = 0$, then $f(x)/x$ is smooth at $0$. if $f$ is analytic, then this is clear by ...
Azur's user avatar
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2 votes
2 answers
127 views

Demonstrating Density Using the Stone-Weierstrass Theorem

In exploring the dense subsets of $C\left(I_n\right)$, I've been particularly focused on the application of a foundational result: $\textbf{Stone-Weierstrass Theorem}$: This theorem posits that for ...
Snowball's user avatar
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1 vote
1 answer
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Proving Density for Function Approximation with Hidden Layer Perceptron

I'm working on a problem related to function approximation within the $L^2\left(I_n\right)$ space of square-integrable functions: Problem Statement: Given a lemma without proof: $\textit{Lemma}$: Let $...
Snowball's user avatar
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0 votes
1 answer
115 views

Square integrable for universal approximation

Let's consider square-integrable functions $f \in L^2\left(I_n\right)$ with the definition of the $\textit{discriminatory}$: $\textbf{Definition:}$. The activation function $\sigma$ is called ...
Snowball's user avatar
  • 1,023
3 votes
6 answers
360 views

An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?

An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers? PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm ...
Joako's user avatar
  • 1,474
2 votes
1 answer
51 views

Compute the correction of a Chebyshev approximation using the Clenshaw summation formula

Assume you have a Chebyshev approximation of a function $f(x)$ evaluated using the Clenshaw summation method, up to polynomial order $N$: $$ f(x) = \sum_{k=0}^{N-1} a_k T_k(x) = (a_0 - y_2)T_0(x) + ...
LladOS's user avatar
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5 votes
1 answer
247 views

A guess on two increasing rational approximations to $\frac{\pi}{4}$

When investigated the Wilf function $$W(z)=\frac{\arctan\sqrt{2\operatorname{e}^{-z}-1}\,}{\sqrt{2\operatorname{e}^{-z}-1}\,},$$ see the preprint [1] below, I proposed a guess which reads that the ...
qifeng618's user avatar
  • 1,846
0 votes
0 answers
37 views

Reducing the number of natural cubic spline interpolation points

Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is ...
Donatas Šimeliūnas's user avatar
0 votes
1 answer
23 views

What's the optimal way to approximate a binomial distribution with a Poisson distribution?

Conventionally, we approximate a binomially-distributed variable $X\sim B(n, p)$ with the Poisson-distributed variable $Y\sim Po(np)$, with the mean of $X$ and $Y$ being identical. However, we could ...
Kyan Cheung's user avatar
  • 3,204
2 votes
2 answers
243 views

Compute the limit of the Log-Sum-Exp function

I am trying to prove that the Log-Sum-Exp function converges to the maximum function, i.e. $$ \lim_{\tau\rightarrow0}\tau\log\left(\frac{1}{N}\sum_{i=1}^N\exp\left(\frac{x_{i}}{\tau}\right)\right) = \...
rcescon's user avatar
  • 286
1 vote
2 answers
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Approximating a random variable by a sequence of random variables [closed]

Consider the triangular hat function: \begin{equation} \varphi(x) = \begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases} \end{equation} It is well ...
user82261's user avatar
  • 1,257
1 vote
2 answers
117 views

Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
Juan's user avatar
  • 33
0 votes
0 answers
42 views

Bounds of polynomial approximation of a function of many variables using Jackson inequality

There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
Masamune's user avatar
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0 answers
75 views

Pade approximation

I am trying to model the Pade approximation of a Lorentzian graph from the taylor series. I am trying to model PA[2/2] from taylor series expansion of order N+M=4th order of derivatives taken at the ...
user1288129's user avatar
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0 answers
26 views

Literature request for polynomial approximation

I would like to learn how to fit a multivariable polynomial through a set of data (surface fitting, volume fitting, and so on). I know how to do it if the polynomial is dependent on only one variable, ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
82 views

Polynomial approximations to $e^{f(x)}$

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a bounded function on some compact subset of the real numbers, i.e. $|f(x)|\leq B$ for every $x$ in the domain $D = [-L,L]\cap \mathbb{R}$. For ...
Cuhrazatee's user avatar
0 votes
0 answers
128 views

B-Spline with increasing knot distance

I'm trying to approximate a function $f(x)$ on $[0, M]$ that, in some sense, begins to rapidly "vary slower" as $x$ increases, i.e. its modulus of continuity (or the variation of its ...
Alex Shtoff's user avatar
1 vote
0 answers
31 views

How does the informativeness of eigenfunction approximations relate to the accuracy of the solution?

Consider the linear eigenvalue problem $$ \hat{M} f = \lambda f, $$ where $\hat{M}$ is a linear operator with eigenfunction $f$ and corresponding eigenvalue $\lambda$. The algorithmic complexity of ...
GeoArt's user avatar
  • 165
2 votes
0 answers
48 views

best polynomial approximations to $f$ vanish at $0$ imply $f$ is an odd function

It is known that if $f\in C[-1, 1]$ is odd/even, then the best polynomial approximation (in $L^{\infty}$ norm) of degree $n$, denoted by $p_n$, must also be odd/even. This has been asked on MSE before....
Yimin's user avatar
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0 answers
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Why does the B-spline constructed from squaring its coefficients appear to converge to the square of the B-spline itself?

Why does the B-spline constructed from squaring its coefficients appear to converge to the square of the B-spline itself? To be clear, I am not sure if this occurs in all situations, but it appears to ...
wyer33's user avatar
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1 vote
0 answers
29 views

What is the error of reconstruction of a smooth function observed only on a fixed grid by projection on a wavelet basis?

Context I'm a PhD student in Statistics and I have evaluations of a $L_2([0,1])$ function $f$, that is $m$ times derivable, on a regular grid of $[0,1]$ $$f\left(\frac{k}{p-1}\right), 0\leq k \leq p-1....
Rocinante's user avatar
1 vote
0 answers
59 views

Theory of sequence appoximation

There's plenty of literature about function approximation, both uniform and pointwise. Moreover, there are typically results on the speed of convergence of a given basis to the approximated function ...
Alex Shtoff's user avatar
0 votes
1 answer
53 views

On a First Order Nonlinear Differential Equation

Consider the ordinary differential equation $\dfrac{\text{d}y}{\text{d}x}=\dfrac{y-3}{x^2+y^2},$ with $y(0)=1$. My question is about determining the graph of $y$. Here is most of the information that ...
Hello's user avatar
  • 2,143
3 votes
2 answers
118 views

Can MLPs represent functions exactly for finite inputs?

The universal approximation theorem states given appropriate depth / width, an MLP can represent any continuous function with arbitrary precision $ \epsilon > 0 $. For discrete functions, $f: \...
rossignol's user avatar
4 votes
0 answers
113 views

Approximating $\sqrt{x}$ by a rational function in the complex plane

Newman (1963) proved the following. Theorem 1. Let $d \in \mathbb{N}$. Define $$p(x) = \prod_{k=0}^{d-1} \left(x+\exp\left(\frac{-k}{\sqrt{d}}\right)\right)$$ and $$r(x) = \frac{\sqrt{x} \cdot (p(\...
Thomas's user avatar
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1 vote
0 answers
38 views

Dirichlet’s approximation Theorem (Simultaneous version): In case Q is not interger

I’m reading “Diophantine approximation” by W.M.Schmidt. At the Chapter 2, Theorem 1E which is Dirichlet’s approximation Theorem of simultaneous version, He proved the theorem using pigeonhole ...
jihyuk seo's user avatar
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0 answers
48 views

Chebyshev approximation for bivariate function

I read the paper. I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one ...
Masamune's user avatar
0 votes
2 answers
56 views

How do I prove that the subset M = {f $\in$ C[0,1] : $\int_0^1f(x)dx = 0$} of C[0,1] is proximinal?

I want to show that the subset $$M = \{f \in C[0,1] : \int_0^1f(x)dx = 0\}$$ is proximinal in the Banach space C[0,1](equipped with the sup norm), that is, for every g $\in$ C[0,1] there exists f$\in$...
14Lucas07's user avatar

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