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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Proving uniform approximation by polynomials when sets are not compact

Here are two problems of the same flavor (and hence I posted them simultaneously) based on the Stone-Weierstrass Approximation Theorem. Let $f$ be continuous on $[1,\infty)$ with ...
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Proving Weierstrass Approximation Theorem on just bounded sets in $\mathbb R$

Suppose $f$ is a continuous function on $\mathbb R$. Show that we can approximate $f$ uniformly by a sequence of polynomials on any bounded subset of $\mathbb R$. My attempt is as follows: ...
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Uniform Convergence of a sequence of polynomials to $e^x$

Show that there exists no sequence of polynomials $P_n(x)$ converging to $e^x$ on $\mathbb R$ uniformly. This is pretty standard but I have come up with a proof of my own, and have not gone ...
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$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

I'm trying to understand a proof in "Interpolation and Approximation by Polynomials" by Phillips. Let me quote (page 253): "For $k\geq 1$ we begin with $$B_{n+k}^{(k)}(f;x)=\frac{(n+k)!}{n!} ...
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show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
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error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize ...
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40 views

Problem with a proof of Mittag-Leffler theorem

I've been going through Rudin's Real and Complex Analysis (3rd edition) but I got somehow stuck at the proof of Mittag-Lefler theorem (Theorem 13.10, page 273). The problem is I can't see why Theorem ...
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properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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48 views

Stone Weierstrass and Runge

Suppose $E(closed)\subset\{z:|z|=1\}$ and let $f(z)$ be a continuous function on the set $E$. I want to show that $f(z)$ can be approximated by polynomials on $E$. I am not exactly sure how to solve ...
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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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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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1answer
44 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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65 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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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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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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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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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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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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54 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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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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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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79 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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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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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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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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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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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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32 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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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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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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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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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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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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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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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 ...