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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Padé multipoint approximants of the exponential function
One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with
$\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
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Approximation of a continuous function by a smooth one on an open set
I'm interested in the following kind of theorems :
Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
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Bounding quantiles of the noncentral chi distribution
I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
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What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
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approximating the $|x|$ function
I'm familiar with Newman's rational approximation of the absolute value function via rational functions. Are there other explicit functions that approximate $|x|$ with exponential error? I was under ...
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Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables
Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
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Optimal $L^2$ bounds of cubic spline interpolation
Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
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Closure of polynomials of a function in $L^2$
Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict ...
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Holes of a compact set in $\mathbb{R}^n$ that do not contain holes of a larger open set
Let $K$ be a compact subset of an arbitrary open set $\Omega\subset \mathbb{R}^n.$ It is said that a connected component $W$ of $\Omega\setminus K$ is $\Omega$-bounded if $\overline{W}$ is a compact ...
user120667
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Finding a tight upper bound of $\int_0^\infty e^{-a\sqrt{1-e^{-x}}-x^2/2} dx$ as a function of $a$, $a>0$
The integral converges as it is easily seen to be upper bounded by $\sqrt{\pi/2}$.
However, Laplace's method does not seem to work out as the maxima of the function $S(x) = -a\sqrt{1-e^{-x}}-x^2/2$ ...
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Approximation error of 1-Lipschitz function on cubical mesh
Let $\Omega = [0,1]^d$ and consider $f : \Omega \to R$ Lipschitz continuous with constant 1.
Consider the regular decomposition of $\Omega$ into $d$-dimensional cubes $\Omega_i$, $i=1 ... k^d$ with $...
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Finding a tight upper bound of $\int_0^\infty e^{-x^2/2-a(1-e^{-x})}dx,\ a>0$, as a function of $a$
Is there a method to find a tight upper bound on the given integral? Note that the integral is upper bounded by $\sqrt{\pi/2}$, and thus converges. I first thought about applying Laplace's method. ...
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A good starting position for maximizing a function with Newton-Raphson / Halley's method
I'm attempting to find the maximum of this function:
\begin{align*}
h(\mathbf{t}) = -\left\{\sum_{i=1}^{n}\lambda_i e^{\boldsymbol{\theta}_i^\intercal \mathbf{t}}\right\} + \boldsymbol{\alpha}^\...
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What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?
I am doing long-running project that involves asymptotic saddle-point estimation of integrals (for flavour, it's this sort of stuff) and I would like to ask if there are established ways in the ...
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Approximately complemented subspaces
Definition:
Suppose $E$ is a subspace of normed space $X$. Then $E$ is approximately complemented in $X$ if for any compact subset $K$ of $E$ and any $\epsilon>0$ there is a continuous linear ...
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Approximating a compact $C^1$ hypersurface without boundary
Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\...
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Literature Request: Finite Dimensional "Approximations" of Linear Operators
I am interested in finding literature pertaining the problem posed by this question, which is the degree to which an operator $A$ on an infinite dimensional (separable) Hilbert $X$ space can be "...
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Integral of exponential of quadratics + exponentials
Eq 2 of this paper states this integral:
\begin{align*}
r^{-\beta} = \frac{1}{\Gamma(\beta)}\int_{-\infty}^{\infty} e^{-re^t + \beta t} dt
\end{align*}
Is there is name for this identity, or the class ...
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Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
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Tight L2 bound on moments approximation and reference
Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.
The error in approximated the moments ...
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Cubic interpolating spline – number of extremum points
Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of ...
- 3,574
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Pade approximation of a rational function
I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated.
So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
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A Lemma on convex domain which is a Lipschitz domain
I am reading the following paper:
https://docs.wixstatic.com/ugd/1de1d9_cd82cb002eaa4eefa9af574eb5efdff2.pdf
I am stuck on the proof of lemma 2.3 on page 6.
I don't see why does the property (i) of ...
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inverse interpolation
Given data points $(x_i,y_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y_i=f (x_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ ...
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A bounded polynomial having bounded coefficients: several variables
Consider the multivariate case for the question "Approximation theory reference for a bounded polynomial having bounded coefficients" (Approximation theory reference for a bounded polynomial having ...
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Literature request: Functional capacities
Are the results of the following book (in French) covered in English in a book or in an article, and if so, could you please provide a reference?
C. Dellacherie, Ensembles analytiques, Capacités, ...
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Scaling a set of reals to be nearly integers
A version of this question was previously asked on MSE. I'll mention progress below.
A geometric construction I'm exploring
leads to a set $R$ of $n$ positive real numbers, for example:
$$
R = \{ \pi,...
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Splines with bounded first derivative?
I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using ...
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A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm
Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
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The infimum over Sobolev norms of compactly supported functions which are 1 on an interval
Let $n\in \mathbb{N}_{0}$. I am interested in the quantity
$\inf\{\|\psi\|_{W^{1,n}(\mathbb{R})}\mid \psi\in W^{1,n}(\mathbb{R}), 0\leq \psi \leq 1, \psi\equiv 1 \text{ on }[-1/2,1/2], \text{ supp}(\...
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Constructive approximation of Hölder functions using kernel functions
Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where
$\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder
functions with norm $L$. I am interested in efficiently ...
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Smoothness Conditions for Planar "Mock-parametric" Spline Interpolation
By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-...
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Variational proof for minimum curvature of cubic splines
Background: Given an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)\in C^2([a,b])$ is a piecewise cubic polynomial on each subinterval $(x_i, x_{i+1})$.
Given a ...
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Steklov means (A paper by H.Johnen and K. Scherer)
I asked this question in MSE about a month ago, and didn't get a reply as of yet.
I'll copy and paste this question here, hopefully to get some response.
I am reading the following paper:
http://www....
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Approximating an infinite Sum
I am interested in finding the approximate answer to the following infinite sum
\begin{equation}
\sum_{l=0}^{\infty}( l+a) \exp^{b{(l+c)}^2}
\end{equation}
in the case where $a>0 , b<0 , c>0$...
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Wrinkling smooth functions
I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized ...
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When (if ever) are the weights from Smolyak (sparse grid) cubature positive?
Are there any $1$-dimensional quadrature rules of arbitrary accuracy, on either $[0,1]$ or $\mathbb{R}$, with any non-trivial weight function, such that the associated $N$-dimensional cubature rule ...
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Approximating Uniform Distribution with Mixture of Gaussians
Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$.
Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$:
$$
\...
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Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
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Approximating dense subspaces of Fréchet spaces
If $H$, $H_0$ are two separable Hilbert spaces and $H$ is continuously and densly embedded in $H_0$, it is possible to construct a sequence of linear operators
$$ P_n : H_0 \to H $$
such that for all $...
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How to choose contour for rational approximation
Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\...
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Polynomial approximations on the Boolean hypercube
Given $\vec{a} \in \mathbb{R}^n$ and $b \in \mathbb{R}$ consider the function $f(x) = Th[\vec{a}.\vec{x}+b]$ on $x \in \{-1,1\}^n$ such that the ``threshold function (Th)" gives $1$ when the argument ...
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How small (in modulus) can a polynomial get?
Question. If $f(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$, is it true that
$$\max\{\,\vert f(x)\vert: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ?$$
Context. This came up while working on ...
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Strong convergence of differential quotient in $L^2(0,T;V^*)$
I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
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What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
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Representing $x$ as a linear combination of higher powers $x^n$
Applying the Müntz–Szász theorem on $[0,1]$ repeatedly, we can represent
$$
x= \sum_{n\geq 2} c_n x^n
$$
as a uniformly convergent series (edit: only over some subsequence, see edits below) on $[0,1]$...
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Lower bounds for finite difference formulas
I'm interested in approximating higher derivatives of a function via values of the function only. I guess the following question has been studied, but I haven't been able to find a reference. I know ...
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stochastic dynamics as approximate deterministic dynamics
Is there a rigorous sense in terms of which a stochastic process may be considered as an approximation to a chaotic but deterministic ODE in a higher-dimensional state space, in a manner that ...
- 3,873
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Approximating a function with sums of powers
One can approximate an analytic $f: \mathbb R\to\mathbb R$ with Chebyshev polynomials $T_n$ or with Taylor polynomials. In applications one usually prefers Chebyshev ones because they would converge ...
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Chudnovsky algorithm and Pi precision
What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.
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