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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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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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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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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18 views

Is there some sense in which Fourier series approximations converge faster than Taylor series approximations?

Suppose I have a periodic function $f(x)$ with period 1 that I wish to approximate. Is there any sense in which if I compute the first $k$ sine and cosine terms of the Fourier series, I will get less ...
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28 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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1answer
52 views

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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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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1answer
28 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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1answer
18 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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Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ with point singularity to be of high rank?

This is a special case of this question. Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Suppose that $df$ is invertible outside a ...
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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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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 do I find a Bezier curve that goes through a series of points?

When someone has the 4 control points P0, P1, P2, P3 of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from P0 at t=0 and end at P3 at t=1 ...
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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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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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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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1answer
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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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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34 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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20 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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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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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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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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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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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 $\mathcal K$ denote a compact set in a metric ...
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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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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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$||\phi-\phi_\epsilon||_{L^1(\mu)}<\epsilon$ 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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“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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How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
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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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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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Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
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162 views

Closed set that is not $\rm proximinal$ set

Every $proximinal$ set must be $closed$, but the opposite is not true. I'm looking for such an example. A $closed$ set that is not $proximinal$ set ?
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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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1answer
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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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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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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 $C(X;\epsilon)=\{z\in\mathbb{C}:\min(|z-w|:w\in X)=\epsilon\}$...
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What is the “right” way of approximating random variables with other random variables?

It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a ...
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50 views

Are of Polynomials of Continuous functions complete?

The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial. I'm curious in polynomials of ...
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38 views

Approximating the basis of a specific function

We are given a continuous function $g: A \to B $, where $A, B$ are compact subsets of $\mathbb{R}$. We define a function $f(x) := g(b_1x)+g(b_2x)+...+ g(b_mx)$, where $b_i < 1$ and $b_ix$ is a ...
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Number of simple functions needed to approximate another function

I have got a function $f$ which maps from one compact space to another. Function $f$ is smooth. I want to approximate it with some simple functions (e.g polynomials). Is there any theory that gives ...