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.
195
questions with no upvoted or accepted answers
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Approximating zero sets of real polynomials with "less complicated" polynomials
Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
15
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0
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516
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Functions approximated by rolling epicycle curves
Imagine a decreasing sequence of (positive) radii $r_1 > r_2 > r_3 > \cdots$
and a series of nested circles $C_1 \supset C_2 \supset C_3 \supset \cdots$
with these radii,
initially each ...
11
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0
answers
200
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Degree of 2-variable real polynomial that’s large on a square and small on a nearby “L”
I’d like to know what’s the minimum possible total degree of a real polynomial $p(x,y)$ such that:
(1) $p(x,y)\in [0,1]$ whenever $x,y\in [0,1]$ with either $x$ or $y$ at most $\epsilon$, and
(2) $p(...
9
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0
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546
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Padé approximations of $e$
The following question came up in the analysis of some algorithm.
Let $R_{s,t}(z)$ be the Padé approximants of $e^z$, and define $r_{s,t} = R_{s,t}(1)$. Using the explicit expression for the error ...
8
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+100
L^2 polynomial approximation in higher dimensions
Let $\mu$ be a measure on $\mathbb{R}^n$ I'm looking for known upper-bounds on
$$\| f-P_m \|_{L^2(\mu)} $$
where $P_n$ is the orthogonal projection of $f$ onto the polynomials of degree less than $m$.
...
8
votes
0
answers
513
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Concave and other bounded functions: Series representation and converging polynomials
Main Question
Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
8
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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}...
8
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682
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The function space defined by deep neural nets
Given a deep net graph and the activation functions on the hidden vertices do we have a description of the function space spanned by it? (even if for some specific architectures and activation ...
8
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637
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Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
8
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583
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convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
7
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Uniform approximation of separately continuous functions on zero-dimensional spaces
For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
7
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On derivatives of polynomials majorized by $\max(1,|x|^d)$
In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question.
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...
6
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Using the Lorentz operators to build polynomials that converge to a continuous function
Questions
Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$.
Find explicit bounds, with no hidden constants,...
6
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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 ...
6
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Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow
Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...
6
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Basis functions for approximation of a convex function on unit simplex
Consider the unit $D$-simplex $S^D=\left\lbrace (x_0, x_1, \ldots, x_D) \in \mathbb{R}^{D+1} \mid \sum\limits_{i=0}^{D}x_i = 1, x_i \geq 0 \right\rbrace$. I have a bounded, convex function $f:S^D\to\...
6
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0
answers
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Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
6
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Which functions can be approximated by piecewise constant functions?
Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in \mathbb{...
6
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0
answers
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what books to read to quickly understand adiabatic approximation
Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...
6
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0
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Polynomial upper approximation with respect to the Gaussian measure
Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and
$$ \lim_{n\to \infty} \int_{\...
5
votes
0
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Standard function spaces with the approximation property
A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
5
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Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?
Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
5
votes
0
answers
257
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$L^p$-convergence of truncated Legendre series approximation to inverse error function $\text{erfinv}$
Bounding the $L^p$-error for an $n$-th order Legendre series approximation
I have function $f\,\colon (-1, 1) \to \mathbb{R}$ where $f \in L^q(-1, 1)$ for any $q \geq 1$. I approximate $f$ using a ...
5
votes
0
answers
197
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On effective constructions in the functional analysis of Volterra's integration operator
Let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra integration operator: $V(f)(x) := \int_0^x f(t) \, dt$.
Is there a universal function $C(L,\varepsilon) < \infty$ such that the following uniform ...
5
votes
0
answers
195
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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 $...
5
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Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?
[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.]
For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...
5
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321
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Approximation by polynomials
The following is a well-known theorem (see e.g. The Chebyshev Polynomial by Rivlin):
If $p(x) = x^n + a_{n_1} x^{n-1} + \ldots + a_0$, then $\max_{-1\leq x \leq 1} |p(x)| \geq 2^{1-n}$ for $n \geq 1$ ...
5
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0
answers
400
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Approximations of negative Sobolev norms
Consider the standard Cahn-Hilliard free energy, augmented by a nonlocal interaction term which measures the $H^{-1}$ norm of a zero-mean function. Could someone point me to a reference where this ...
4
votes
0
answers
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Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes
Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
4
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Bounds on the expectation of a function of a hypergeometric random variable: A "Jensen gap"
Main Question
Let $f:[0,1]\to [0,1]$ be continuous, let $B_n(f)$ be the $n$-th degree Bernstein polynomial of $f$, and let $r\ge 3$.
Given certain assumptions on $f$, what is an explicit and tight ...
4
votes
0
answers
102
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Asymptotics in the Chebyshev-type optimization problem
Let $g(x)\colon [-2,2]\to \mathbb{R}$ be a continuous function. Let $f_n(x)$ be a polynomial of degree $n$ such that $\log |f_n(x)|\leqslant ng(x)$ for all $x\in [-2,2]$. Then the maximal possible ...
4
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0
answers
103
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Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
4
votes
0
answers
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Covering number of smooth functions from $\mathbb{R}^d$ to $\mathbb{R}$
Let $(\mathcal{X},d)$ be a space of function $f: \mathbb{R}^d \to \mathbb{R}$ where $d=\| \cdot \|_\infty$ (i.e., $d(f)= \sup_{x\in \mathbb{R}^d} |f(x)|$ ).
Let $D_\alpha f= \frac{\partial^\alpha}{ \...
4
votes
0
answers
479
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analytic approximations of the min and max operators
Question:
What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here.
For any $\...
4
votes
0
answers
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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$ ...
4
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0
answers
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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 ...
4
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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 ...
4
votes
0
answers
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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 ...
4
votes
0
answers
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Algebras and $\sigma$-algebras associated to random variables
Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^...
4
votes
0
answers
709
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Estimating overshoot in spline interpolation
Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...
4
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The closures in $C^0(\mathbb C,\mathbb C)$ of the set of integer valued polynomials
This question is closely related to the thread The closures in $C^0(\mathbb R,\mathbb R)$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients. (Recall that a ...
4
votes
0
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The proximality of low rank function approximation
The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question:
Q1: Is there for a given integer $n$ always a best ...
4
votes
0
answers
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Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets
I was involved into this subject when I answered
this
question from MSE. Trying to generalize my answer, I am thinking about a following
Question. Let $X$ and $Y$ be metric spaces. When each ...
3
votes
0
answers
162
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Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
3
votes
0
answers
355
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A conjecture on consistent monotone sequences of polynomials in Bernstein form
A Conjecture
In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
3
votes
1
answer
196
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Smooth cut-off in homogeneous Besov space
Given a Littlewood-Paley decomposition
$$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$
where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...
3
votes
0
answers
139
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Chebyshev Equioscillation Theorem in presence of extra conditions
Let $P_\ell$ be polynomials of degree $\ell$. For $f \in C[0,1]$, define the minimax error $E_\ell(f) = \min_{p \in P_\ell} \max_{x \in [0,1]} |f(x) - p(x)|$. We know that for the above scenario the ...
3
votes
0
answers
34
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Do higher-order splines with Lipschitz derivatives exist on finite sets?
Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$.
If $n=m=1$ then it's easy to see that:
$$
...
3
votes
0
answers
61
views
How to find or approximate (e.g. using method of steepest descent ) integral?
Can you give any advice on how to find or approximate the following integral
$$
F(t,y) = \int_{0}^{y}\frac{i e^{-\frac{3 t^2 \left(x^2+1\right)}{2 \left(9 x^2+1\right)}-i \frac{4 t^2 x}{9 x^2+1}}}{\...
3
votes
0
answers
289
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Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$
Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...