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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56 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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41 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 ...
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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
9 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 ...
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1answer
100 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
65 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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39 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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117 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 = ...
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1answer
43 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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29 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 ...
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52 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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0answers
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 ...
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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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2answers
46 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
15 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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71 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
86 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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20 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
31 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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18 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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1answer
53 views

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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2answers
40 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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54 views

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
49 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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13 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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35 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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43 views

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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15 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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28 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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0answers
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 ...
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1answer
33 views

Approximation of $L^1$ function with compactly supported smooth function with same mass and same uniform bounds

Recently, I have asked this question. Now, I even want to make this better. Given $f\in L^1(\mathbb{R})$ with $0\leq f\leq 1$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such ...
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23 views

Sign of approximation error and remainder (residual)

In the Wikipedia article Taylor series it is said that: The error incurred in approximating a function by its $n$th-degree Taylor polynomial is called the remainder or residual and is denoted by ...
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1answer
26 views

Inverse problems with Graphical Approximation and Graphs

Suppose an inverse problem with graphical approximation for the system where only a small subset of system features are known hence undetermined scenario. The system can be represented by a graph. ...
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25 views

References on Inverse Problems, Approximation theory and Algebraic geometry

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined ...
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17 views

approximation of weakly differentriable bochner functions

Given a function $u\in L^2(0,T;H^1(\Omega))$ with $u_t\in L^2(0,T;(H^1(\Omega))^*)$. Can we approximate $u$ by functions $u^k$ with $$u^k=\sum\limits_{i=1}^{n(k)}c_i^k\phi_i^k,\text{ where } c_i^k\in ...
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16 views

Expansions and Approximations of Functions

When looking at series expansions of functions, the main cases I've worked with have been Taylor series and Fourier series. Whilst the latter can arguably be viewed as a subset of the former (e.g. ...
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54 views

If $f\in C[0,1]$ and $A\subset[0,1]$ is finite, can $f$ be approximated uniformly by polynomials that coincide with $f$ on $A$?

Let $f\colon[0,1]\to\mathbb{R}$ be continuous and $A$ a finite subset of $[0,1]$. Given $\epsilon>0$ is there a polynomial $p$ such that $$ |f(x)-p(x)|\le\epsilon\quad\forall ...
26
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422 views

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
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25 views

Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...