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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9 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
27 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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14 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 ...
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1answer
20 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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42 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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6 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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23 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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32 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 ...
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1answer
27 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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27 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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12 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
35 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 ...
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162 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 ...
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1answer
26 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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21 views

How to do a linear approximation with several parts

if i have partial data set of $ \langle x,y\rangle\in \mathbb{R}$ for a given function $f$, and i want to approximate it by $n$ partial linear functions how would i calculate those linear functions, ...
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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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38 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
38 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 & ...
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12 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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38 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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33 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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42 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 ...
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1answer
23 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 ...
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5 views

Are there other names for multilayer perceptrons or multidimensional interpolants based on Kolmogorov's approximation work?

Are there other names for multilayer perceptrons that are used outside of the neural net community? At its core, multilayer perceptrons form a multidimensional interpolant of the form $$ ...
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23 views

min-max polynomial approximation with sign definite error

I am looking for an algorithm for approximating a function $f(x)$ on a finite interval by a (generalized) polynomial, $p(x)$ such that p(x) is greater than equal to $f(x)$ the error ...
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24 views

Is there a nice way to solve this Ramanujan-like approximation?

I have managed to approximate $l=\frac{2 \sqrt{4 \pi ^2 A^2+W^2} E(\epsilon )}{\pi }$ with parameter $\epsilon =k=\frac{2 \pi A}{\sqrt{4 \pi ^2 A^2+W^2}}$ alike Ramanujan as $l \approx ...
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40 views

Archimedes' Apprxomation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
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1answer
36 views

Proof of Weierstrass' second theorem using the Fejér operator

Weierstrass' second theorem states the following: Let $f$ be a real continuous $2\pi$-periodic function (write $f\in C_{2\pi}$). Then for all $\epsilon>0$ there exists a trigonometric polynomial ...
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1answer
17 views

Solving a statistics equation

Suppose $X$ is a random variable which follows a Poisson distribution, such that, for some positive integer $m$, $$X \sim Po(0.01m)$$ Find the least value of $m$ such that $$P(X \ge 1) > 0.9$$ ...
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1answer
31 views

Strict convexity and best approximations

Let $V$ be a normed vector space. It is said to be strictly convex if its unit sphere does not contain nontrivial segments. A subset $A \subset V$ is said to have the unicity property if for any $x ...
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26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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1answer
60 views

Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
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1answer
30 views

Given $u \in L^1$, is there approximating sequence $u_n \in L^\infty$ uniformly bounded in $L^p$?

Let $u \in L^1(U)$ where $U$ is a bounded domain. Is it possible to find a sequence $u_n \in L^\infty $ converging to $u$ in $L^1$ such that the $u_n$ are uniformly bounded for all $n$ in some $L^p$ ...
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1answer
48 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
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1answer
51 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
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1answer
65 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
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1answer
48 views

Logistic function approximation of the real valued Riemann $\zeta(x)$ function

Given the function: $$f(x)=\dfrac{a}{1-b\exp(-cx)}+d$$ where: $a = 0.7071$, $b = 2.21$, $c = 0.7672$, $d = 0.2942$, I found the following inequality: $$|\zeta(x) - f(x)|\lt \epsilon$$ for ...
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1answer
40 views

Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous ...
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43 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
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27 views

the best approximate element

$X$ denotes the set $\{ f \in C[-1,1] | f ~ is~ continuous~ differentiable~ on~ [0,1]\}$, $Y$ denotes $\{ f \in X | f ~satisfies~ $f '(t) = f(t-1)$~ on~ [0,1]\}$. For any $f\in X$, does there ...
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13 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
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50 views

Upper bound or approximate form for the CDF of Hypo exponential random variable

The CDF of hypo exponential random variable (sum of $n$ independent exponential random variables $X_{i} $ with different rates $\lambda_{i}$) is given by I seek for an upper bound or an approximate ...
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2answers
72 views

$(1-x)^y ≈ e^{-xy}$

Here is an approximation I often see in biology articles but don't really understand: $$(1-x)^y ≈ e^{-xy}$$ I think this $e^{-xy}$ closely approximates $(1-x)^y$ whenever $x$ is small. Can you help ...
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1answer
247 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
3
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1answer
25 views

Show that a subset $V \subseteq C[a,b]$ is a Haar subspace

Let $C[a,b]$ be the set of continuous functions on $[a,b]$, then a linear subspace $V \subseteq C[a,b]$ of finite dimension $n+1$ is called an Haar subspace iff one of the following equivalent ...
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52 views

Cebysev (Tschebyscheff-) Approximation

I have to find the best Tschebyscheff-Approximation of $x(t) = \sin(t)$ on the Interval $[0,d]$ by a line where $0 < d < 2\pi$. But I have no idea how to perform such an approximation, do you ...
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3answers
34 views

How to calculate the degree of Lagrange polynomial to satisfy a given error?

I need help. I have $f(x)=sin(x)$. If I want to use Lagrange polynomial to make an approximation of $f(x)$, what should be the degree of that polynomial if I work in the interval $[0,\pi]$, and the ...
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1answer
26 views

Complete orthonormal system

Let $H$ a linear space with inner product. An orthonormal system $\{e_1, e_2, \dots \}$ is called complete in $H$ if $x=0$ is the only element that satisfies the relations: $$(x, e_n)=0, \ \ \ ...
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29 views

Solve Van der Pol equation by Padé approximation

I want to solve the Van der Pol equation: $$f''+ \mu \, (f^2-1)f'+f=0, \quad f = f(t),$$ by Padé approximation. I know the solution should be the combination of $\sin{t}$ and multiplied by $\mu$, ...