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0
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0answers
43 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 $C^0$ ...
-4
votes
0answers
13 views

Good estimates on the truncation of the exponential series. [closed]

What are the good (analytic) upper bounds we have on the series $\sum_{k=D}^\infty \frac{a^k}{k!}$ in terms of $a$ and $D$?
3
votes
0answers
30 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 ...
1
vote
0answers
15 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 ...
0
votes
0answers
85 views

Explicit twisted Padé approximants

This is a follow-up of Twisted Padé approximants Let $z\in\mathbb Z_p$ with $v_p(z)>0$. One puts $f_z(x)=(1+z)^x$ for all $x\in\mathbb Z_p$. I try to determine the twisted Pade approximants ...
5
votes
2answers
153 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 ...
2
votes
1answer
76 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, ...
1
vote
1answer
148 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 ...
4
votes
2answers
264 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 ...
0
votes
1answer
91 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 ...
11
votes
0answers
428 views

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 ...
1
vote
0answers
61 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 ...
0
votes
0answers
32 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 ...
1
vote
1answer
43 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
1answer
55 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 ...
1
vote
0answers
89 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 $z$ ...
0
votes
0answers
20 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
1answer
89 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 ...
1
vote
0answers
15 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
45 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 ...
22
votes
2answers
5k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
2
votes
0answers
82 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
0answers
21 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 ...
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
0answers
115 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 ...
1
vote
0answers
62 views

How do I find Coefficients in a Multivariate ChebyShev Polynomial Approximation? [closed]

How do I do a Multivariate ChebyShev approximation? Let $\vec{x} = x_{0}, x_{1}, ... , x_{n}$ Let $\vec{a} = a_{0}, a_{1}, ... , a_{n}$ Let $\vec{b} = b_{0}, b_{1}, ... , b_{n}$ Let $f(\vec{x})$ ...
0
votes
2answers
138 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 ...
2
votes
0answers
51 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
124 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 - ...
2
votes
0answers
27 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 ...
5
votes
0answers
184 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 ...
5
votes
0answers
60 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 ...
11
votes
2answers
906 views

Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...
1
vote
1answer
165 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 ...
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 ...
1
vote
0answers
42 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) = ...
6
votes
1answer
117 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 ...
5
votes
0answers
134 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 ...
1
vote
0answers
33 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 ...
7
votes
1answer
80 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} = ...
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 ...
14
votes
4answers
680 views

Finite interpolation by a nondecreasing polynomial

Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences of $n$ real numbers. It is well known that there are polynomials that "interpolate" in that ...
10
votes
2answers
243 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
82 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
45 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 $W^{1,\infty}$ ?
5
votes
1answer
334 views

Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite. There are at least two ...
3
votes
1answer
198 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 ...
2
votes
1answer
185 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 ...