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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How to approximate a sharply peaked function with deltas?

I have the function $$ f(x)=\cases{\frac{1}{\sqrt{\sinh^2(a)-\sinh^2(ax)}} & $-1<x<1$ \\ 0 & otherwise} $$ Where $a \in \mathbb{R}^+$. Here is a plot of $f$: It would be very convenient ...
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Solve transcendental equation : $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \ll 1$ and at $\alpha\gg 1$?

To solve this transcendental equations approximately : Preivous: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \leq 1$ and at $\alpha\geq 1$. Edit: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 ...
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Can Euler's totient function be made continuous enough for an ANN?

I am way out of my comfort zone here, so please be gentle! I have been reading papers that try to use Artificial Neural Networks to approximate the Euler's Totient function of form $\varphi(n) = (p-1)(...
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Bounding matrix-vector product norm

Let $A,B : \mathbb{R}^{n} \to \mathbb{R}^{n}$ be diagonalizable matrices with $\lambda$ being the eigenvalue of maximal absolute value between them. Let $v \in \mathbb{R}^{n}$. Running a few ...
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Jacobi $sd(u,m)$ hyperbolic approximation for $m \to 1$

I am following a paper (Fink 1976); the details are a mathematical problem. We are given a solution (paraphrased from the paper for clarity) $$y^2 = \alpha m_{1n} sd^2(u,\sqrt{m_n}),$$ where $sd$ is a ...
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21 views

Reverse-time martingale for non-polynomial approximating functions

Let $f(\lambda):(0,1)$ → $(0,1)$ be a continuous function. Given a coin with unknown probability of heads of $\lambda$, sample the probability $f(\lambda)$. One way to do so is to build randomized ...
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34 views

Convexity of $A = \{\alpha \sin^2(x)\in C[−\pi,\pi]|\alpha \in\mathbb{R}\}$

Show the set $A = \{\alpha \sin^2(x)\in C[−\pi,\pi]|\alpha \in\mathbb{R}\}$ is convex. I know to show a set is convex I also have to show any two points $x,y\in A$, then $\lambda x +(1-\lambda)y \in ...
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19 views

Approximation of convex function on product space

I am working on a problem where I need the following property that I guess should be true but I am not able to prove it. I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\...
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How can I get a function that is “approximately similar” to the part I care about in another function?

I have a function f(x)=$(a+bx)e^{-cx}, x\ge0$ where $a$, $b$, and $c$ are constants, and $c>0$. How can I get a function that is "approximately the same" as the part after the hump? I ...
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Recursive functional differential equation.

I'm trying to find a polynomial approximation for functions under an interval $(n,m)$. However, I'm stuck at trying to solve the following: $$P_{n+1}(x) = \int P_n(x) \; dx - \frac{\int_{n}^{m} \int ...
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What strategy can I use to choose points for approximating an expensive-to-compute function?

Suppose: I have some function $f(x)$ which is very expensive to compute I do not have a way to calculate its derivative (except as a numerical approximation given nearby points $f(x)$) I have enough ...
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Finding a series for a polynomial function that approximates a function under an interval (n,m).

Taylor series can be used for approximating a function near a point. However, if I want to approximate a function with a polynomial function under an interval, Taylor series fail. My idea for a ...
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28 views

Approximation error with Bernstein polynomials at a specific point.

Let $f:[0,1]\to \mathbb R$ be function, twice-differentiable in the interior such that $f''(0) \to \infty$. Let $B_nf$ denote its approximation with the $n$ Bernstein polynomials. Fix an $x_0 \in (0,1)...
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Approximate (continuous) functions by step functions - Convergence Rate

I want to approximate a measurable and bounded function $S:[0,1]^2 \rightarrow \mathbb{R}$ by step functions. So, assume we have a uniform partition $$ \{I_j\}_{j=1}^n, \textit{ } j=1,...,n $$ of $...
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1answer
33 views

Multivariate piecewise linear interpolation

Let $f:X\rightarrow E$ be a uniformly continuous function into some Banach space $E$ ($\dim(E)\geq 1$) from a (non-empty) compact subset $X\subseteq \mathbb{R}^n$ where $n\geq 1$. Why, for every $\...
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An approximation of a d-dimensional function using an order q<d function

Suppose we have a function $f:\mathbb{R}^d\to\mathbb{R}$. Using the notation of https://www.maths.unsw.edu.au/sites/default/files/amr08_5_0.pdf, which defines an order $q$ function $g:\mathbb{R}^d\to\...
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Why is this function approximation correct?

I want to show that the following expression: $$\left | \frac{ 1-\rho } {1-\rho e^{i\mu -\sigma^2/2} } \right |^2 $$ Is very well approximated by: $$\frac{ 1 } {\big (1-\frac{\sigma^2}{2\ln{\...
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3answers
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Taylor expansion of a function

I would like to approximate the function $f(x)=\frac{2x}{1-e^{-2x}}$ analytically for both small and large $x$. But when I use the formula for the Taylor expansion, I run into the problem that the ...
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Can a convex function be uniformly approximated by convex combination of simpler convex functions?

According to universal approximation theorem, any continuous function can be uniformly approximated by neural network with single hidden layer. Similarly, can a convex function $f: \mathbb{R}^n \...
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Approximating a Fourier transform

Suppose the Fourier transform $\hat{f}(k)$ (with $k \in \mathbb{R}^d$) is given, and one intends to get some information about its position-space counterpart $f(x)$. When the analytical calculation of ...
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Can a function be approximated by an infinitly often differentiable function with common values at the boundary of the Intervall

Let $ f:[0,1] \to \mathbb{R} $ be a $ C^1 $-function. Does for every $ \epsilon > 0 $ exist an $C^{\infty}-$function $ g:[0,1]\to \mathbb{R} $ such that $ g(0)= f(0) $, $ g(1) = f(1) $ and $ \vert \...
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Approximating musical pitch function (the function $2^x$)

I have an old digital synthesizer and I would like to understand how its pitch control was implemented. A perfect representation of a given pitch in an equal temperament scale with the concert pitch ...
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Approximation theory. Technical lemma

$H -$ Hilbert space, $D \subset H$ and closure $\overline{span D} = H$. The set $D$ is called dictionary. Let $$\rho(D):= \inf_{x\in H, |x| = 1} \sup_{g \in D} |\langle x,g \rangle|.$$ I need to prove ...
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Lower-bound for Hoffman constant of linear system of inequalities in terms of singular-values of coefficient matrix

Let $A$ b an $m \times n$ matrix with rows $a_1,\ldots,a_m$ and $b$ be an $m$-dimensional vector. Define the set $S_{A,b} := \{x \in \mathbb R^n \mid a_j^Tx \le b_j\;\forall j \in [m]\}$, assumed to ...
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18 views

P-NP Approximation algorithm

Define an independent set of a graph $G = (V, E)$ to be a subset $S$ of vertices such that $V-S$ is a vertex cover of $G$. Is every 2-approximation algorithm for finding a minimum vertex cover also a ...
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8 views

Approximation algorithm for vertex cover

Consider the independent set of a graph $G = (V, E)$ to be subset $S$ of vertices for which $V-S$ is a vertex cover of $G$. Is each of $2$-approximation algorithm for finding a minimum vertex cover ...
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Approximation Algorithm question

Define an independent set of a graph $G = (V, E)$ to be a subset $S$ of vertices such that $V-S$ is a vertex cover of $G$. Is every $2$-approximation algorithm for finding a minimum vertex cover also ...
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1answer
22 views

Expanding convolution integrals for sharply-peaked functions.

Question: I have integrals like $$ I(x) = \int_0^\infty K(y) F(x,y) dy $$ where $K(y)$ is some kernel function that is sharply peaked at some special value $y = y_0$ and otherwise trends toward zero ...
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52 views

Existence of low-pass filters

For $h=(h_k)_{k=0}^{n-1}\in \mathbb{R}^n$ call $\hat{h}(\omega)=\sum_k h_ke^{-j\omega k}$ and take $\gamma_h(\omega,\epsilon)\! :=\! \frac{\min_{f\in [0,\omega]} |\hat{h}(f)|}{\max_{f\in[\omega+\...
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1answer
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A question about two definitions of covering numbers

I am reading the book "Greedy approximation"(Page 208) written by V. Temlyakov and thinking about the following question of covering numbers: Let $X$ be a Banach space and let $B_X(y,r)$ ...
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What is a sharp bound on the difference between $f$ and its linear approximation on $[a,b]$?

So I have the following problem. Let $f:[0,1] \rightarrow [-1,1]$ such that $\lvert f(x) - f(y) \rvert \leq \lvert x - y \rvert$ for all $x,y \in [0,1]$. An example of such a function is drawn in ...
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Are there particular cases of Muntz theorem which can be proved with an elementary way?

The Muntz theorem states that the space $S := span\{x^{\lambda_i}\}_{i \geq 0}$ is dense in $(C[0,1], \| .\|_{\infty})$ if and only if $\sum_{i = 0}^{\infty} \frac1{\lambda_i} = \infty$ (where $\...
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Formalizing a “density argument” (from a differentiable lipschitz function to a non-differentiable lipschitz function)

Let $Z=f(X_1,...,X_n)$ where $f : [0,1]^n \rightarrow \mathbb{R}$ is a separately convex (convex with respect to each of its variables) and 1-lipschitz function, and $X_1,...,X_n$ are independent ...
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Approximating compactly supported $L^2$ functions with Schwartz functions “from within”?

Crossposted from MathOverflow. It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz ...
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1answer
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A question about the property of covering numbers

I am reading the book "Multivariate approximation"(Page 321 - 322) written by V. Temlyakov and thinking about the following property of covering numbers: Let $X$ be a Banach space and let $...
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Are error bounds on Bernstein-form polynomials also error bounds on their Bernstein coefficients?

Let $f(\lambda)$ be a continuous function that maps $[0, 1]$ to $(0, 1)$. An answer to another question on this site gave useful error bounds for approximating $f$ with polynomials. Many of them ...
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Proof of Mergelyan's theorem on a compact set $K$ whose complement has a finite number of components

The simple form of Mergelyan's Theorem is that if $K$ is a compact set whose complement is connected and $f$ is continuous on $K$ and holomorphic in the interior of $K$, then it can be uniformly ...
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1answer
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Is there a tighter error bound for this convergent version of Stirling's series?

I have written an implementation of the log factorial function which employs a convergent version of Stirling's series. This is the formula in the bottom of page 12 of R. Schumacher, "Rapidly ...
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Examples of (families of) functions that are badly approximatable by polynomials but well approximatable by splines

As the title says, I am looking for concrete examples of functions which are badly approximatable by polynomials (i.e. a slow convergence rate), but well approximatable by B-splines (hopefully, ...
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1answer
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Approximation for Beta distribution

I've heard that, when the parameters $\alpha$ and $\beta$ of the beta distribution approach infinity, the distribution becomes approximately normal. But I have only seen a proof of such fact for the ...
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1answer
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Problem with the density of polynomials.

Problem: Suppose that $X \subset \mathbb{R}^n$ is compact. Let $Y = \{y_1,...,y_p\}$ be a finite subset of $X$. Let $\mathscr{P} = \{p \in \mathbb{P}[x_1,...,x_n]: p(y_i) = 0$ for all $p\}$ (the ...
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Inverse laplace transform of $\left(s(\sqrt{\frac{s+q}{s+p}}-1)\right)^k$

I am working on finding the inverse Laplace transform of this function: \begin{equation} g(s) = \left(s(\sqrt{\frac{s+q}{s+p}}-1)\right)^k \end{equation} where $q>p>0$ are real numbers, $k$ is a ...
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What are ways to compute polynomials that converge from above and below to a continuous and bounded function in $[0,1]$?

My interest is to take a coin of unknown bias $\lambda$ and use it to produce a coin of bias $f(\lambda)$. This is called the Bernoulli Factory problem, and only certain functions $f$ can be simulated ...
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1answer
62 views

How to prove this inequality using Taylor approximation?

I'd like to prove the following via Taylor approximation : $$x > 0 \implies \sin x > x-\frac{x^3}{3!}$$ I tried to estimate the error but I found $E_3(x) \leq |\frac{x^4}{4!}| $, which doesn't ...
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How suboptimal is $x_{\mathrm{ls}}=\min _{x}|A x-b|_{2}$ for the Chebyshev problem $\min _{x}|A x-b|_{\infty}$?

The least-squares problem $$\min _{x}|A x-b|_{2}$$ has a closed-form solution: $$x_{\mathrm{ls}}=\operatorname{argmin}|A x-b|_{2}=\left(A^{T} A\right)^{-1} A^{T} b$$ But the corresponding Chebyshev or ...
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The difference between f and the optimal function h* is orthogonal to the space the optimal function comes from - How can I interpret this visually?

I'm kind of confused about this theorem. I know and understand the proof, but I "can't see it". If I have a 2D function f, and it's optimal function is also a 2D function, thus they are from ...
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1answer
57 views

How do I approximate the function?

Let $f$ be a continuous function on the unit circle $T=\{|z|=1\}$. Show that $f$ can be approximated uniformly on $T$ by a sequence of the polynomial in z if and only if $f$ has an extension $F$ that ...
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10 views

Approximate nearest neighbor algorithm using jaccard distance

Given have $h_\pi(A) = \text{argmin}_{a\in A} \pi(a)$, where $A$ is a set and Jaccard distance as $d = 1 - J$. I want to create an approximate nearest neighbor search using $h_\pi$ and $d$ however, I'...
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36 views

Optimally approximating the sign function by functions with compactly supported Fourier transform

I'm looking for a systematic way to approximate the sign function $$sgn(x)=\begin{cases}1&\text{ if }x\geq 0\\-1&\text{ if }x<0,\end{cases}$$ by functions $f$ whose (distributional) Fourier ...
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22 views

How to determine order of the remainder term of a series for specific problems?

Background: In the text book I have, in the 'expansion of integrands' part (perturbation theory) the author kept determining the order, but I can't understand how. For integration of $\sin \epsilon x^...

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