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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Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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28 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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1answer
56 views

Approximating nice functions with wild ones

Let $X$ and $Y$ be toplogical spaces, and call a function $f:X\to Y$ wild if the preimage $f^{-1}(\{y\})$ is dense in $X$ for every $y\in Y$ -- or, equivalently, if the image of every nonempty open ...
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1answer
23 views

How to solve $ \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0$

How to find a solution to the following equation \begin{align*} \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0 \end{align*} where $c$ and $A$ are some constants such that $c\ge 1$ ...
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30 views

Trace zero not needed for $H^2$ regularity if $V_N\subset H^2$ is finite dim?

Reading Evans and this note after asking this question, I have been thinking about the estimates for interior/global regularity in Evans, 6.3.1, theorem 1, and thoerem 4 in 6.3.2, of the form ...
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23 views

Is there any objective criterion to determine the adequate degree of polynomial approximations?

I carry out a polynomial function approximation to a load duration curve (monotone decreasing function). I can approximate the load duration curve by using polynomial degrees from 4 to 12. I need an ...
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23 views

Least Square Approximation Using Legendre Polynomials

Obtain a fourth degree least squares polynomial for $f(x) = 1/|x|$ over $[-1,1]$ by means of Legendre Polynomials I got stuck when trying the integral over the given interval. Is there another way ...
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28 views

Polynomial approximation for a function implies an approximation to its square root?

Assume $f:[0,1] \rightarrow \mathbb{R}$ satisfies $f(t)\geq 0, f(0)=0$ I am looking for a machinery, which given a polynomial approximation of $f$ of a certain degree, determines the highest order ...
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18 views

Normal approximation to Poisson-binomial and beta-binomial distributions

I am looking for normal approximations (preferably without any extra expansion terms) for Poisson-binomial and beta-binomial distributions. In particular, something better than the standard normal ...
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52 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
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52 views

Can the triangle function approximate the Gaussian curve for complex numbers?

I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this: their respective functions are: $$ y_1(x) = t(x) = max(0, 1 - |x|)$$ $$ y_2(x) = e^{ - ...
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1answer
9 views

Degree theorem for Runge's approximating rational functions

Suppose that $f$ is analytic on an open set $D\subset\mathbb{C}$, and one uses Runge's theorem to obtain a sequence of rational functions $\{r_n\}$ which approach $f$ uniformly on compact subsets of ...
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1answer
29 views

Approximations using Poisson and Binomial distributions

The probability of a bit error in a communication line is $10^{-5}$ per bit. Suppose we examine a string of $1000$ independent bits. Calculate the probability of $0$, $1$, $2$, and $3$ errors in the ...
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16 views

Confused about Landau-notation and inequality

Let $f$ be a real valued function and $|f(x)| \le x^2\cdot C + o(x^3)$ as $x\to 0$, where $C \ge 0$ is a constant independent of x. Is it true that there is a $x_0$ such that for all $x\in [0,x_0]$ ...
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1answer
51 views

Closed Form Solution for Minimization involving Standard Normal CDF and PDF

Could someone please advice and provide detailed steps regarding any possible closed form solutions or other suggestions regarding solving a minimization problem of the type shown below? Here, $\phi$ ...
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46 views

Solve constrained system of linear equations from samples of a reference function

I have a system of $2n$ linear equations in $2n$ unknowns represented by the standard matrix equation: $$Ax = b$$ Where the solution vector $x = (p_1, ..., p_n, q_1, ..., q_n)$ represents real ...
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55 views

Weierstrass theorem and necessary condition

This is $(7.36)$ exercise of Hewitt & Stromberg - Real and Abstract analysis and i can't figure out the construction. Let $X$ be any noncompact subset of $\mathbb{R}$. Find a separating family ...
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21 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
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25 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise in every point where $f$ is continuous if $f:[0,1]\to \mathbb ...
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1answer
70 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
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1answer
35 views

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge's theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic ...
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25 views

Approximation of Sobolev functions by Polynomials

If $\Omega$ is star-shaped with respect to a ball, then Dupont and Scott show in "Polynomial approximation of functions in Sobolev spaces" that $$ \inf_{p\in \pi_{k-1}(\Omega)}\|u-p\|_{L^\infty}\leq ...
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25 views

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge's theorem (having poles in some prescribed set $A$). For ...
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27 views

Image by continuous function of approximation

If $g_{h}$ is an approximation of order $k$ of $g$, if $f$ is continuous (not linear), what can we say in general about the order of the approximation $f(g_h)$ of $f(g)$ ?
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25 views

Order approximation for rational polynomial

I have this fraction: $\frac{(-12a^3)d^3 + (4wa^3 - 16a^2)d^2 + (5wa^2 - 8a)d - a^2w^2 + 2aw - 1}{(- 12wa^4 + 12a^3)d^3 + (4a^4w^2 - 20a^3w + 16a^2)d^2 + (4a^3w^2 - 11a^2w + 7a)d + a^2w^2 - 2aw + 1}$ ...
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22 views

Linearization around an equilibrium point.

I am trying to understand linearization around an equilibrium point. This is new to me. So I would like to 'see' how it works basically and see how important it is to choose a right equilibrium point ...
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1answer
32 views

approximation of $x^2$ in hilbert spaces

use the least squares to find the best linear approximation to $f(x)=x^2$ on [-1,1]. that is find the line $y=a_0+a_1x$ that minimizes $\int_{-1}^1|f(x)-y(x)|^2$ solution I used the theory of ...
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55 views

Is there a meaningful way to approximate a discrete random variable?

Is there a meaningful way to find a continuos approximation of a discrete random variable? Thoughts for the $L^2$ case If $X \in L^2$, then we may want to consider the subspace $V = C^1 \cap L^2$ ...
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22 views

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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33 views

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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12 views

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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34 views

$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

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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106 views

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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1answer
15 views

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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1answer
110 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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1answer
54 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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43 views

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
56 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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3answers
124 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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1answer
25 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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25 views

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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41 views

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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23 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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1answer
37 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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1answer
32 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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53 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
74 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
31 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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2answers
156 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
95 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 ...