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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An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
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Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Set $$...
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bounds on derivatives of mollifiers/mollified functions

Consider the standard mollifier $$ \phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1. $$ such that $\int\phi(x) = 1$. Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am ...
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Approximation of the indicator function of an interval by polynomials

Suppose $f:[-1, 1]\rightarrow\mathbb{R}$ is a polynomial. I am curious what the minimal degree of $f$ can be such that for $0<a<b<1$, $f$ satisfies the following two properties: 1) $\forall ...
ignescent's user avatar
8 votes
2 answers
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Best constant approximation in $L^p(\Omega)$

For $\Omega$ a bounded open set of $\mathbf{R}^d$ and $f\in L^p(\Omega)$ the infimum \begin{align*} \inf_{C\in\mathbf{R}} \|f-C\|_p \end{align*} is reached (by compactness). For $1<p<\infty$ ...
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Upper bound on Lp distance of functions before and after change of variables

Setup I am trying to upper-bound the difference between two functions: one before the change of variables and the other after. For example, let $r \in \mathbb{N} \cup \{\infty\}, 1 \leq p < \...
diadochos's user avatar
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Family of funcitons that approximates uniform density on an ellipsoid

Given an nondegerate ellipsoid $E$ in $\mathbb{R}^d$, described as $E = \{x\in\mathbb{R}^d: (x-x_0)^TQ_0(x-x_0)\leq 1\}$ and let $\chi_E$ be the characteristic function supported on $E$. I am thinking ...
Min Wu's user avatar
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Error bounds for spline interpolation. Hall and Meyer's conjecture

Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some ...
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The $L_\infty$ norm of the derivative of the $L_2$ spline projector

A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
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Approximation of measured-valued function by continuous functions

For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e., $$ \int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty. $$ Let $\mu$ be a probability measure on $R^d$ such that $$ \int_{R^d}\int_{R^d}(|z|^2\...
Wenguang Zhao's user avatar
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Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
Wenguang Zhao's user avatar
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Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$

I am trying to prove equations (3) given in this paper http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf. The authors use taylor series to approximate function $f(q) = \left(1 + q \...
Emil Bjelski's user avatar
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Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$. Question Given $\epsilon> 0$, find a "low-degree" ...
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Convergence rate of cardinal series (Whittaker-Shannon interpolant)

Given $f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series $$ s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \...
user14717's user avatar
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Multiple series calculation

Let $n$ be a positive integer. I would like to find a numerical evaluation of the convergent (!) series $$ S_{n,s}=\sum_{k\in \mathbb Z^{n}}\frac{1}{(1+\vert k\vert^{2})^{s/2}},\quad s> n, $$ where ...
Bazin's user avatar
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Interpolation Inequality's Proof

Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$: \begin{equation} \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
Elena Villalón Sepúlveda's user avatar
3 votes
0 answers
241 views

Interlacing sequences by polynomials?

Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
Turbo's user avatar
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Closed form expression for $Tr\left[ (\mathbf{DW})^k \right]$

Given the $N \times N$ diagonal matrices $\mathbf{D}$ and $\mathbf{W}$ as defined below $ \begin{split} \mathbf{DW} &= \left[ \begin{array}{cccc} \beta_{1} & 0 & \cdots & 0 \\ ...
Felipe Augusto de Figueiredo's user avatar
6 votes
2 answers
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An expansion from Ramanujan related to birthday problem

A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ...
Upc's user avatar
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Closed subvariety that is unique in its small analytic neighborhood

Let $Y$ be some smooth projective variety over $\mathbb C$ with $\dim Y \geq 2$. For a closed sub-variety $X \hookrightarrow Y$, consider the following property: There is some small open neighborhood ...
sawdada's user avatar
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10 votes
2 answers
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Reference request: Extensions of Wiener's Tauberian Theorem

Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
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Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set $$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
Asaf Shachar's user avatar
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Padé Approximants of Power Series with Natural Boundaries

Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
MCS's user avatar
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Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$

If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$ An approximate solution of $\phi$ ...
Mahmoud Hassan's user avatar
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Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...
Asaf Shachar's user avatar
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Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It may be better to move this to a separate question. Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
Vesselin Dimitrov's user avatar
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1 answer
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Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?

Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
Vesselin Dimitrov's user avatar
1 vote
2 answers
335 views

Trotter-Kato approximation theorem for uniformly continuous approximants

Let $E$ be a $\mathbb R$-Banach space $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $E$ with generators $(\mathcal D(A_n),A_n)$ and $(\mathcal D(A),A)$, ...
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Specific L1 piece-wise linear approximation of the convex function of one variable

Consider a convex function $f(x)$ of one variable $x$ on some interval $[a,b]$. Question: What is known about the L1 approximation of $f(x)$ by piecewise linear functions? How to construct ...
Alexander Chervov's user avatar
2 votes
1 answer
410 views

Best approximation of a compactly supported density by a single Gaussian

Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow. Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...
JohnA's user avatar
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1 answer
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Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
Wenguang Zhao's user avatar
2 votes
0 answers
147 views

Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that $$...
Wenguang Zhao's user avatar
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0 answers
74 views

Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
Sam OT's user avatar
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1 answer
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Approximation of a compactly supported function by Gaussians

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
JohnA's user avatar
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4 votes
1 answer
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Construct a dense family in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ based on distance functions

Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function. Suppose $\mathcal{F}$ is a ...
John's user avatar
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0 answers
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Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
Chilote's user avatar
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1 vote
0 answers
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Transcendental functions generating almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
Almost user's user avatar
-1 votes
1 answer
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On probabilistic extension for Bernstein polynomials

Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
ZUN LI's user avatar
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0 answers
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Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...
NN2's user avatar
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2 votes
1 answer
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Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...
dohmatob's user avatar
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0 answers
61 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user avatar
1 vote
1 answer
63 views

Name of a function space

For a real function $f$ on $\mathbb{R}$, define $e_n(f)$ to be the infimum of the $L_1$ distance between $f$ and piecewise constant functions on the subdivision of $\mathbb{R}$ into intervals of ...
alesia's user avatar
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2 votes
1 answer
457 views

Approximation of a two-variable function by tensor products

Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function. We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...
Richard's user avatar
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4 votes
0 answers
227 views

Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel

Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
Minkov's user avatar
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0 answers
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Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
Ramen's user avatar
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4 votes
1 answer
674 views

Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
Chaos's user avatar
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2 votes
1 answer
145 views

How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?

How far away is $$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$ from $$\max_{0 \leq t \leq 1} |W(t)|$$ In other words, if you simulate a Wiener process over a finite ...
cgmil's user avatar
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6 votes
1 answer
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Universal approximation theorem for whole $\mathbb{R}^d$

The well-known universal approximation theorem states that neural network with one hidden layer can approximate any continuous function on every compact subset of $\mathbb{R}^d$. My question is ...
almnagako's user avatar
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3 votes
1 answer
166 views

Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type $$\int_C f(z) M(\lambda g(z)) dz$$ for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
Mr_3_7's user avatar
  • 105
3 votes
1 answer
197 views

Simple but entangled inequalities

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold? 1) $G(x)\le x$, 2) $G(1)<1$, 3) $F(x)>0$ if $x>0$, ...
Yaakov Baruch's user avatar

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