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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Pi Appoximation: Simpler Solution to Limit?

May be a ridiculous question, but I wanted to see if MSE had "simpler" proofs for Viete's approximation (specifically, using an equation derived from Viete's formula) of $\pi$: $$\lim_{x \to \infty}2^...
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20 views

Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
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34 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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116 views

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number ...
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32 views

The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $ 4x^3-3x = \cos \alpha $. Now, by expressing $ x $ as $ x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
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263 views

A curious algebraic fraction that converges to $\frac{\sqrt{2}}{2}$

I have noticed that the algebraic fraction $\frac{3a+2b}{4a+3b} $ Gives better and better approximations to $\sin 45^\circ = \frac{\sqrt{2}}{2} $ For $ a = b = 1$ we get $5/7 \approx 0.714 $ ...
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34 views

Approximating Minkowski Sum of 3 dimensional Convex Polytopes by Sampling

Let $P_1,P_2...P_r$ be a set of convex polytopes with $n_r$ vertices in 3 dimensions. These polytopes basically represent uncertainties of '$r$' number of 3d-points respectively in space. The global ...
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22 views

Refinement of the trapzoid rule

Suppose that $f:[y,x]\to\mathbf{C}$ is a continuously three times differentiable function, where $y<x$ are integers. Show that $$\sum_{y\leq n\leq x} f(n)=\int_y^x f(t)\,dt+\frac{1}{2}f(x)+\frac{...
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1answer
63 views

A curious approximation to $\cos (\alpha/3)$

The following curious approximation $\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$ is accurate for an angle $\alpha$ between $0^\circ$ ...
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43 views

Separation of integral by approximation

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x \sqrt{...
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1answer
18 views

Approximation of ODE [duplicate]

A problem I should solve is the following IVP: I need to find an approximation for small $\varepsilon$ of $$y'' + y + \varepsilon y^3 = 0, y(0) = 1, y'(0)=0$$ My approach so far was: I assume that I ...
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1answer
10 views

Does monomials approximate the constant function in the sense of $L^2$?

I'm trying to show that $\{x,x^2,x^3,\dots\}$ approximate the constant function in the sense of $L^2[0,1]$-convergence; i.e. that there exists a sequence of polynomials $p_n$ with $p_n(0) = 0$ such ...
3
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1answer
102 views

Close approximation for absolute value function

I made a very acurate approximation function for $\sqrt{n^{2}+1}$ It is $\sqrt{n^{2}+1}\approx\frac{2n(n^{2}+1)}{2n^{2}+1}+\frac{2n^{2}+1}{n(4(2n^{2}+1)^{2}+1)}$ From this I can make a very close ...
2
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1answer
72 views

Function Approximation by Wavelet method

If $ f(x)\in{L^{2}(R)}$ , then $ f(x) $ can be projected into the subspace $ V_{j}$ as, $$ P_{j}f(x) = \sum\limits_{k\in{Z}}c_{j,k}\phi_{j,k}(x) $$ where $ k\in{Z}$. The projection equation is ...
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41 views

Algebraic approximation to differential equation

Suppose I have a first order differential equation of the form: $\frac{dx}{dt} = \frac{1}{\tau}f(x,t)$ where $f(x,t)$ is a nonlinear function. In the limit where the time constant $\tau$ is small, ...
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128 views

A generalization of an integral related with $\zeta(2)$

It is pretty well-known (and not difficult to prove) that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = \int_{0}^...
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1answer
44 views

Linearspan of Gaussians dense in Schwartz space

as the title already says I am trying to show that the linear span "A" of the gaussians $e^{\frac{-|x|^2}{2}}$ and their translations/ dilations are dense in the Schwartzspace. This is the space of ...
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30 views

Finding the inverse of a function involving logarithms

Let $A \asymp B$ mean that there exists universal constants $m,M >0$ such that $mA \leq B \leq MA$. Let $k,n \in \mathbb{N}$ be such that $\log n \leq k \leq n$. I want to prove that $$ k \log(\...
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1answer
53 views

Runge's Approximation Theorem

This is a homework problem, so my apologies for a seeming lack of motivation. I want to understand what's going on in this problem more than I want someone to just write down the solution, so if ...
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24 views

Approximation of a continuous bounded function on $\mathcal{P}_2(\mathbb{R}^d)$ by Lipschitz functions

Hello everyone I am currently struggling with the following problem. Consider a bounded, measurable and continuous function $f: \mathcal{P}_2(\mathbb{R}^d) \rightarrow \mathbb{R}$ where $\mathcal{P}_2(...
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1answer
21 views

How to solve this question using approximation theory?

I am asked to find the first three terms in the taylor series of the function $$ f(x)=(x-1)\ln x $$ around $x_0=0$. Then to find the maximum error in my approximation in the interval $[0.5,1.5]$. ...
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47 views

Prove that $|x|$ can be uniformly approximated in [-1,1] by polynomials?

I know how to aproximate it, is it enough to actually aproximate it to prove it can be aproximated? I just noticed there is a suggestion here at the end. Sorry for lateness Do Taylor arround the ...
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1answer
31 views

Bound for the max value of a Lagrange polynomial

Given some Lagrange polynomial $L$(x) that interpolates over the points $x$0, $x$1,..., $x$n with values in the set $A$ = {$a$0, $a$1,..., $a$n } on some interval [a,b], show that the max value that $...
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1answer
16 views

Does a closed subalgebra of $C(X)$ which separates points and vanishes nowhere contain 1?

Let $X$ be a compact Hausdorff space and $C(X)$ be the real valued continuous functions on $X$. Suppose that $\mathcal A$ is a (topologically) closed subalgebra of $C(X)$. Is it true that if $\mathcal ...
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72 views

Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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1answer
93 views

If a function is odd/even, then its best polynomial approximation is also odd/even. [duplicate]

If $f$ ∈ $C([a,b])$ is an even/odd function, then show that the best approximation among the polynomials of degree n is also even/odd. I'm almost certain that to show this directly, you should ...
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21 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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33 views

Given n points, show there is a unique polynomial through all of them

if x0, x1,..., xn are distinct points in [a,b] and $A$ = { a0, a1,..., an } ∈ Rn+1, show there is a unique polynomial $p$A of degree at most "n" such that $p$A(xj) = aj for each "j". Then, show ...
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29 views

Slope of linear approximation to a function?

Assume $f$ ∈ $C([a, b])$ is twice continuously differentiable and $f$ ''(x) > 0 on [a, b]. Show that the best linear approximation (polynomial of degree one) $p$ to $f$ has the slope $p$'(x) ...
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21 views

How well are certain functions approximated by simple functions?

Let us restrict ourselves to functions on the unit circle. It is well known that any function on $L^2(S^1)$ can be approximated arbitrarily well using simple functions. Now suppose $f \in L^2(S^1)$ ...
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A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence $\...
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1answer
32 views

Richardson extrapolation - deriving methods for forward difference

I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as $$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ...
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21 views

How to obtain an analytic approximation of this wavelet-looking function?

I would like the approximating function to be infinitely differentiable. I have tried fitting (1) a polynomial and (2) a combination of sin, cos, and exp functions to approximate the function shown ...
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Approximating polynomials over $\mathbb{C}$ with an entire function

Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n \...
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42 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
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31 views

How could I solve this equation: $ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1} $ for $x$?

I want to have a solution for $x$ in this equation. $$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$ Thanks !
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Is the Preimage of an Open Disk a Continuous Action

Let the kernel-like operator, $K$, be defined as $K(f,R) = \text{Closure}({\{x; \lvert f(x)\rvert < \frac{1}{R}\})}$ where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function on the complex ...
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3answers
50 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
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14 views

Finite element approximation in semi-norm $|\dot|_{H^1}$ and $L^2$

Let $\Omega$ be a polyhedral domain in $\mathbb{R}^n$. Consider the Poisson equation $-\Delta u=f$ in $\Omega$ subject to the following non-homogeneous boundary condition $u= g_D$ on $\Gamma_D$; $\...
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36 views

Integral involving double exponentials

I am trying to estimate the growth in time of a certain quantity. In order to do this, I have to evaluate the following integral: $\int_0^{+\infty} \exp(-b x)\exp(-c\,t \exp(-a x \log x)) dx$ with $...
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40 views

Validity of an Approximation

I am considering the approximation of the following integral: $$\int_{-W/2}^{W/2}\exp[-p( x + iq)^2 ] dx \approx \int_{-\infty}^{\infty}\exp[-p( x + iq)^2 ] dx = \sqrt{\frac{\pi}{p}}$$ For large ...
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1answer
14 views

Higher Order Polynomial Interpolation

I am trying to approximate some log and exp functions in my code. I have implemented linear and cubic splines, but I want more accuracy. I am thinking about biquadratic splines (4th order, quartic), ...
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Uniform approximation by even polynomial

Proposition Let $\mathcal{P_e}$ be the set of functions $p_e(x) = a_o + a_2x^2 + \cdots + a_{2n}x^{2n}$, $p_e : \mathbb{R} \to \mathbb{R}$ Show that all $f:[0,1]\to\mathbb{R}$ can be ...
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16 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
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29 views

Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and $...
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19 views

Show that the Bernstein operator is not a projection

I'm currently trying to show that the Bernstein operator is not a projection, but I can't find a good counter-example to show that it's not a projection. I was thinking about starting with some ...
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13 views

are there any other properties like sub-modularity which can help compute the maximization problem of a set function

Now I have a set function $f$ (it's monotone), which inputs a node set $S$, and output a real value $r=f(S)$. And I want to compute $$ \max_{S,|S|\leq k}f(S). $$ where k is given number. And I know ...
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65 views

Calculate integral curve through a discretely sampled vector field

given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
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1answer
46 views

Stronger Whitney approximation for smooth manifolds

If $M$ and $N$ are smooth manifolds, then any continuous map $f:M\rightarrow N$ is homotopic to a smooth map $g:M\rightarrow N$. If $f$ is smooth on a closed set $K\subseteq M$, the homotopy can be ...