The approximation-theory tag has no usage guidance.

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400 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 ...

**9**

votes

**0**answers

386 views

### 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**

votes

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64 views

### numerical methods to compute approximated Stiefel-Whitney class

Let $0<\epsilon<1$, $k\geq 4$ be an integer, $S^2$ be the unit sphere in $\mathbb{R}^3$, $d$ the Euclidean distance in $\mathbb{R}^3$ and
$$
...

**7**

votes

**0**answers

121 views

### 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 $|p(x)| \leq (1+|x|)^d$. ...

**6**

votes

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209 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 ...

**6**

votes

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219 views

### 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} ...

**6**

votes

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444 views

### 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 ...

**5**

votes

**0**answers

166 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

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24 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 ...

**5**

votes

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107 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 ...

**5**

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176 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 ...

**5**

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278 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 ...

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272 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 ...

**5**

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274 views

### 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**

votes

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151 views

### 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 ...

**5**

votes

**0**answers

329 views

### 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

75 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 ...

**4**

votes

**0**answers

73 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 ...

**4**

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123 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 ...

**3**

votes

**0**answers

47 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

**0**answers

142 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**

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222 views

### Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...

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35 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

**0**answers

96 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

**0**answers

26 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 ...

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votes

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67 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 ...

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77 views

### 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 ...

**2**

votes

**0**answers

512 views

### 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 ...

**2**

votes

**0**answers

87 views

### Greedy interpolation of functions

Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation:
Set $r_0 = f$.
for $k = 0,1,\ldots,$
Find the location of the global ...

**2**

votes

**0**answers

195 views

### Is there an absolutely continuous function $f$

Is there an absolutely continuous function $f$ satisfying
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1,
$$
which is not $C^{1}$?

**2**

votes

**0**answers

153 views

### Rational interpolation: Error bounds for coefficients

The following question was asked on MSE, but might be more suitable here.
Assume there is a rational function
$$
f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}}
$$
of type $(m,n)$ with ...

**2**

votes

**0**answers

107 views

### On Artin-Hironaka lemma and Galois theory

Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point.
Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...

**2**

votes

**0**answers

131 views

### A.G. Vitushkin's “Easily representable families of functions” - can it be generalized?

Background
In his monograph "Estimation of the complexity of the tabulation problem" (translated into English as "Theory of the Transmission and Processing of Information") Vitushkin studies ...

**2**

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**0**answers

157 views

### Worst-case error and Cramer-Rao Lower Bound - is there any mathematical relation between them?

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...

**2**

votes

**0**answers

178 views

### Markov-type inequalities with arbitrary exponents

By a Markov-type inequality I mean an inequality of the form
$$
\| p^{(k)} \| \leq \lambda_{k,n} \| p \|,\quad \forall p \in U_n,
$$
for some $\lambda_{k,n} > 0$, where $U_n \subset L^\infty[-1,1]$ ...

**2**

votes

**0**answers

103 views

### Noisy bases for linear functions

For any $x \in \mathbb{R}^n$, the following statement is trivially true:
There exists a set $I \subset \mathbb{R}^n$ with $|I| \leq n$ such that for any $x' \in \mathbb{R}^n$, if $x \cdot y = x' ...

**1**

vote

**0**answers

50 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**

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**0**answers

35 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**

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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 ...

**1**

vote

**0**answers

77 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 ...

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**0**answers

25 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 ...

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vote

**0**answers

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 ...

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107 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.
...

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41 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**

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55 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 $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

**1**

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**0**answers

33 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 ...

**1**

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**0**answers

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, ...

**1**

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**0**answers

98 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 ...

**1**

vote

**0**answers

38 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 $k$ ...

**1**

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**0**answers

67 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 ...