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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59 views

What is a good way to sample data points on an interval as to avoid Runge's phenomenon but have a deterministic sampling scheme?

I recently asked: Is there an analogous Gibbs phenomena to approximating sinusoidal but with polynomial terms? because I noticed that at the edges, polynomial interpolation of equidistance points ...
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414 views

Hermite polynomials approximate of a function and its derivatives

Given a differentiable function $f\in C^{(n)}(-\infty, \infty)\cap L^2(-\infty,\infty)$ with Gaussian measure $\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ and its Hermite polynomial expansion $f_n=\sum_{i=0}^...
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47 views

Find first degree polynomial approximation to function in $L^2[0,1]$

Let $u(t)=t^2$. Find the best approximation $v(t)$ of the form $v(t) = ct+d$ ($c,d\in\mathbb{R}$) to $u(t)$ in $L^2[0,1]$. We can define the set $V:=\{ct+d:c,d\in\mathbb{R}, t\in[0,1]\}$. $$\inf\...
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Best approximation to $t^2$ in first-dgree polynmial in $L^1[0,1]$

Let $u(t)=t^2$. Find the best approximation $v(t)$ in the form of $v(t)= ct+d $ (with $c,d\in\mathbb{R}$) to $u(t)$ in $L^1[0,1]$. So we need to find $$\inf\limits_{c,d\in\mathbb{R}} \int_0^1 \left|...
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91 views

Best constant approximations in $C[0,1]$, $L^1[0,1]$, $L^2[0,1]$

Let $f(t)=t^2$. Find the best approximation to $f(t)$ over $[0,1]$. Consider the Banach space $(C[0,1],\|\cdot\|_\infty)$. Then the best constant approximation, in my understanding, should be $g(t)\...
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When using taylor expansion for an expression with multiple terms, how do I determine the limit for accurate approximation?

I was once taught that, for using the Taylor series to expand $ln(1+x)$ the limit/range for having an accurate approximation (when the series "converges") is $|x|<1$. So the range of values for $ln(...
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Does the function $U=\frac{kx}{(x+x_{0})^2}$ reduce to a simple harmonic potential energy function for small oscillations around $x=x_0$?

The question is from a physics problems book, A particle of mass $m$ moves in a potential energy function given by $$U=\frac{kx}{(x+x_{0})^2}$$ where $x$ denotes the position and $x_0$ is a ...
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An error formula for linearization

Question: Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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What is the relation between Chebyshev and Taylor polynomials?

I just read about Chebyshev polynomials and that they are used in approximations. I don't fully understand them yet. What is the relation between Chebyshev polynomials and Taylor expansions?
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Bounding sum of samples of a Gaussian

Suppose we have $K$ points $x_1,\ldots,x_K$ in $\mathbb{R}^d$ and let $$f(x)=\sum_{k=1}^K \exp(-\lambda \Vert x-x_k \rVert^2).$$ Can we uniformly bound $f$ independent of $K$? It is okay to use ...
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118 views

First order approximation and small angle approximation

I am given $\omega\left ( k \right )=\left ( \frac{C+H}{m}\pm \frac{1}{m}\sqrt{C^{2}+H^{2}+2 CH Cos\left ( k a \right )} \right )^{1/2}$. It is mentioned that this reduces to $\omega\left ( k \...
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42 views

Making an approximation

How does $\sqrt{\frac{2(cos(x)-1)}{cos(x)}} \approx x$? At first I thought it was an example of binomial approximation but I was unable to approximate is to just $x$. I could only get $$\left(\frac{...
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Approximating a function's derivative in machine precision

Let's say I have a very complex analytical function $f$, in which finding its derivative at point $Q$, i.e. $f'(Q)$ is not practical. So we resort to finding the derivative numerically without ...
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169 views

Details of proof of convergence of Bernstein polynomial approximation

In Durrett's Probability Theory and Examples https://services.math.duke.edu/~rtd/PTE/PTE4_1.pdf, he gives the following. I have two questions: how do we get the underlined line? We apply Jensen's ...
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38 views

How to approximate a level curve?

Let $G$ be a $C^\infty$ function $G:\mathbb{R}^2\rightarrow\mathbb{R}$, and let $C:=G^{\leftarrow}(c)$, i.e. $C$ is a level set of $G$. I know that $C$ is bounded (which implies that it's a closed ...
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67 views

Finding approximation methods for a function and its derivative

I need to find numerical methods to approximate a map $y=h(x)$ where $h: R^n \to R^m$ on the compact sets $D_x \times D_y$. More specifically, I need to find appropriate methods that ensure that, for ...
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775 views

Bounding error of Padé approximation

I'm trying to understand how one would understand the error of a given Padé approximation for a function. For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a ...
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53 views

Show that the sequence $\{x_n\}_{n=1}^\infty$ does not converge in $ \ell^2 (\mathbb{N}). $

For each $n \in \mathbb{N}$, define a sequence $x_n=\{ x_n(k) \}_{k=1}^\infty$ by $$ x_n(k) = \begin{cases} \frac{1}{n} \;\;\; \text{ if $1\leq k \leq n^2$}\\ 0 \;\;\; \text{ otherwise.} \end{...
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Pade approximation of a rational function

I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated. So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
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107 views

Error in approximation of sum of products

Let $a,k,N$ be positive integers with $k\leq n-a$, and $0< a<n$. Let $$\theta(k) = \prod_{j=0}^{k-1}\left(1-\frac{a}{n-j}\right)$$ It is clear that $\theta(k) \leq \prod_{j=0}^{k-1} (1-\frac{a}{...
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Is it possible to find a McLaurin Series for the following function?

Since the limit as $y$ approaches $0$ is $\infty$ for the function $g(y$) = $\frac{1}{\sqrt{1+y^2} - \sqrt{1-y^2}}$, can we say the MacLaurin Series in this case does not exist? And if this is case, ...
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Weighted polynomial approximation on the half-line

Let's $w : \mathbb R_+ \to \mathbb R_+^*$ a continuous function (I will only be interested in the case $w : t \mapsto e^{-t}$). We use $w$ to define the Banach space $C^0_w(\mathbb R_+)$ of continuous ...
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Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
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179 views

Approximation of multivariate functions by a polynomial

Is there any reference about approximating multivariate functions with polynomials ? I search on google, but I fail to find my goal. Does any one come across a book detailed in this manner.
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Is the best subspace-constrained approximation in $\mathbb{R}^n$ independent of the norm chosen?

Let $U \subset \mathbb{R}^n$ be a subspace, let $\lVert \cdot \rVert_A$ be any norm on $\mathbb{R}^n$, let $a \in \mathbb{R}^n$. Because of Bolzano–Weierstrass, there exists a $v \in U$ such that $$ \...
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74 views

Linearisation of a Matrix Function

Can one linearise a function of matrices in a similar way to a scaler function i.e $$f(x) = (x^{-1} + c^{-1}) \approx f_a(x) = (c + a)^{-2}\left(c^2x + a^2c\right)$$ (Where $f_a(x) $ is the the ...
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92 views

Pointwise approximating identity by compact operators

Does there exist a sequence of compact operators (not necessarily linear) $T_n: H^1(\mathbb{R}^N)\to L^2(\mathbb{R}^N)$ such that, for every $u\in H^1(\mathbb{R}^N)$, $$ \lim_{n\to\infty} n\|T_n(u) - ...
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164 views

Approximating distribution and delta method

Suppose i.i.d r.v $X$ has density of $\frac{3x+1}{8}$ on interval $(0,2)$, define $Y = \pi X^2$. Questions are: If $Z = \pi \bar{X}^2$, where $\bar{X}$ is arithmetic mean of all $X_s$ If $\bar{Y}$ ...
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183 views

Patterns appearing in irrational approximation graphs

I'd like to know more about some patterns I found in graphs corresponding to irrational numbers. Here's the graph for $\sqrt 2$ for example First, I'll try to explain most naturally the function that ...
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37 views

$\lim_{n\to \infty} \min_{a,b \in \mathbb{C}}\bigg[\frac{1}{2\pi}\int_{-\pi}^\pi\bigg|\sqrt{|x|^3}-a\sin(nx)-b\sin([n+1]x)\bigg|^2dx\bigg]$

Let us define $R_n$ by:$$\min_{a,b \in \mathbb{C}}\bigg[\frac{1}{2\pi}\int_{-\pi}^\pi\bigg|\sqrt{|x|^3}-a\sin(nx)-b\sin([n+1]x)\bigg|^2dx\bigg]$$ Calculate $\lim_{n\to\infty}R_n$ My attempt: Let ...
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1answer
56 views

Choosing which points/nodes to interpolate through?

I have an expensive scalar-valued function $f$. If needed, you can assume it's single-variate. I want to approximate the function on the interval $[0,1]$, so I evaluate it at several points $x^n \...
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135 views

Showing that a best approximation exists in a finite-dimensional subspace?

I have some questions on a proof that for every element of a normed space $X$, there exists a best approximation of this element in a finite-dimensional subspace $U$. Here is the proof: Let $\varphi \...
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363 views

Mathematical symbol describing variables that increase together

If I have a pair of functions increasing together or decreasing together, I certainly have: \begin{equation} min(|\Delta F|,|\Delta G|) > 0 \iff \frac{\Delta F}{\Delta G} > 0 \end{equation} I ...
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65 views

Approximating functions by Taylor.

I want to apply the Taylor for approximating function $$\begin{align}f(t,x,v)=f(t,x_{k},v_{k})+\frac{\partial f}{\partial x}(t,x_{k},v_{k}) (x-x_{k})+ \frac{\partial f}{\partial v }(t,x_{k},v_{k})(v-...
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About a function approximating the $\arctan(x)$

I found in a paper this function: $$f(x)=\frac{8x}{3+\sqrt{25+\left(\frac{16x}{\pi}\right)^2}}$$ is a good approximation of the $\arctan(x)$. If we consider the difference function: $$d(x)=|\arctan(x)-...
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denseness of smooth functions in space of lipschitz continuous functions

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, let $C^{0,1}_0(\Omega)$ be the set of all Lipschitz continuous functions on $\Omega$ that vanish on the $\partial \Omega$. Let $C^{\infty}_c(\...
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1answer
1k views

Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous real-...
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How to approximate a negative exponential distribution?

In Hierarchical Web Caching Systems: Modeling, Design and Experimental Results by Hao Che, Ye Tung and Zhijun Wang, the authors used $$ g(t) = \frac{\exp(- (t - \tau)/(T - \tau))}{T - \tau} $$ to ...
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What is the exact meaning of 'in the first approximation' in the context of applied mathematics?

In the applied mathematics textbooks or papers, I often see the phrase 'in the first approximation'. For example, substitution of Eq.(1) into the boundary condition (2) results in Eq.(3) describing ......
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Does exist some classes of functions dense in the Hölder space?

I am dealing with $f:[0,T]\to\Bbb R$, $\;\alpha$-Hölder continuous functions, with $\frac12<\alpha\le1$ such that $f(0)=0$, nowhere differentiable. Let us denote the space of such functions with $\...
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49 views

Polynomial approximation over two intervals

I want to approximate the function $f(x) = \frac{1}{x - z}$, $z \in \mathbb{C}$, on two intervals $[a,b] \cup [c,d] \subset\mathbb{R}$ using polynomials. If $z$ was real and $b = c$, Bernstein's ...
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163 views

Can a continuous, locally Lipschitz and bounded map be approximated by globally Lipschitz functions?

Let $(X,\| {\cdot}\|)$ be a Banach space, and let $C_b(X)$ denote the space of all, bounded, continuous real-valued function on $X$ with the supremum norm $\|{\cdot}\|_{\infty}$. Suppose that $f\in ...
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123 views

Is there an analytic approximation of the following function

Let $\psi$ be a function on $\mathbb{R}$ satisfying $\phi(x)\geq 0$ for any $x$ and $\psi(x)=0$ when $|x|\geq 1$. $\int_{-1}^1\psi(x)dx=1$ $\int_{-1}^1x\psi(x)=0$. $|\psi'''(x)|\leq B$ for a ...
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278 views

How does one find a polynomial approximation of a non-analytic function?

I have a function $$f(x) = \left\{ \begin{array}{lr} 0 & 0 \leq x < 1/3\\ q(x) & 1/3 \leq x < 2/3 \\ 1 & 2/3\leq x \leq 1 \end{array} \right.\...
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43 views

Limit on minimum involving best $L^2$ approximaion and Fourier series

Let $f:\mathbb R \to \mathbb R$ be the periodic continuation of the function $\sqrt{\left|x \right|^3}$ on the interval $[-\pi,\pi)$. For every $n \in \mathbb N$ let us denote: $$\lambda_n = \...
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73 views

Numerical Approximation Rules

I was not exactly sure what to title this question, but I would appreciate if someone can confirm my understanding of the left hand, right hand, trapezoidal, midpoint, and simpsons approximations. ...
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45 views

Control higher derivatives of approximating sequence

Let $\Omega \subset \mathbb R^d$ be a bounded open set. Let $H_0^1(\Omega)$ be the usual Sobolev space. From the definition it follows, that each $u \in H_0^1(\Omega)$ can be approximated by a ...
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72 views

Least Squares Approximation of a Function on a Interval Different of the Limits of Integration of the Scalar Product of the Space

Suppose we want to approximate a function $f(x)$ on a interval $[c,d]$ by, say, a linear polynomial $p(x) = a_0 + a_1x$ using the scalar product $$\langle f,g\rangle = \int_a^bw(x)f(x)g(x)\,dx,$$ ...
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735 views

Relative error $\sim$ absolute error of logs: always true?

If $f(x) \sim g(x)$ have a relative error $\sim h(x)$, is it always true that $\ln f(x)$ and $\ln g(x)$ have an absolute error that it also $\sim h(x)$? For instance, $\left(1+\frac{1}{x}\right)^x \...
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180 views

approximating an integral/hypergeometric function

I am looking to approximate the following integral for small $z$: $\int_0^{\infty}dy \frac{1}{z} e^{-y/z} \frac{w e^{-y}}{s + w e^{-y}}$ . The integral can be solved in general to be a ...