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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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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27 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] ...
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+100

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 ...
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1answer
21 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 ...
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84 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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1answer
78 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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33 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 ...
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72 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 ...
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48 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 ...
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1answer
52 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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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
40 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
56 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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29 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
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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43 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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27 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 ...
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33 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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36 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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167 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
30 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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42 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
42 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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16 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
46 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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36 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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47 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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24 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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44 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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44 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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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
32 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
64 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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32 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
51 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
54 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
72 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
49 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
49 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 ...