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.
580
questions
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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 (...
8
votes
1
answer
596
views
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
$$...
2
votes
0
answers
1k
views
bounds on derivatives of mollifiers/mollified functions
Consider the standard mollifier
$$
\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 ...
3
votes
1
answer
943
views
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 ...
8
votes
2
answers
383
views
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$ ...
1
vote
1
answer
119
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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 < \...
1
vote
1
answer
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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 ...
2
votes
0
answers
114
views
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 ...
3
votes
0
answers
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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 ...
1
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0
answers
63
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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\...
-1
votes
1
answer
115
views
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
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
0
votes
0
answers
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Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$
I am trying to prove equations (3) given in this paper
http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf.
The authors use taylor series to approximate function
$f(q) = \left(1 + q \...
11
votes
2
answers
992
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Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$
For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$.
Question
Given $\epsilon> 0$, find a "low-degree" ...
2
votes
0
answers
34
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Convergence rate of cardinal series (Whittaker-Shannon interpolant)
Given $f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series
$$
s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \...
2
votes
2
answers
235
views
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 ...
-1
votes
1
answer
368
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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\...
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 \...
2
votes
0
answers
477
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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 \\
...
6
votes
2
answers
537
views
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 ...
3
votes
0
answers
178
views
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 ...
10
votes
2
answers
522
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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 ...
2
votes
0
answers
74
views
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
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...
8
votes
0
answers
167
views
Padé Approximants of Power Series with Natural Boundaries
Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
2
votes
1
answer
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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 $\phi$ ...
4
votes
0
answers
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views
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 $(\mathbb{D}^n)^o$ ...
9
votes
1
answer
473
views
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 ...
10
votes
1
answer
539
views
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 ...
1
vote
2
answers
335
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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 $(\mathcal D(A),A)$, ...
1
vote
0
answers
143
views
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 ...
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 ...
1
vote
1
answer
192
views
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(...
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
$$...
2
votes
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. ...
10
votes
1
answer
776
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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 ...
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 ...
6
votes
0
answers
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views
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 ...
1
vote
0
answers
233
views
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 ...
-1
votes
1
answer
82
views
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\...
0
votes
0
answers
255
views
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 ...
2
votes
1
answer
441
views
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 ...
0
votes
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 ...
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 ...
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 \...
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 ...
1
vote
0
answers
110
views
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$-...
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} \, ,$$
with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$.
Following ...
2
votes
1
answer
145
views
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
How far away is
$$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$
from
$$\max_{0 \leq t \leq 1} |W(t)|$$
In other words, if you simulate a Wiener process over a finite ...
6
votes
1
answer
821
views
Universal approximation theorem for whole $\mathbb{R}^d$
The well-known universal approximation theorem states that neural network with one hidden layer can approximate any continuous function on every compact subset of $\mathbb{R}^d$.
My question is ...
3
votes
1
answer
166
views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
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 $x>0$,
...