Questions tagged [approximation-theory]

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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Approximate formula from data (7 inputs, 1 output)

I'd like to approximate a formula which 'fits' my data. The formula should take 7 inputs and produce a single ouput. The input variables are listed in columns A to G of my data (see image below). The ...
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Approximation of dynamic systems

I was looking for a formal approach to simplify models of dynamic systems. Say we have a dynamic system given by $\frac{dx}{dt} = f(t,x,u), ~~~~x(t_0) = x_0$ $y = g(x)$ We know $f$ and $g$ but ...
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Zolotarev number and commuting matrices

Recently in a post (link) upper bounds on the singular values $\sigma_j(X)$ of a matrix $X$ have been considered. To restate the central observation, it says that if $AX−XB=F$ for $A$ and $B$ normal ...
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71 views

Show that there exists 2 different best approximations

Let $F$ be a linear space with a norm $\lVert \cdot \rVert$ which is not strictly convex. Show that there is a function $f \in F$ and a subspace $S \subset F$, so that $f$ has different best ...
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Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

I would like to use mathematical tools to prove that $$9.9998\lt \frac{\pi^9}{e^8}\lt 10$$ With an on-line calculator I got $$ \frac{\pi^9}{e^8}\approx 9.9998387978$$ But I do not know any ...
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119 views

Uniform approximation of second derivative via Bernstein polynomial

Let $f:[0,1]\mapsto(0, +\infty)$ be a continuous function. Define by $$ B_k(f,t):=\sum_{j=0}^k f(j/k) {k \choose j} t^{j} (1-t)^{k-j}, \quad t \in [0,1] $$ the associated Bernstein polynomial. For $...
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168 views

Prove that a set is not strictly convex

I want to prove that the set $\{ f:||f||_{\infty}=1 \}$ where $f$ belongs to the space of continuous functions on $[a,b]$ is not a strictly convex set. As a counterexample, I'm asked to use $f(x)=x$ ...
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Approximating $\frac{\frac{N}{2}!\frac{N}{2}!}{(\frac{N}{2}-m)!(\frac{N}{2}+m)!}$ without using logs

This question came up in a recent problem. It basically states: In the limit of $N\gt\gt m \gt\gt 1$ show that $$\frac{\frac{N}{2}!\frac{N}{2}!}{(\frac{N}{2}-m)!(\frac{N}{2}+m)!} \approx \exp\left({-\...
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approximate bijective function such that the inverses are bijective and “easily” computable

I have a infinitely differentiable, bijective function $f:[0,1]\to[0,1]$, and I would like to approximate this function by a series of other functions $T_i$ (think Taylor) – with the conditions that ...
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68 views

Approximation for the following integral needed

I have following integral $$\int_0^{\infty}e^{-ax-bx^m}dx$$ where $a>0, b>0, m>1$. I can get an approximation for the above integral when $b$ is small. However, I want to get an expression ...
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110 views

What are some applications of “separable” spaces?

A separable space is a space that contains a countable dense subset. For example, the space of continuous functions $C[a,b]$ is separable. Are there some practical applications arising out of this ...
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Second order central difference = first order central difference applied twice?

Approximating the 1st order derivative via central differences can be written as $ \delta_{2h}u(x) =\frac{u(x+h) - u(x-h)}{2h} \approx u'(x) .$ What is the main issue with applying again a central ...
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How does one derive Radial Basis Function (RBF) Networks as the smoothest interpolation of points?

I was reading/watching CalTech's ML course and it said that one could derive the RBF Gaussian kernel from the solution to smoothest interpolation that minimizes squared loss. i.e. one can derive the ...
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71 views

Best approximation of a function out of a closed subset

I'm studying approximation theory and I saw this exercise on Rivlin book an introduction to the approximation of functions: Prove that if $V$ is a normed linear space, $W$ a finite-dimensional ...
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Clarification for definition of admissible: $\Delta\in (K)$

I am reading through the following book: E.M. Nikishin, V.N. Sorokin: Rational Approximations and Orthogonality, Translations of Mathematical Monographs, vol. 92, Amer. Math. Soc., Provindence RI, ...
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90 views

Almost locality of cubic spline interpolation

The natural cubic interpolating spline is the unique $C^2$, interpolating cubic spline, endowed with two extra boundary conditions. Obtaining this spline, denoted by $s(x)$, involves the inversion of ...
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89 views

construction of entire function by using Runge's Approximation theorem

I have been trying the some exercises given in the book of "Robert E green and Krantz".But I am unable to do it,the exercise is following Construct a sequence of entire function $f_j$ such that $f_j$ ...
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Can the remainder of a Taylor expansion be estimated from the coefficients?

Given a formula for the coefficients $c_n\in\mathbb C$ of a not analytically known function $f:\mathbb C\to\mathbb C, z\mapsto f(z)$'s Taylor series, is there any way to estimate the remainder term of ...
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84 views

Linear programming or mixed integer linear programming approximation

This may well be a stupid question. Given a one-dimensional non-convex/concave piecewise linear $\mathbf R\to\mathbf R$ function, is there a way to translate its minimization problem into a linear ...
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177 views

Curve fitting N points with n(fixed) quadratic curves

I essentially have a constrained curve fitting problem that I need to solve efficiently. The following problem arises when performing practical calibration of RSSI (signal strength), providing ...
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56 views

Relationship between the Prime and Triangular numbers ${T_n\over P_n}\sim {n\ln{\pi}\over 2\ln{P_n}}$

I was observing $T_n:=1,3,6,10,15,...$ and $P_n=2,3,5,7,...$ of these sequences. $T_n={n(n+1)\over 2}$ is the triangular numbers and $P_n$ is the prime numbers We came acrossed this relation ...
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calculate the derivative of a function containing factorial

I got an expression for free energy F in a book as: $$F(N_1) = \frac{N_1N_2}{N}W-kT(\log N!-\log N_1!-\log N_2!),\tag1\label1$$ where $N=N_1+N_2$. As what the book later introduced, the derivative of ...
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138 views

Fejer monotone with respect to the convex hull of $C$

Let $C$ be a nonempty subset of a real Hilbert space $H$. Let $\lbrace z_n \rbrace_{n=1}^{\infty} \subset H$ be Fejer monotone with respect to $C$. Show that $\lbrace z_n \rbrace_{n=1}^{\infty} $ is ...
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187 views

Any insight on the half reciprocal Fibonacci sequence?

Define $R_n=R_{n-1}+\frac{1}{R_{n-2}} $ with $R_0=R_1=1$ Define $K_n=\frac{1}{K_{n-1}}+K_{n-2} $ with $K_0=K_1=1$ These are all limits I've found using Python but no basis of proof for these limits ...
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112 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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40 views

Analytic Decay Rate of Polynomial-Gaussian Product

Any analytic techniques I can use to solve this problem? I'm basically looking for some scale for which this function decays: $$ f(x)=x^k e^{-\frac{x^2}{4\lambda^2}} $$ I'm currently trying the ...
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129 views

Show that $T$ is a contraction.

Let $A$ and $B$ be nonempty, closed and convex subsets of a Hilbert space $H$. Let $\alpha, \beta \in (0,1)$ such that $\alpha + \beta <1 $. Define $T:H \rightarrow H$ by $$ Tx = \alpha P_A x + \...
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106 views

If one proves uniform convergence of a function does that imply convergence in every other $L^p$ norm?

I was wondering about the truth of the following: $$ \text{if} \lim_{n\to \infty} \| f_n - f \|_{\infty,S}=0 \implies \forall p\in \{ 2,3,4,...\}, \lim_{n\to\infty} \| f_n - f \|_{p,S} = 0$$ ...
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385 views

A generalization of an integral related with $\zeta(2)$

It is well-known 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 = \int_{0}^{+\infty}\frac{x^2}{e^x-1-x}\,dx \...
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Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist ...
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Approximation of $1-e^{-z}$

I am interested in approximating the function $1-e^{-z}$. I want to approximate it by a family of functions $f_n$ that are entire and $f_n(2\pi k i) = 0$ for all $2\pi k i$ with $|k|\le n$ and $f_n(2\...
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Show as $n$ get large that $e^{\frac{-n^2}{2K}}$ is an approximation of $\frac{k!}{k^n(k-n)!}$

I'm asked to show this on a problem I'm working on. $e^{\frac{-n^2}{2K}}$ is an approximation of $\frac{k!}{k^n(k-n)!}$ when $n$ is large. However in this class we've never gone over $e$, so the ...
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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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Fourier series approximation of $f \in H_p^s$ rate of convergence

$H_p^s$ is a standard $L^2$ Sobolev space of periodic functions on $[0, 2\pi]$, with $\|f\|^2_{H_p^s} = \sum_{j = 0}^s \|f^{(j)}\|_{L^2}^2$. Let $$f_N(x) = \sum_{|n| < N} \hat{f}_n \frac{1}{\...
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104 views

Orthogonal Projection of a function onto $M$ [closed]

Let $I_1, · · · , I_N$ be pairwise disjoint intervals whose union is $[0,1]$. Let $$M = \lbrace g ∈ L^2([0,1]) :\text{ g is constant on $I_n$ } \forall n \rbrace.$$ Suppose $f \in L^2([0,1])$. ...
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What is a good way to sample data points on an interval as to avoid Runge's phenomenon but have a deterministic sampling scheme?

I recently asked: Is there an analogous Gibbs phenomena to approximating sinusoidal but with polynomial terms? because I noticed that at the edges, polynomial interpolation of equidistance points ...
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392 views

Hermite polynomials approximate of a function and its derivatives

Given a differentiable function $f\in C^{(n)}(-\infty, \infty)\cap L^2(-\infty,\infty)$ with Gaussian measure $\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ and its Hermite polynomial expansion $f_n=\sum_{i=0}^...
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47 views

Find first degree polynomial approximation to function in $L^2[0,1]$

Let $u(t)=t^2$. Find the best approximation $v(t)$ of the form $v(t) = ct+d$ ($c,d\in\mathbb{R}$) to $u(t)$ in $L^2[0,1]$. We can define the set $V:=\{ct+d:c,d\in\mathbb{R}, t\in[0,1]\}$. $$\inf\...
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37 views

Best approximation to $t^2$ in first-dgree polynmial in $L^1[0,1]$

Let $u(t)=t^2$. Find the best approximation $v(t)$ in the form of $v(t)= ct+d $ (with $c,d\in\mathbb{R}$) to $u(t)$ in $L^1[0,1]$. So we need to find $$\inf\limits_{c,d\in\mathbb{R}} \int_0^1 \left|...
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88 views

Best constant approximations in $C[0,1]$, $L^1[0,1]$, $L^2[0,1]$

Let $f(t)=t^2$. Find the best approximation to $f(t)$ over $[0,1]$. Consider the Banach space $(C[0,1],\|\cdot\|_\infty)$. Then the best constant approximation, in my understanding, should be $g(t)\...
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When using taylor expansion for an expression with multiple terms, how do I determine the limit for accurate approximation?

I was once taught that, for using the Taylor series to expand $ln(1+x)$ the limit/range for having an accurate approximation (when the series "converges") is $|x|<1$. So the range of values for $ln(...
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Does the function $U=\frac{kx}{(x+x_{0})^2}$ reduce to a simple harmonic potential energy function for small oscillations around $x=x_0$?

The question is from a physics problems book, A particle of mass $m$ moves in a potential energy function given by $$U=\frac{kx}{(x+x_{0})^2}$$ where $x$ denotes the position and $x_0$ is a ...
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An error formula for linearization

Question: 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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169 views

What is the relation between Chebyshev and Taylor polynomials?

I just read about Chebyshev polynomials and that they are used in approximations. I don't fully understand them yet. What is the relation between Chebyshev polynomials and Taylor expansions?
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Bounding sum of samples of a Gaussian

Suppose we have $K$ points $x_1,\ldots,x_K$ in $\mathbb{R}^d$ and let $$f(x)=\sum_{k=1}^K \exp(-\lambda \Vert x-x_k \rVert^2).$$ Can we uniformly bound $f$ independent of $K$? It is okay to use ...
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116 views

First order approximation and small angle approximation

I am given $\omega\left ( k \right )=\left ( \frac{C+H}{m}\pm \frac{1}{m}\sqrt{C^{2}+H^{2}+2 CH Cos\left ( k a \right )} \right )^{1/2}$. It is mentioned that this reduces to $\omega\left ( k \...
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42 views

Making an approximation

How does $\sqrt{\frac{2(cos(x)-1)}{cos(x)}} \approx x$? At first I thought it was an example of binomial approximation but I was unable to approximate is to just $x$. I could only get $$\left(\frac{...
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157 views

Approximating a function's derivative in machine precision

Let's say I have a very complex analytical function $f$, in which finding its derivative at point $Q$, i.e. $f'(Q)$ is not practical. So we resort to finding the derivative numerically without ...
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144 views

Details of proof of convergence of Bernstein polynomial approximation

In Durrett's Probability Theory and Examples https://services.math.duke.edu/~rtd/PTE/PTE4_1.pdf, he gives the following. I have two questions: how do we get the underlined line? We apply Jensen's ...
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37 views

How to approximate a level curve?

Let $G$ be a $C^\infty$ function $G:\mathbb{R}^2\rightarrow\mathbb{R}$, and let $C:=G^{\leftarrow}(c)$, i.e. $C$ is a level set of $G$. I know that $C$ is bounded (which implies that it's a closed ...