The tag has no usage guidance.

learn more… | top users | synonyms

0
votes
1answer
112 views

Twisted Padé approximants

Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$ $$f(z)=\sum_{k\...
9
votes
3answers
555 views

Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
2
votes
1answer
196 views

Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$. Let $\epsilon\in[\frac{1}2,1)$. Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ }f(S_1)\subseteq[1-\epsilon,1+...
4
votes
0answers
86 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in S^{D-...
3
votes
1answer
185 views

Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, $$|u|_{W^{...
5
votes
1answer
147 views

Asymptotics of Fresnel integrals

It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ...
1
vote
0answers
115 views

Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it. ...
1
vote
2answers
212 views

Generalized Schwarz Lemma for near-zeros

In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma: Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain $...
0
votes
0answers
50 views

Approximation property of Fréchet if range is restricted to an embedded Hilbert space

Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
2
votes
1answer
208 views

Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
1
vote
1answer
154 views

How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and $a=(K-1)d$,...
3
votes
0answers
103 views

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 ...
20
votes
3answers
1k views

A “better” rational approximation of pi?

$355/113$ is a good fractional approximation of $\pi$, because we use six digits to produce seven correct digits of $\pi$. $$\frac{355}{113} = 3.1415929\ldots$$ Let $R$ be the ratio of the number of ...
1
vote
1answer
118 views

Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$. Thanks!
2
votes
1answer
120 views

cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline} F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, x_{3k+1},x_{3k+2},x_{3k+3})\in\...
5
votes
1answer
222 views

Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully). Suppose $g$ is holomorphic on $\mathbb{C}^...
1
vote
0answers
43 views

Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
1
vote
0answers
67 views

Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
9
votes
2answers
294 views

Approximation theory on the disc

Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...
1
vote
0answers
34 views

Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $...
4
votes
2answers
279 views

Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous function $f$ such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is an interpolation polynomial ...
6
votes
0answers
226 views

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 ...
1
vote
0answers
73 views

Bounds on degree from bounds on derivatives

Let $f(x)\in \Bbb R[x]$ and $r(x)\in\Bbb R(x)$. Supposing we have information about the values taken by $f(x)$ and $g(x)$ in certain intervals and also can bound their derivatives in these intervals, ...
4
votes
0answers
86 views

Estimating polynomial approximation error in high dimension

Question Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...
5
votes
1answer
299 views

Jackson's theorem for partial sum of Fourier series

There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies $$ |f(x) - ...
1
vote
0answers
127 views

Questions related to a previous question about interpolation based on non-decreasing polynomials

Let $n$ be a positive integer greater than $2$. Let $X(1),X(2),\cdots,X(n)$ and $ Y(1),Y(2),\cdots,Y(n)$ be two strictly increasing sequences of n real numbers each, listed in order of increasing size....
2
votes
1answer
203 views

Maximum of a mollified/convolution function

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function $$\tilde{f}(x)=\int_{-\...
0
votes
1answer
170 views

Uniform approximation of increasing function in $C^{\infty}$

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function $g\...
3
votes
3answers
922 views

Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?
1
vote
1answer
102 views

How to formulate approximation from above?

(This is perhaps a stupid question. If so, please give me a hint and a down vote.) I have a sequence of Banach spaces $X_{1}\supset X_2\supset ...$ and a sequence of elements $x_j\in X_j$ ($j=1,2,..$)...
3
votes
1answer
130 views

Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...
0
votes
1answer
133 views

About the maximum degree of multivariate polynomial interpolation

It is well known that in the univariate case, to interpolate $k$ points in $\mathbb{R}$, we need to use a polynomial of degree $k-1$. My question is about multivariate polynomial interpolation in ...
0
votes
0answers
77 views

on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$ Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $...
1
vote
1answer
47 views

Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...
4
votes
1answer
460 views

Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...
2
votes
1answer
124 views

using polynomials as lower / upper bound?

I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials $p_1(...
1
vote
1answer
57 views

Error estimates for $L^2$ approximation of multivariate $C^k$ functions in terms of norms on derivatives

Let $\Omega \subset \mathbb{R}^d$ be compact and convex, and let $f \in C^{k+1}(\Omega)$. Let $P_k$ be the set of multivariate polynomials up to degree $k$ on $\Omega$. I am looking for any results in ...
3
votes
0answers
179 views

$\epsilon$-covering number of a set of rank-2 matrices

Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...
3
votes
1answer
280 views

Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$

Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z_1, z_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by $\frac{1}{z_1}+\frac{1}{z_2}$...
1
vote
1answer
186 views

Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
5
votes
0answers
286 views

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 ...
0
votes
0answers
35 views

Inapproximability of logarithmic factor of independent set

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
1
vote
0answers
43 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
4
votes
2answers
127 views

Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys $$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$ with additional condition that $\mathbb{E}X^k$ does not ...
4
votes
0answers
124 views

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 ...
0
votes
2answers
180 views

Markov-Bernstein like inequalities for monotone polynomials

Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
1
vote
1answer
250 views

a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it? to clearly express the problem assume that $$ z(\mathbf{a,B,c})=\mathop {\inf} ...
1
vote
0answers
59 views

interpolation and approximation [closed]

Given a function $f$ in C^k[a,b], can we always construct function $g \neq f$ such that $g(x) \ge f(x)$ for all $x \in [a,b]$, $f^{(m)}(a)=g^{(m)}(a)$ and $f^{(m)}(b)=g^{(m)}(b)$ for $m=0,1,\dots, k$ ...
1
vote
0answers
73 views

Global Approximation via Convex Combination of Local Approximations

I recently faced the problem of efficiently approximating a very large set of data points and, neither having a model of the empiric function, nor of the error distribution, my method of choice would ...
5
votes
0answers
288 views

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{...