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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Division-free differentiation of $\frac1x$

Given $a>0$, $~b=\frac1a$ and $c>0$ is it possible to calculate $\frac{1}{a+c}$ in such a way to avoid divisions? The solution can be approximated, but the percent error must be less than 1% ...
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Approximation algorithm for mine packing problem [on hold]

Problem: In the Mine Packing Problem, we are given an undirected graph G = (V, E), and wish to find a set of vertex-disjoint trees of depths 1 (all leaves connected directly to the root). The goal is ...
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A polynomial in $g$ approximates every $f$ iff $g$ is injective

Prove or disprove the following statement: There exists a continuous function $g$ defined on $[a,b]$ with $g(x)\neq x $ for at least one $x\in[a,b]$ such that for every continuous function ...
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43 views

Approximation of a continuous function by the polynomial of a continuous function

Prove or disprove that there does not exist a real valued continuous function $g$ on $[0,1]$ with $g (x ) \neq x$ for all $x \in(0,1)$ such that given any $\varepsilon > 0$ and any real valued ...
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59 views

Need some good problems on Weierstrass Approximation Theorem

I know the Weierstrass Approximation Theorem, and I know its proof. I however till now have not really found any good application of the theorem except in one problem where it is given that if $f$ is ...
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16 views

About approximation by Haar polynomials

I'm reading about Haar functions, and I found the statement of a theorem which says that if $f$ is a continuous function on $\mathbb{T}$ and $\varepsilon >0$, then there exists a Haar polynomial of ...
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the set of Best Approximation is a norm-closed convex set.

Let $V$ be a normed vector space. Let $W$ be a proper closed subspace of $V$. Let $M$ be the set of best approximations in $W$. Prove that $M$ is a norm-closed convex set. I've shown that the ...
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60 views

Bound of the derivative of polynomials. [closed]

How do I show that for every positive integer $d$ there exists $C(d)>0$ such that for every polynomial $p(x)$ of degree $\leq d$, $$\max_{x \in [0,1]}|p'(x)| \leq C(d) \max_{x \in [0,1]}|p(x)|$$
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Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
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18 views

Simpson's rule is not good enough for the best approximation in L2 problem

The problem came from my computation methods (practice) class. It was to write a program which does the following: Original problem statement: We have a [0; 1] segment. Let us divide it into $2^n$ ...
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28 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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28 views

Stone Weierstrass theorem generalization

Find all such functions $g:[a,b]\to [a,b]$ such that $g$ is continuous. For any continuous function $f:[a,b]\to \mathbb{R}$, given $\varepsilon >0$ there is a polynomial $P_{\varepsilon}(t)$ ...
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2answers
46 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
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2answers
53 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...
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1answer
30 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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152 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...
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1answer
76 views

Orthogonality of functions related to Legendre polynomials

If $q\in P^{0}_{k}(I)$, i.e $q$ is a polynomial of degree $\leq k$ that vanishes at two end points of the interval $I=(0,1)$ and ...
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76 views

Series Expansion of the determinant for a matrix near the identity.

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally we ...
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33 views

Differentiable approximation $f(x) = x$ for $x>0$ and $0$ otherwise.

I would like to find a twice continuously differentiable approximation of $$f(x)= \begin{cases} 0 & x\leq 0 \\ x & x>0. \\ \end{cases}$$ Are there any approximations ...
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2answers
54 views

Differentiable approximation of the absolute value function

Are there any good approximations of the absolute value function which are $C^2$ or at least $C^1$? I've thought about working with exponentials and then adding in more terms to keep the function from ...
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On Proximinal sets

A subset $K$ of a Hilbert space $H$ is called proximinal if every $x\in H$ has a vector of minimum norm. Question: How do I show that if $K$ is proximinal and bounded, then $K$ also has ...
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Approximating continuous functions by steps functions: Proof that the approximation error monotonically decreases as the number of intervals increase

Let $f$ be a continuous function defined on a compact set, $f: X \subset \mathbb{R} \rightarrow \mathbb{R}$. Let $\mathcal{P}_k = P_1,\ldots,P_k $ be partitions of $X$ such that $\mathcal{P}_k$ is an ...
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Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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27 views

why conventional approximation method is true?

why the text book method for finding the fitting curve is right ? we have n data we want to approximate with a polynomial of degree m $P_m(x)$ (m < n-1). and of course $E = \sum_{i=1}^m ...
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using approximation to find interpolantion function?

my question is can we actually find the interpolating polynomial if we solve the approximation problem for degree m = n-1 ( where we have n data ).(i know we usually solve the approximation problem ...
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1answer
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Approximation theory in multiple dimensions (reference request)

I understand the following from approximation theory: if $f(x)$ is a well-behaved function on some interval $[a,b]\subset\mathbb{R}$ then for any tolerance $\varepsilon$, there exists an $N$th-degree ...
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31 views

How to prove an inequality $\left| {g(j + 1)} \right| \le 5/4$ in Stein's method for Poisson approximation

The following is a lemma in Barbour, A. D., Holst, L., & Janson, S. (1992). Poisson approximation. Oxford: Clarendon Press,p7. For $j=1,2,...$ and $\lambda > 0$, we have $\left| {g(j + ...
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1answer
22 views

Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following: Let $f(x)$ be a multivariate positive and continuous function. Can we approximate ...
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38 views

Polynomial Approximation of Holomorphic Functions

Consider $\Omega \subseteq \mathbb{C}$ be open. Let $f:\Omega \rightarrow \mathbb{C}$ be holomorphic on $\Omega$. For any closed ball $B[a;r]$ in $\Omega$ does there exist a sequence of polynomials ...
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Approximation of continuous even functions

If $f$ is an even function in [-1,1], how do I show that it can be approximated by sequence of polynomials of $p_n(x^2)$? [The question is followed by a hint saying we could consider $f(\sqrt{x})$ in ...
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Calculating Approximate Distribution from Trials

Update: Now answered below; if anyone has any better answers, then please let me know - I'd be most appreciative. :) My question is the following, and here is the setup (fairly standard Polya's ...
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1answer
38 views

approximating with a class of indicator functions: any theorems?

Let $G$ be a class of $\it{indicator}$ functions where $g\in G$ implies that $g : X \rightarrow \{0,1\}$, where the domain $X$ is a compact subset of $\mathbf{R}^n$. For example: $$ g(x) = 0 \text{ ...
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a countable dense subset of Lipschitz functions

Suppose $(X,d)$ is a metric space and let $\mathcal{L}$ be the space of bounded Lipschitz functions on $X$. Let $D$ be a countable dense subset of $X$ and consider the set of functions ...
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Analysis of block-greedy algorithms for function approximation?

I consider the problem of selecting a final basis set $\{\phi_{c_j}\}_{c_1}^{c_n}$ approximation of function $f \in \cal{H}$ in a Hilbert space that minimizes $L_2$ error. One can use a greedy ...
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1answer
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Series approximation to $\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du$

I have figured out by graphing that, for small $x$: $$ \int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du\approx\log(1/x)+\pi/2+O(x) $$ However, I am unable to prove that this is the case. As $x\to 0$ ...
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after hajek, can we guarantee that annealing has reached a solution in the best 1%?

hajek showed in http://web.mit.edu/6.435/www/Hajek88.pdf that there are conditions under which an annealing process is guaranteed to find the global minimum. these constraints are pretty tight, but ...
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48 views

Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
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Is there Stone-Weierstrass theorem for piecewise continuous functions

There is a version of the theorem for continuous mappings $f$ from a compact metric space $X$ to real numbers $R$. Namely, if an algebra $A$ of continuous functions separates points and is ...
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49 views

stone weierstrass approximation theorem for simple functions: does it exist?

The most general version of the theorem (I have seen) states that a function from a compact metric space into reals can be approximated by an algebra of functions that maps in the same way, is ...
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Ratio of fpras approximations

If I need to compute the ratio $\frac{A}{B}$ and if there exists an FPRAS that approximates the numerator and the denominator separately, that is, $\exists A_{fpras},B_{fpras}$: $Pr(A(1-\epsilon)\le ...
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1answer
40 views

Is there a standard way to obtain an approximation piecewise-linear function for a function

I am trying to find a generic way to get an approximation function for a given function. (I will be doing it programmatically eventually). What I want to obtain is a set of pairs, mapping the x-axis ...
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Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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Approximate $1 + (c^2 - 1) e^{-x^2/2}$ with a power law

Take the function in the title, and plot it in log-log scale with $c = 10^{-4}$: It is clear that besides the initial plateau there is a range where it is possible to approximate the function with a ...
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Approximating a function in an integral equation

Let we have the values of the functions $F(t), H(t)$, defined by the following equations, at a finite set of points $t\in \{t_0, \cdots, t_n \} \subset[a,b]$, $$F(t)=t.\int_0^\infty ...
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1answer
33 views

Lower Bound for Bernstein Approximation

If have to do solve the following problem: Let $f(t) = | t - \frac{1}{2} |$ be defined on $[0,1]$ and let $B_n ( t )$ denote the $n$-th Bernsteinpolynomial for $f$, i.e. $$ B_n ( t ) = \sum_{i = 0}^n ...
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176 views

Atan2 Faster Approximation

I am using atan2(y, x) for finding the polar angle from the x-axis and a vector which contains the point (x,y) for converting Cartesian coordinates to polar coordinates. But, in my program which will ...
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Approximating a continuous function with an exponential sum [closed]

For any continuous, real-valued function in the interval $[-R,R]$, show that it can be approximated by a function of the form $h(x) = \sum_{n=0}^N a_n e^{\nu_n x},$ where $\nu_n \in \mathbb{Z}$, with ...
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47 views

Using Galerkin's method to find an approximate solution

How we can use Galerkin's method to find an approximate solution of \begin{align} x''(t)+ tx(t) &= 1, \\ x(0) &= x(1) = 0, \end{align} using $t(1-t)$ and $t^2(1-t)$ as expansion function? ...
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39 views

How to prove that the subset of polynomials $A=\{\sum_{k=0}^n{a_kx^{2k}}:a_i\in\mathbb{R},n\in\mathbb{N} \}$ is dense in $C[0,1]$

I have $$A=\left\{\sum_{k=0}^n{a_kx^{2k}}:a_i\in\mathbb{R},n\in\mathbb{N} \right\}$$ and I have to prove that A is dense in $C[0,1]$ with respect to the supremum norm. My efforts in trying to solve ...
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Existence of smooth function using Runge's theorem

Let $\Omega$ be a domain in $\mathbb{C}$ and let $h\in C^{\infty}\left(\Omega\right)$. Show that there exist $u\in C^{\infty}\left(\Omega\right)$ such that $u_{z\bar z}=h$, and if ...