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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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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18 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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33 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
25 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
16 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
28 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
51 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
29 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
37 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
49 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
64 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
45 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
27 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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38 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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26 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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0answers
35 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
69 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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238 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 ? ...
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1answer
24 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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2answers
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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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
24 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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26 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 $\mu$, ...
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1answer
23 views

Approximation theory and proximinal sets

The question is to give an example such that the finite union of proximinal sets is not proximinal. I have no idea to construct any example to suit this problem, will anybody help me?
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1answer
37 views

Smooth function and mollifiers with $\mathcal{C}^{\infty}$ continuation to a given function

We consider a function $f : [0 ; 1] \mapsto \mathbb{R}$, which is $\mathcal{C}^{\infty}$ on the open interval, and positive. For example $f(x) = 1 + x$. ($f$ is a nice function, not a nasty beast.) I ...
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1answer
28 views

Discrete Function Approximation Error - Which type? (Applied math, signals)

I have two functions, one derived via software, and we can call it the exact function, $f_{exact}$. The other is a result I got through hardware, and we can call it the approximation, $f_{approx}$. ...
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21 views

Collection of monotone functions on compact subinterval is closed in uniform norm

Let $C([a,b])$ be the space of continuous functions on $[a,b]$ in the uniform topology.Suppose that $[c,d]\subset[a,b].$ Show that $M=${$f\in C([a,b]):f$ is monotone on $[c,d]$} is a closed set. I ...
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2answers
52 views

Asymptotic approximation of binomial theorem

Binomial theorem is a very popular theorem that: $$(x + y) ^ n = \sum_{i=0}^n {n \choose i}x^i y^{n-i}$$ I am looking for any papers (the newer the better) where I can find any informations about ...
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1answer
27 views

Link between Chebyshev polynomials and best approximants

I'm reading Interpolation and Approximation by Davis, more specifically "Best Approximation" Chapter VII. Let $n \in \mathbb N$. Let $C[a,b]$ denote the set of continuous real functions over ...
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2answers
28 views

Multiplication of asymptotic approximation

If I know that: $a = (1 - O(\frac{1}{n}))$ and $b = (1 + O(\frac{1}{n}))$, what is the asymptotic approximation of $a\cdot b$? Is answer $ab = (1 - O(\frac{1}{n^2}))$ correct or it is still $ab = (1 - ...
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2answers
51 views

Using random numbers to evaluate $\sum_{k=1}^{10000}e^{\frac{k}{10000}}$

I tried using the Monte Carlo Method to approximate the sum $\sum_{1}^{10000}e^{\frac{k}{10000}}$. First I genarating 100 random numbers in (1, 10000). Then by the strong law of large numbers: ...
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1answer
71 views

Uniqueness of best approximation in strictly or uniformly convex normed linear spaces

I am studying approximation theory, just to learn basically, and in different books I saw a theorem about uniqueness of best approximation. One book uses strict, the other uses uniform convexity with ...
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1answer
34 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For ...
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0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
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2answers
50 views

Approximation of difference of harmonic numbers

Harmonic number $H_n$ is equal to $$H_n = \sum_{i=1}^n \frac{1}{i}$$ Asymptotic expansion of harmonic humber is $$(1) H_n = \ln n + \gamma + \frac{1}{2n} - O\left(\frac{1}{n^2}\right)$$. Very popular ...
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19 views

Operator norm convergence of interpolation operator

Let us denote with $S_h$ the piecewise linear finite element subspace of $H^1$ and with $I_h:H^1\rightarrow S_h$ a nodal interpolation operator defined as follows: For each node $p_i$ in the FE mesh ...
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18 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [duplicate]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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54 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ ...
2
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1answer
58 views

Proof of the theorem of optimal approximation!

I am looking at the proof of the following theorem: $$$$ Let $\widetilde{H}$ subspace of $H$,where $H$ is an Euclidean space, and $x \in H$. $y \in \widetilde{H}$ is the optimal approximation of $x$ ...
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34 views

Approximating a $C^1$ function by piecewise affine maps

Let $\Omega\subset\mathbb R^n$ be an open and bounded domain and let $f\in C^1(\bar\Omega,\mathbb R^m)$. I would like to approximate $f$ by a function $u:\bar\Omega\to\mathbb R^m$ that is piecewise ...
2
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1answer
32 views

Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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35 views

Approximating functions of two variables by polynomials

Let $f:[0,1]^2\rightarrow\mathbb{R}$ be a real-valued continuous function and $\epsilon>0$. Suppose there are integers $m_1,m_2,n_1,n_2\in\mathbb{N}$ and polynomials $p_{m_1,n_1}$ and $p_{m_2,n_2}$ ...
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1answer
92 views

Approximation of a function with a polynomial of degree n.

Let $f\in{L_{loc}^{1}}$ such that $\int_{a}^{b}{f(x)}{\phi}^{(n)}(x)dx=0$ for all $\phi\in{C_{0}^{\infty}}(a,b)$. Then how do we show there exists a polynomial $P(x)$ of degree less or equal to $n-1$ ...
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28 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme ...
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2answers
64 views

What is the probability that a student knows the answer given that he has answered it correctly? [closed]

A large class in stochastic processes at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer is ...
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1answer
27 views

Generalized Linear Least Squares

I've run across a problem which asks me to calculate a best fit line through data using a 'generalized linear least squares' approach where, instead of minimizing the residual: $\vec{r} = \vec{b} - ...
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3answers
250 views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
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0answers
32 views

Integral involving wavelet

Let $\hat{\psi}_m$ be a Fourier transform of Daubechies wavelet of order $m$ and $\chi_I$ is a characteristic function of interval $I$. How to bound from above the following integral $$ ...