The tag has no usage guidance.

learn more… | top users | synonyms

3
votes
2answers
63 views

Pade approximation of gaussian distribution to given precision

Apologies if the question is too elementary here. For a certain computational application I need to approximate Gaussian distribution $e^{-x^2}$ with specific absolute precision (within $10^{-7}$ ...
5
votes
1answer
148 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 ...
2
votes
1answer
45 views

Cubic splines convergence?

I am looking for a basic, classical, result on approximating a smooth function using cubic and linear splines. Is there a reference on the convergence, in some sense, of the splines to the function of ...
0
votes
0answers
38 views

Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function. We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...
0
votes
0answers
110 views

Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
8
votes
1answer
373 views

Polynomial approximations of curves

This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit. For ...
3
votes
1answer
80 views

Regularization by mean curvature flow

I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the ...
5
votes
0answers
63 views

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 $M$ ...
0
votes
0answers
52 views

approximating smooth functions by non-smooth ones, in the distribution topology

The classical Stone-Weierstrass theorem gives a necessary and sufficient condition for a class of continuous functions on a compact to approximate a larger class of continuous functions in $C^0$ ...
3
votes
0answers
40 views

covariant derivative of manifold-valued function and logarithm map

Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon U\...
1
vote
0answers
26 views

Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result: Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in \mathbb{...
2
votes
1answer
98 views

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, ...
5
votes
2answers
182 views

$L^{\infty}$ polynomial approximation

In short: For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial? Jackson (1911) proved that there is a best approximating polynomial in the $L^{\...
1
vote
1answer
150 views

Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression: $$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor 1/...
1
vote
0answers
71 views

Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous function, $ \omega $ be a modulus of continuity and assume $$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$ whenever $ x,h \in \...
1
vote
1answer
45 views

On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...
1
vote
0answers
90 views

approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
1
vote
1answer
57 views

Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...
0
votes
0answers
36 views

Approximation of a volume-preserving Hölder homeomorphism by diffeomorphisms?

Is it known whether a volume-preserving Hölder homeomorphism of an arbitrary manifold can be approximated by a volume-preserving diffeomorphism? The answer is clearly no if the volume-preserving ...
0
votes
1answer
92 views

approximation of products of polynomials

I am wondering whether the following can be proved: Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...
0
votes
1answer
114 views

Interpolation of a series of data points via Chebyshev approximation?

first of all: english is not my native language, so there might be differences between what I meant and what you understood. Sorry for that in advance. As a research project, I try to comprehend and ...
1
vote
0answers
17 views

Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
0
votes
0answers
21 views

Bounding the error of the optimal solution from an approximated objective

Let $f$ be a smooth convex function defined in a bounded region $X$ and its Hessian is bounded $m I \le |\frac{\partial^2 f(x)}{\partial x^2}| \le M I$ for some $M>m>0$. Let $x^*=argmin_{x\in X} ...
0
votes
0answers
46 views

Approximation by polynomials with coefficients sum

In this paper Erdos p.1176 remarked that if the coefficients of $f(z):\sum_{v=0}^{n}a_{v}z^v$ are all real,then $\sum_{v=0}^{n}|a_{v}|$ is maximal for $f(z)=\pm T_{n}(z)$,where $T_{n}(z)$ is Chebyshev ...
2
votes
0answers
84 views

Error term for Euler-MacLaurin summation formula when applied to infinitely smooth functions?

A function $f(z,x)$ is tempered if all of the following are true: $f(z, x)$ is infinitely differentiable in $z$ $f(z,x)$ is defined for all $z,x \in \mathbb{R}$ Every derivative of $f(z,x)$ is ...
1
vote
1answer
28 views

Approximation with a more regular function and an inequality constraint

The motivation of the question comes from a geometric problem: can we approximate a $C^{1,\alpha}$ set $\Omega$ with positive curvature (in distributional sense) from inside with $C^2$ sets with ...
1
vote
0answers
148 views

Approximation of semicontinuous functions by continuous (or smooth) functions with closed form

I'm looking for a sequence $(f_{\epsilon})_{\epsilon>0}$ of continuous (or smooth) functions approximating a semicontinuous function $f$. Here, for approximation, pointwise convergence is fine. For ...
0
votes
2answers
139 views

Explicit analytic function with modulus asymptotic to $\Re z+\Im z$

Is there a simple and explicit continuous function $f\colon[0,\infty)^2\to\mathbb C$ such that $f$ is analytic on $(0,\infty)^2$ and $|f(x+iy)|/(x+y)\to1$ as $x+y\to\infty$, where $(x,y)\in[0,\infty)^...
2
votes
0answers
56 views

How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...
2
votes
0answers
131 views

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as $$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
2
votes
0answers
28 views

Uniform convergence of the best $L_1$ approximations by polynomials

Let $P_n$ be the vector space of real multivariate polynomials in $d$ variables of total degree no more than $n$ and let $f \in C(X)$, where $X \subset \mathbb{R}^d$ is compact. Let $(p_n)_{n=1}^\...
5
votes
0answers
188 views

Constructive approximation of Lipschitz functions

There are a number of theorems in classical functional analysis about approximation of Lipschitz functions by smooth functions. I was wondering if there are any similar constructive and explicit ...
1
vote
1answer
176 views

Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting: Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$. Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
5
votes
0answers
72 views

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\...
3
votes
1answer
69 views

Minimum degree of a nonnegative polynomial uniformly approximating two constant values on two disjoint closed intervals

This is a one-dimensional problem over $\mathbb{R}$. Given $y_0, y_1 \ge 0$ with $y_0 \neq y_1$, and closed intervals $I_0$ and $I_1$ with $I_0 \cap I_1 = \emptyset$, define a partial function $f(x) = ...
1
vote
0answers
34 views

How to treat equation with alternating square of frequency?

Let's have equation $$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$ Here $$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$ and functions $A(t), B(t)$ have ...
5
votes
0answers
139 views

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
votes
1answer
83 views

Approximation theoretic question about operator norm

Let $\|M\|:=\sup_{u:\|u\|=1}\|Mu\|$ be the operator norm induced by the Euclidean distance. Suppose $A$ is a $k\times k$ symmetric matrix with $A_{ij}>0$ for all $i,j$ and $\sum_{i,j} A_{ij} = 1.$...
0
votes
1answer
45 views

Separation of peaks

Could you please give any reference to literature on "separation o peaks", i.e. approximation of a numerically given function by a linear combination of two or several Gaussians with unknown ...
10
votes
2answers
251 views

A generalization of Chebyshev polynomials

What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$. Now suppose ...
1
vote
0answers
85 views

Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$. Let $...
1
vote
0answers
29 views

Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables? In the 2-dimensional case the ...
6
votes
1answer
119 views

Polynomial interpolation of binary word signal

Let consider a binary word $x_1 \ldots x_n$ (finite sequence of elements of $\{0,1\}$. I want to construct a polynomial $P$ that interpolates the points $(i, x_i)$ for $i \in \{1\ldots n\}$ , such ...
1
vote
0answers
48 views

Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...
3
votes
0answers
53 views

Are functions whose partial derivatives are simple functions dense in $W^{1,\infty}$?

In a 2D domain, are the functions whose partial derivatives are simple functions dense in $W^{1,\infty}$ ?
3
votes
1answer
225 views

Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?
3
votes
1answer
207 views

Chebyshev Polynomials

Given $$-\frac{1}2<a<\alpha<0<\beta<b<+\frac{1}2$$ $$+\frac{1}2<c<\gamma<1<\delta<d<+\frac{3}2$$ I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f([a,b])...
5
votes
1answer
139 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, ...
6
votes
2answers
176 views

Accuracy of the truncated Hausdorff moment problem

For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that $$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$ In other words, $M_s$ ...
5
votes
0answers
182 views

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 ...