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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L1 distance from a trigonometric susbspace

How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in \mathbb{Z}\...
Maciej Skorski's user avatar
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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 \...
Nikola Kovachki's user avatar
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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 ...
Rajesh D's user avatar
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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 ...
oliversm's user avatar
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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 ...
Vesselin Dimitrov's user avatar
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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 $...
Vesselin Dimitrov's user avatar
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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 ...
Marc Nardmann's user avatar
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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 $n \geq 1$ ...
Navin Goyal's user avatar
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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 ...
Nilima Nigam's user avatar
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Does NP = "epsilon-P" (PTAS / BPP)?

Some NP-complete optimization problems, like the knapsack problem, have a solution reachable in polynomial time that is guaranteed to be within arbitrary ε of the optimum answer. (aka PTAS - ...
Sai's user avatar
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Can SO_n(R) be approximated arbitrarily well using a discrete subgroup?

Let $G := SO_n(R)$ be equipped with the Euclidean metric on vectors of length $n^2$. Is it true that for any $\epsilon >0$, there is a finite subgroup of $G$ which intersects every metric ball of ...
Hari's user avatar
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When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", &...
AmirHosein Sadeghimanesh's user avatar
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Relative error approximation by polynomials

For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define $$ D(f,g) = \sup_{x \in [-1,1]} \left|{\frac{f(x)-g(x)}{f(x)}}\right| $$ (in this context, let's take $0/0$ to be $0$ ...
slimton's user avatar
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Estimate on sum of squares of multinomial coefficients

I am interested in approximating the sum of the squares of the multinomial coefficients, i.e. $a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$ or more general, $a_\...
Liss's user avatar
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Approximating a fraction with a given denominator

Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits). I want to approximate the fraction: $$\frac{M}{N} \sim \frac{k}{L+r}$$ where $r$ is at most $L$. In ...
mtheorylord's user avatar
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Lower-bound for $\Pr[X \geq m]$ subject to $E[X]>m$ where $X$ is a binomial random variable

Given an integer number $m>0$ and a real number $\alpha\in [1, 2]$, I am interested in finding a lower-bound for $\Pr[X\geq m]$ subject to $X \sim \text{Binomial}(n, m\alpha/n)$. For large values ...
Melika's user avatar
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automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...
Exodd's user avatar
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Approximation of a $C^{\infty}_c$ function by tensor products

Suppose that $f \in C^{\infty}_c ( \mathbb{R}^2 )$, i.e. $f$ is a $C^{\infty}$ function with compact support defined on $\mathbb{R}^2$. The following link Approximation of smooth compactly supported ...
Richard's user avatar
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2 answers
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Markov-type functions

I'd like to have some informations about Markov-type functions (or Cauchy-type): \[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\] $\gamma$ is a positive measure with compact support $\...
Thomas's user avatar
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Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions: 1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)? 2) if yes, can ...
joaopa's user avatar
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On the set of good approximators in the sense of Dirichlet's theorem

This question came up when thinking about an older question that hasn't been answered as of now. Let $\mathbb{N}$ be the set of positive integers. If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N}$ is ...
Dominic van der Zypen's user avatar
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1 answer
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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, ...
Matti Kiiski's user avatar
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1 answer
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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 ...
Amir Sagiv's user avatar
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Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
Amir Sagiv's user avatar
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Distance to finite degree polynomials for BV functions

A result of Jackson establishes for lipschitz functions $f\in\text{W}^{1,\infty}(0,1)$ the bound $$\inf_{p\in\mathbf{R}_n[x]} \|f-p\|_\infty\lesssim \frac{1}{n}\|f'\|_\infty,$$ where $\mathbf{R}_n[x]$ ...
Ayman Moussa's user avatar
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Schrodinger's equation over a randomized grid

I am interested in solutions to $$ \frac{d}{dt} \Psi = -iH \Psi $$ for $H$ hermitian and time independent. This boils down to evaluating $$ \Psi(t) = e^{-iHt}\Psi_0 $$ at points of interest $t_n$. I ...
dranxo's user avatar
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1 answer
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Interpolation by a function whose second derivative is bounded

I don't know if this is an easy question for specialists in the field. Consider the following interpolation problem : let $\varepsilon >0$, $X$ be a finite set of real numbers and $g$ be a real-...
Ewan Delanoy's user avatar
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4 votes
1 answer
575 views

Explicit and fast error bounds for approximating continuous functions

Main Question This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
Peter O.'s user avatar
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1 answer
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Smooth approximation of the $\max\{0,x\}$ function with controlled derivatives

Motivation/Hand-Wavy Question: In this post, it was asked what the best local approximation of $f(x):=\max\{0,x\}$ is by a polynomial of a given degree; with the answer provided by Chebyshev's ...
ABIM's user avatar
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4 votes
1 answer
407 views

Series expansion for gaussian-like function

I need a series expansion to describe a general gaussian-like (bell shaped) function. I couldn't find a rigorous definition of "bell shaped" online but in essence the function should have ...
john smith's user avatar
4 votes
2 answers
260 views

Approximated solutions of SEIR models

Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof – $\dot{S} = - N$ $\dot{E} = + N - E/\lambda$ $\dot{I} = + E/\lambda - I/\delta$ ...
Hans-Peter Stricker's user avatar
4 votes
1 answer
130 views

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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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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1 answer
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Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
user521337's user avatar
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4 votes
1 answer
461 views

Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space. ...
Asaf Shachar's user avatar
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4 votes
1 answer
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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 ...
guy's user avatar
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4 votes
1 answer
455 views

Polynomial interpolants in quadrature points and L2 convergence spectral rate

We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|...
Amir Sagiv's user avatar
  • 3,544
4 votes
1 answer
259 views

Approximation of $e^x$ by rational functions

Define a sequence of polynomials: $p_0(x)=1$, $p_1(x)=2+x$, and, for $n\ge 1$, $p_{n+1}=(4n+2)p_n(x)+x^2p_{n-1}(x)$ so that the first few are $1, x+2, x^2+6x+12, x^2+12x^2+60x+120$. Is there an ...
user47804's user avatar
  • 211
4 votes
1 answer
903 views

Lebesgue constant as condition number of polynomial interpolation

Let $T = \{ x_0,\ldots,x_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X_T$ be the associated interpolation operator on $C[a,b]$: it takes a function $f \in C[a,b]$ into ...
Back's user avatar
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4 votes
1 answer
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Minimizing the L1 norm of odd-term trigonometric polynomial

I want to minimize the $L_1$ norm of a finite odd-term trigonometric polynomial: $$\min_{a_k} \int_{0}^{1} |\sin(2\pi t)+\sum_{k \in \{3,5,7,...,2K-1\}}a_k\sin(2\pi kt)| \, \mathrm d t$$ Obviously, ...
Sven Ole Aase's user avatar
4 votes
1 answer
228 views

Expectation over Pareto Sums

Given $K$ iid random variables $x_i$ with uniform distribution on $(0,1]$ and a constant $\alpha > 0$, the random variable $x_i^{-\alpha/2}$ is Pareto-distributed with scale parameter $1$ and ...
August.xxz's user avatar
4 votes
1 answer
203 views

Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
Benoît Kloeckner's user avatar
4 votes
0 answers
98 views

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)}^...
Drew Brady's user avatar
4 votes
0 answers
542 views

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 ...
Peter O.'s user avatar
  • 637
4 votes
0 answers
102 views

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 ...
Fedor Petrov's user avatar
4 votes
0 answers
103 views

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{...
Pierre's user avatar
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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}{ \...
Lisa's user avatar
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0 answers
477 views

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 $\...
Aidan Rocke's user avatar
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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 $(\mathbb{D}^n)^o$ ...
Asaf Shachar'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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