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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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) = \...
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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 ...
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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
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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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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 ...
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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, ...
Nikola Ristic's user avatar
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From convergence of orthogonal projection to orthogonal series expansion in reproducing kernel Hilbert spaces.

Introduction: Let $\mathcal{H}$ be a Hilbert space of functions $\Omega\to\mathbb{R}$ with reproducing Kernel $K:\Omega\times\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$, where $K$ is ...
Max Stuthmann's user avatar
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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
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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
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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 ...
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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....
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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 ...
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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
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Question about existence of certain apporximating cdf

Consider a cumulative distribution function $F_0 \in C^{\gamma}[0,M]$ where $C^{\gamma}$ is the space of continuous functions which are Holder continuous of degree $\gamma$. Does there exist a ...
Grandes Jorasses's user avatar
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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
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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 ...
Bananas's user avatar
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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
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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(\...
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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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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
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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$\in$...
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How to approximate $\ln \left( {{e^{{x_1}}} - {e^{{x_2}}}} \right)$?

In the Max-Log-Map algorithm for channel decoding, the approximation $\ln \left( {{e^{{x_1}}} + {e^{{x_2}}}} \right) \approx \max \left( {{x_1},{x_2}} \right)$ can be considered because $\ln \left( {{...
Tuong Nguyen Minh's user avatar
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Convergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on a subset of $\mathbb{R}$

Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let \begin{equation} \text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt. \end{equation} given projections of $\text{erf}(x)$ onto the first $k$ ...
Cuhrazatee's user avatar
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Uniformly Continuous version of Kirszbraun Theorem?

Context The Kirszbraun theorem states that if $H_1$ and $H_2$ are Hilbert spaces, and $E \subset H_1$, then any Lipschitz function $f: E \to H_2$ may be extended to a Lipschitz function $\widehat{f}: ...
Daniel P.'s user avatar
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1 answer
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Which big-Omega meaning re Ramsey number r(4, t)?

In https://arxiv.org/pdf/2306.04007.pdf, Mattheus and Verstraete prove that the Ramsey number $r(4,t) = \Omega(t^3/\log^4 t)$ as $t \rightarrow \infty$. Which of the two incompatible definitions of $\...
murray's user avatar
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Finding best approximation of polynomial via max norm

How can I find best approximation of a polynomial $f(x) = -2x^3+3x^2-4x+5$ with respect to max norm by a polynomial of degree 2 on segment [1,5]. As I have understood, if I have a polynomial $g(x) = ...
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Quantitative approximation by shallow ReLu networks

Let $f: I \to \mathbb R$ be an $L$-Lipschitz function with compact support. Show that for all $\epsilon > 0$ there is a function $\phi$ given by $$ \phi(x) = \sum_{i=1}^W c_i \mathrm{ReLu}(a_i x + ...
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A counter example that I cannot find an analytic branch of Logarithm

Let $A(F)$ be defined as the class of functions that are Analytic in $F^o$ and continuous on $F$. I am reading a paper with the following assertion: Let $F$ be an Arakelyan set. Then for any $f\in A(...
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Taylor expansion of the gradient: how to use big O notation?

Let $f:\mathbb{R^n}\to\mathbb{R}$ be a continuously differentiable function and let $\nabla f$ be its gradient. I am interested in approximating this gradient around some point $x_0$ using a Taylor ...
user_lambda's user avatar
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How to Calculate Small Order

Question Let $c$ be constant. As $x \to 0$, $$ f = cx^2 + o(x^2). $$ Then, $$ f^2 = c^2 x^4 + o(x^4). $$ What I think \begin{align*} f^2 &= c^2 x^4 + 2cx^2 o(x^2) + o(x^4) \\ &= c^2 x^4 + o(x) ...
ytnb's user avatar
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Finding relative error between two curves

I want to find the relative error between the curve $f(x)$ and some approximation curve $g(x)$ in $[a,b]$. Is it alright to use the integral as shown below: $$\frac{\int_{a}^{b}|f(x)-g(x)|dx}{\int_{a}^...
winsmoretti's user avatar
1 vote
1 answer
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Approximate summation formula of time to count numbers from 1 to N

When calculating how much time it takes to count from $1$ to $n$, it is normally used the approximation that it takes about $1s$ to say a number out loud, so it would take $n$ seconds, but there's a ...
Wagner Martins's user avatar
1 vote
2 answers
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Interesting pattern in Fourier coefficients

I was playing around with Fourier coefficients and I observed something very interesting The above is for the function $3x^2+2x^3$ for $100$ terms($-\pi$ to $\pi$ periodic). The alternate coefficients ...
DatBoi's user avatar
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Nuclear $C^*$-algebras have the WEP-property.

Exercise: Let $A$ be a nuclear $C^*$-algebra. Show that $A$ has Lance's WEP-property, i.e. show that there exists a ucp map $\Phi: B(H_u) \to A^{**}$ such that $\Phi(a)= a$ where $A\subseteq A^{**}\...
Andromeda's user avatar
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Can low rank/low variable first order logic formulas approximate higher order one?

I say a formula $\Phi$ labels a model $g$ as $1$ iff $g \models \Phi$. Otherwise the formula labels the model $-1$. I use $\Phi[g]$ to denote the label that $\Phi$ assigns to $g$.I also assume that $g$...
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Question about divergences of functions Vs divergences of their expansions

I have the following function $$f(a,x)=\frac{1}{(1-a\sqrt{1-x^2})}$$ For small x (i.e. $x<<1$), I can Taylor expand around $x=0$, according to $$f(a,x)=\frac{1}{1-a}-\frac{ax^2}{2(1-a)^2}+\...
schris38's user avatar
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Relation of $L^1$ and "$L^{1/2}$"

For any continuous $g:[0,\frac{\pi}{2}]\rightarrow (0,+\infty)$ and $\forall \varepsilon >0$, is there a contiuous $h(t):[0,\frac{\pi}{2}]\rightarrow (0,+\infty)$ such that $$ h(\frac{\pi}{2})\le\...
Enhao Lan's user avatar
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Optimal Choice of Regression points for Minimizing the Approximation Error when solving a PDE with Function Approximation

I want to solve a high-dimensional PDE $F(\mathbf{x}, v ,\triangledown v , \triangledown^2 v )$ $$ -\dot{V_{t}}(\mathbf{x}) - A (\triangledown v_t (\mathbf{x})) = f(\mathbf{x}) , \quad \mathbf{x} \...
François's user avatar
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Behavior of a squared-integrable function close to a specific value of $x$.

Is there a way to prove that an $L^2(\mathbb{R})$ function $f$ close to $x=0$ is ${o}(x^{-1})$, in the sense that $$ \lim_{x\to 0} \left(|x{f(x)}|\right)=0? $$ I am working on a proof involving $L^2(\...
Gateau au fromage's user avatar
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Smooth approximation of the identity in $\mathbb{R}^d$ with bounded derivative

I would like to produce, for every $n \in \mathbb{N}$, a function $f_n: \mathbb{R}^d \to \mathbb{R}^d$ with the following properties: $f_n \in C^\infty(\mathbb{R}^d; \mathbb{R}^d)$, $f_n(x)=x$ for ...
Metric01's user avatar
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Stone-Weierstrass theorem applied to Wiener processes: Does it require always a polynomial of infinite order?

This is a conceptual question: I know beforehand I don't have enough background to understand a detailed answer, so please keep it as simple as possible. A few days ago I discover in Wikipedia the ...
Joako's user avatar
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Simple skewed Gaussian

I recently reported this on Quora but I'd like to share it here too to try to get more people to give their opinions about this. Basically I designed a simple equation for skewed Gaussian curves. For ...
David Sánchez's user avatar
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1 answer
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Asymptotic solution of an integral equation

Consider an integral equation of the form $$\sigma_B(\Lambda)+\int_{-B}^{B} K\left(\Lambda-\Lambda^{\prime}\right) \sigma_B\left(\Lambda^{\prime}\right) d \Lambda^{\prime}=f(\Lambda)$$ where the ...
user824530's user avatar
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Is there a vector $y$ such that $\|P_Sy\|$ is constant for all sets $S$ of size $K$?

Let $\Phi\in \mathbb{R}^{m\times n}$ be a matrix with $m\leq n$ and it satisfies the restricted isometry property (RIP) of order $K>1$. Let $S\subset [n]$ be any subset of size $K$, let $\Phi_S$ be ...
Samrat Mukhopadhyay's user avatar
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Approximation of pressure gradient term in relativistic hydrodynamics equations to calculate vertical height of a thin rotating flow

In studies of rotating fluid flows around a relativistic star, the vertical height, or (half)-thickness, of the flow is usually obtained from the vertical component of the Euler equation. The Euler ...
Richard's user avatar
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2 answers
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Fourier series vs polynomials in approximation

I am looking into uses of Fourier series. I learned that it can be used to approximate functions (link). However, I think there are ways to approximate functions with polynomials (not limited to ...
Hayatsu's user avatar
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Sufficient criterion for a binary string to have dense orbit

I was recently asked the following question: Does the map $$\mu : [0, 1) \rightarrow [0, 1)\\ \hspace{105px}x \hspace{5px}\rightarrow \hspace{5px}2x \mod 1$$ have some point, say $\alpha$, whose ...
CCC's user avatar
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How do I find the power law of divergence of second derivative of an integral? [closed]

I have the integral of the following: $$I(g) = g \int_{-\pi}^{\pi} dk \sqrt{1+g^{-2} + 2g^{-1} \cos(k)},$$ whose second derivative ($\partial_g^2 I(g)$), I know diverges as $g \rightarrow 1$. How do I ...
purestate's user avatar
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2 answers
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Estimate intersection of exponential and linear function

I would like to solve a function of the form $a^x = bx + c$ for $x$, but I read it can not be solved algebraically (transcendental equation). My attempts basically fall flat at $\log_a(a^x - bx) = \...
stimulate's user avatar
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Approximation of differentiable functions

Do there exist a sequence $e_n:\ [-1,1]\to [-1,1]$ of functions such that, for every function $f\in C^s([-1,1])$ we have the following approximation property: $$\inf_{g\in \text{span}(e_1,\dots e_n)}\|...
Davide Maran's user avatar
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