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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2
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1answer
164 views

Why does the sup norm make the results of approximation theory independent from the unknown distribution of the input data?

I was reading the paper "Why and When Can Deep – but Not Shallow – Networks Avoid the Curse of Dimensionality: a Review" and I was trying to understand the following statement in section 3.1: On ...
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1answer
106 views

If one proves uniform convergence of a function does that imply convergence in every other $L^p$ norm?

I was wondering about the truth of the following: $$ \text{if} \lim_{n\to \infty} \| f_n - f \|_{\infty,S}=0 \implies \forall p\in \{ 2,3,4,...\}, \lim_{n\to\infty} \| f_n - f \|_{p,S} = 0$$ ...
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0answers
63 views

Approximation of $1-e^{-z}$

I am interested in approximating the function $1-e^{-z}$. I want to approximate it by a family of functions $f_n$ that are entire and $f_n(2\pi k i) = 0$ for all $2\pi k i$ with $|k|\le n$ and $f_n(2\...
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33 views

Show as $n$ get large that $e^{\frac{-n^2}{2K}}$ is an approximation of $\frac{k!}{k^n(k-n)!}$

I'm asked to show this on a problem I'm working on. $e^{\frac{-n^2}{2K}}$ is an approximation of $\frac{k!}{k^n(k-n)!}$ when $n$ is large. However in this class we've never gone over $e$, so the ...
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61 views

Fourier series approximation of $f \in H_p^s$ rate of convergence

$H_p^s$ is a standard $L^2$ Sobolev space of periodic functions on $[0, 2\pi]$, with $\|f\|^2_{H_p^s} = \sum_{j = 0}^s \|f^{(j)}\|_{L^2}^2$. Let $$f_N(x) = \sum_{|n| < N} \hat{f}_n \frac{1}{\...
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1answer
104 views

Orthogonal Projection of a function onto $M$ [closed]

Let $I_1, · · · , I_N$ be pairwise disjoint intervals whose union is $[0,1]$. Let $$M = \lbrace g ∈ L^2([0,1]) :\text{ g is constant on $I_n$ } \forall n \rbrace.$$ Suppose $f \in L^2([0,1])$. ...
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1answer
57 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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1answer
37 views

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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1answer
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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1answer
89 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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43 views

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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1answer
33 views

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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170 views

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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1answer
116 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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1answer
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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3answers
250 views

Possible Pythagorean relation with Golden Ratio $ \phi^2+e^2 \approx \pi^2$

While study Numerics and playing with famous constants ($e$, $\pi$, Golden ratio) I came across the following relation $$ \color{blue}{1.6^2+2.7^2 = 9.85\approx 3.14^2}$$ This is nothing special but ...
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1answer
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Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

I would like to use mathematical tools to prove that $$9.9998\lt \frac{\pi^9}{e^8}\lt 10$$ With an on-line calculator I got $$ \frac{\pi^9}{e^8}\approx 9.9998387978$$ But I do not know any ...
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144 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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1answer
37 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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64 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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2answers
477 views

Can the remainder of a Taylor expansion be estimated from the coefficients?

Given a formula for the coefficients $c_n\in\mathbb C$ of a not analytically known function $f:\mathbb C\to\mathbb C, z\mapsto f(z)$'s Taylor series, is there any way to estimate the remainder term of ...
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3answers
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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1answer
394 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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121 views

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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2answers
42 views

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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1answer
164 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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1answer
248 views

$\rm Proximinal$ and $\rm \varepsilon-proximinal$ set

When $K$ is a nonempty subset of a metric space $M$, $\forall x \in M$ let $P_K(x)=\{y \in K: d(x,y)=d_K(x)=\inf_{k \in K}d(x,k)\}$ (metric projection) $\bullet$ The set $K$ is $proximinal$ in $M$ ...
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1answer
106 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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0answers
156 views

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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173 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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1answer
21 views

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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1answer
72 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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0answers
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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181 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
55 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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2answers
134 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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1answer
64 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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1answer
349 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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128 views

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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154 views

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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0answers
67 views

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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1answer
161 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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1answer
91 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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1answer
117 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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1answer
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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2answers
266 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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0answers
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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1answer
283 views

Why are Cartoon like functions named like this?

I just saw an interesting talk about shearlets on PyData Berlin. One point when defining shearlets is the use of so called cartoon like functions, which are ...