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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79 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ ...
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21 views

fredholm integral equation endpoint singularity

I am trying to solve numerically the following: $$ f(x)=\int_0^1\dfrac{1+tx}{(1-tx)^3}f(t)dt$$ Are there any quadrature rule or any other method to handle this singularity? Plz suggest any expansion ...
0
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1answer
32 views

Looking for methods for approximating an iterative equation regarding primes

In a previous question, I was looking for an equation for counting the number of the number of integers between $1$ and $x$ that have a prime factor besides $2$ or $3$. There were 2 iterative ...
2
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1answer
36 views

example of convergence to exponential type of function

I want to prove the existence of $n \in \mathbb{Z}^+$ and $a_0,a_1,\dots,a_n \in \mathbb{R}$ such that $$\bigg| \left( \sum_{k=0}^n \dfrac{a_k}{x^k} \right) - \exp \left( \dfrac{\sin(e^x)}{\sqrt{x}} ...
2
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1answer
30 views

Natural cubic spline interpolation error estimate

I am looking for an error estimation for natural (one with $s''(a) = s''(b) = 0$ boundary conditions) cubic spline interpolation on an evenly spaced grid. The best result I've found was $O(h^2)$ ...
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0answers
18 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
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0answers
11 views

Approximation of ordering functions

I am looking for smooth approximations of the functions $f_k^n:[0,1]^n\mapsto [0,1]$, $k=1,..,n$ ordering the entries of vectors $x=(x_1,..,x_n) \in [0,1]^n$, in the sense that $f_k^n(x)$ yields the ...
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2answers
131 views

Miklos Schweitzer 2014 Problem 8: polynomial inequality

Look at problem 8 : Let $n\geq 1$ be a fixed integer. Calculate the distance: $$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over polynomials with degree less than $n$ with real ...
0
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2answers
39 views

Weierstrass Approximation Theorem

Does it matter what the interval is in the Weierstrass Approximation Theorem? Is it possible that the interval be any possible numbers within the function f(x)? How much the interval matter?
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27 views

Additive and Multiplicative Error in $n!$ Approximation

Let $S(n)=\sqrt{2\pi n}\big(\frac{n}{e}\big)^n$ be the approximation of interest to $n!$. What are good lower and upper bounds on the following two functions $$(1)\mbox{ }|S(n)-n!|?$$ $$(2)\mbox{ ...
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8 views

Strategies for approximating fourier transform of $k$-th power of the $n$-th derivative of a function

For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I ...
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2answers
48 views

differentiability of $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$

Let $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$ if $(x,y)\neq (0,0)$ and $0$ if $(x,y)=(0,0)$. Determine the points in which $f$ is differentiable I know that $f(x,y)$ is differentiable at ...
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5answers
93 views

Approximating $\log x$ with roots

The following is a surprisingly good (and simple!) approximation for $\log x+1$ in the region $(-1,1)$: $$\log (x+1) \approx \frac{x}{\sqrt{x+1}}$$ Three questions: Is there a good reason why this ...
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0answers
54 views

Uniqueness of the best linear approximation

Let $f:D\subset \mathbb R^2 \to \mathbb R$ where $D=${$(x,y)\in \mathbb R^2: y=x$}, $(x_0,y_0)\in D$ and $f(x,y)=x$. Let $l_1, l_2, l_3 : \mathbb R^2 \to \mathbb R$ linear functions so that ...
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15 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...
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31 views

Approximating a set of convex quadratic inequalities by a convex polytope

I have a convex set of the form $$Z = \{x|x^TQ_ix+b_i^Tx+c_i\le0,i=1,\ldots,m\}$$ where each $Q_i\succeq0$, that I wish to approximate by a set of the form $$\hat Z = \{x|Ax\le b\}$$ We can further ...
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36 views

Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
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1answer
28 views

When is it not safe to apply the approximation (1+a)^N = 1+Na (for a<<1)?

For example, consider the following equation where $|k|<<1$, $N$ is a positive integer ranging from 1 to 100,000) and $k,N$ are both real. $$ \left(\frac{1+k}{k} \right)\left[(1+k)^N-1 \right] ...
3
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84 views

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
0
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1answer
30 views

Bounding error of Padé approximation

I'm trying to understand how one would understand the error of a given Padé approximation for a function. For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a ...
3
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1answer
93 views

Best approximation for a normed vector space $X$

I am self-studying functional analysis. As far as I know, a best approximation of $X$ by a closed subspace $C \subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach ...
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106 views

Approximation of a $L^1$ function by a dominated sequence of continuous functions

Consider $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$ and the Lebesgue measure on it. Denote by $L^1(\mathbb{T})$ the set of integrable functions on $\mathbb{T}$ and by $C(\mathbb{T})$ the set of ...
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1answer
34 views

How to interpolate a function with Gaussian functions?

A function $f:[a,b]\to \mathbb{R}$ is given (we can assume it is continuous or differentiable) and we want to interpolate it with Gaussian functions $$\phi_i(x) = e^{-\alpha(x-x_i)^2}\text{,}$$ where ...
3
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1answer
79 views

Reference request: approximation theory

I am somewhat intrigued by the ideas behind approximation theory. So, I would like to know (1) what are some thorough clear reference books to get acquainted with approximation theory; (2) what are ...
2
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1answer
50 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
3
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1answer
57 views

Can we find a real number $k$ such that the fractional part of sequence $k^n(n\in N)$ monotonically increasing?

Can we find a real number $k$ such that the fractional part of sequence $k^n(n\in N)$ monotonically increasing? This link may be helpful , it says that Hardy and Littlewood have shown that ...
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0answers
22 views

Estimating upper bound

Let the following Cauchy Problem be $\displaystyle\cases{ y'(t)=f(t,y(t)) & \cr y(0)=\eta }$ for $t\in[0,T]$ Define the approximation $y_n$ of $y(t_n)$ as: ...
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1answer
52 views

how to find the best polynomial approximation to discontinuous function on floating point domain

I have several discontinuous functions on $f : \mathbb{R} -> \mathbb{R}$ for which I wish to find the best/Chebychev/minimax polynomial approximations in some interval $[a,b]$ for use in a computer ...
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1answer
65 views

Explicit formula for the implicit Euler method

Given the problem; $\displaystyle\cases{ y'(t)=y^2(t) & \cr y(0)=1 }$ for $t\in[0,1]$ Using the implicit euler method, find an explicit formula to get $y_{n+1}$ HINT: The ...
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38 views

Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
0
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1answer
22 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
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44 views

Guess a property of the integral average value function

Let $f$ be a function that is defined on $[a,b]$ and integrable on $[a,b]$. Def1. $$\hat f(x)=\begin{cases} f(x),&x\in[a,b], \\ f(a),&x<a, \\ f(b),&x>b, \end{cases}$$ ...
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7 views

Approximation of $C^1$ by $C^1_b$

Can we approximate a function which is $C^1$ with functions that are $C^1$ with bounded first derivative? Thank you in advance.
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30 views

Integral Inequality bounded by $\sup \{|f''(x)|-f''(y)| , |x-y|\leq r\}$

Let $f \in C^2 [a,b]$. Define $$\omega_2(r)= \sup \{||f''(x)|-|f''(y)|| \, : \, |x-y|\leq r\}$$ we can prove that $\omega_2(r)$ is continuous. The following lemma is given in Convergence rates of ...
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40 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
1
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1answer
37 views

Kolmogorov n-width of N+1 dimensional ball

For a normed linear space $\mathscr{X}$, let $\mathscr{A}\subset\mathscr{X}$ and $\mathscr{X}_N$ any $N$-dimensional subspace of $\mathscr{X}$. Define the $n$-width of $ \mathscr{A}$ in $\mathscr{X}$ ...
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30 views

Can the Fourier Series be made “ Shorter ”?

I have tried to give only the intuitive part of my question and haven't included many specific details. Please help me frame it more precisely. I have inluded the symbol (*) where I need more details. ...
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13 views

Decomposing a matrix into a finite set of elements

I'm looking to approximate real, symmetric matrix $\mathbf{A}$ of size $N$ with a unique restriction. First, it is known that the matrix has zero mean $\left < A_{ij} \right>=0$. Next, it is ...
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1answer
40 views

Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
10
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1answer
169 views

Function for which trapezoidal rule outperforms midpoint rule for every $n$

Is there a continuous elementary function $f:[0,1]\to [0,\infty)$ such that for every $n$ the trapezoidal approximation to $\int_{0}^{1}f(x)\,dx$ with $n$ trapezoids is strictly better than the ...
0
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1answer
32 views

Bounding an approximation error of Bernstein polynomials

I have to show that for $ 0 \leq x \leq 1 $ \begin{align*}\sum|f(x)-f(k/n)|p_{nk}(x)\leq(2M/\delta^2)\sum_{k=0}^n (x-k/n)^2p_{nk}(x)\end{align*} Important to know is that $p_n(x)$ and $p_{nk}(x)$ are ...
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20 views

Can a logistic function(x+y) be approximately factored into $f_1(x)$ and $f_2(x)$?

I need to somehow factor the logistic function $$\frac{1}{1+e^{-(\theta-\beta)}}$$ into $f_1(\theta)$$f_2(\beta)$ approximately... namely $\frac{1}{1+e^{-(\theta-\beta)}} \approx ...
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0answers
43 views

Stability of transfer functions with internal delay

I would like to know what the best method is for finding stability of transfer functions that have internal delays. Basically I have a transfer function of the form: $\frac{f(s) e^{-st}}{g(s) + h(s) ...
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1answer
43 views

Appoximation of Lipschitz functions by $C^1-$functions

I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
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24 views

How do I approximate $f''(x)+(E-U(x))f(x)=0$ for a piecewise $U$ and find $E$?

I am trying to approximate the solution to the equation $f''(x)+(E-U(x))f(x)=0$ where $U(x) = \begin{cases} \frac{U_0}{m}x-U_0 & \text{for $-m<x<0$} \\ \frac{-U_0}{m}x-U_0 & ...
0
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0answers
18 views

Integral of Hypergeometric Function with polynomial, power, exponential and logarithm function

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it?
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1answer
53 views

Find a polynomial using minimax approximation

Find a polynomial with the maximum 1. degree which best approximates the $f(x)=e^x$ function in terms of minimax approximation in $[0,1]$.
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37 views

What does “*the best approximation $A_{a}(x)$ of a function $f(x)$*” mean?

I have a general question. What does "the best approximation $A_{a}(x)$ of a function $f(x)$" mean? $A_{a}(x)$ stands for the approximation of $f$ at $a$ and then you get a function of $x$ which you ...
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2answers
48 views

Proof that the best linear approximation to $f(x)$ near $a$ is given by the linear function $L_{a}(x) = f(a) + f '(a)(x-a)$

The title basically says everything. The formula for linear approximation appears to be right intuitively but is there a proof for it? Secondly, is there also a proof why to put the $\frac{1}{2}$ in ...
2
votes
1answer
25 views

An abstract a priori estimate in finite element method

Let $V$ and $K$ be Banach spaces (with norms $\|\cdot\|_V$ and $\|\cdot\|_K$ resp.) and suppose that there is a compact linear embedding $K\hookrightarrow V$. Furthermore, let $P_n$ be a family of ...