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0
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0answers
30 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 ...
0
votes
0answers
64 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [migrated]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
1
vote
3answers
106 views

Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
-1
votes
2answers
429 views

Approximating a subspace by sampling a base without replacement

Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...
1
vote
1answer
69 views

Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question: For any bivariate ...
-1
votes
0answers
23 views

Is the Interpolation unique?

Is the solution(Interpolation coefficient) of (2) in the paper"The shape parameter in the Gaussian function II" unique? ...
-2
votes
1answer
58 views

Approximation of locally Lipschitz function by globally Lipschitz functions? [closed]

Let $f(x)=|x|^mx$ for $m \geq 0.$ $f$ is differentiable and locally Lipschitz. Is it possible to approximate $f$ by a sequence of globally Lipschitz functions $f_n$? Can we write down $f_n$ ...
1
vote
0answers
64 views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
10
votes
0answers
151 views

Hilbert's Theorem on $L_2$ norm of polynomials in $\mathbb{Z}[X]$ - Explicit construction and a converse?

Consider the set of polynomials with real coefficients as a vector space with the following inner-product: $\langle f, g \rangle = \int_{a}^{b} f(x)g(x) dx$. Hilbert showed, in a paper from 1893, ...
1
vote
1answer
68 views

Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$. Can ...
2
votes
0answers
117 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
2answers
63 views

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 ...
0
votes
0answers
24 views

Threshold approximation with positive coefficients polynomials

Consider the family of polynomials $\{p_n\}$, each of degree $n$, such that for every polynomial $p = \sum_{i}c_{i}x^{i}$, $c_i \ge 0$ for every $i$ and $\sum_{i}c_i = 1$. Let $T$ be the ...
1
vote
2answers
62 views

Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions

The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large. ...
0
votes
3answers
222 views

Approximating higher dimension step function

Let $s \in R^{n}$ (meaning $s$ is $n \times 1$ vector), where $n$ is the dimension of the vector. The ideal sliding term, $\nu$ is taken to be: \begin{equation} \nu = \frac{s}{\|s\|} ...
12
votes
1answer
1k 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 ...
0
votes
1answer
121 views

Approximation of the sum involving binary entropy function

Given the following sum: $S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$ where $H$ is the binary entropy function defined as: $\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $. Let ...
5
votes
1answer
164 views

Approximating an iteratively defined function

Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ...
2
votes
1answer
196 views

How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds?

The odd order continued fraction approximants for $\ln(1+X)$ are $$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$ In "Some bounds for the logarithmic function", ...
2
votes
0answers
58 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
0answers
179 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 $C^{1}$?
4
votes
1answer
107 views

Do interpolation nodes have to be dense?

Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval. For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates ...
0
votes
0answers
72 views

Absolute convergence of double Fourier-Haar series

Let $\left\{\chi_n\right\}_{n\geq 0}$ be a complete orthonormal (wrt a measure $\mu$) set on $[-1,1]$. Given a function $f:[-1,1]^2\rightarrow \mathbb{C}$, what smoothness requirements on $f$ are ...
7
votes
4answers
3k views

Approximation with continuous functions

Is it true that for every function $\mathbb{R} \to \mathbb{R}$ there exists a sequence of continuous functions $f_n(x): \mathbb{R} \to \mathbb{R}$ such that for any $x \in \mathbb{R}$ $f_n(x)$ ...
5
votes
0answers
62 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 $|p(x)| \leq (1+|x|)^d$. ...
10
votes
4answers
618 views

Using Quotient of Prime Numbers to Approximation Reals

We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime. Question ...
5
votes
6answers
544 views

best approximation to the LambertW(x) or exp(LambertW(x))

what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000
1
vote
1answer
160 views

What's the asymptotic behavior of this function at large distance? [closed]

This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where ...
2
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0answers
29 views

Norms of B-spline coefficients

In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...
0
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0answers
48 views

Uniform approximation of tangent cone

Let $\mathcal{C}$ be a convex and closed set in $\mathbb{R}^n$ containing $0$. The tangent cone of $\mathcal{C}$ at $0$ is given as, \begin{equation} T_{\mathcal{C}}(0)=Cl(\text{cone}(\mathcal{C})) ...
4
votes
1answer
274 views

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 ...
1
vote
1answer
186 views

Approximation of a given function by rational functions

Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of ...
5
votes
1answer
387 views

Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.

Suppose that a continuous function $f$ on the line and satisfies $$ |f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1] $$ ...
1
vote
0answers
72 views

The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion: \begin{equation*} B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}. ...
2
votes
2answers
391 views

Multivariate Bernstein polynomials for approximation of derivatives.

If I have a $C^\infty$ function $f: [0,1]^n \to \mathbb{R}$ then its Bernstein polynomials $$ B_m(x) = \sum_{k_1,\dots,k_n=0}^m f\left(\frac{k_1}{m}, \dots, \frac{k_m}{m}\right) \prod_{i=1}^n ...
2
votes
0answers
111 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 ...
0
votes
0answers
90 views

Bound on a sum of Laguerre polynomials

I am trying to find an asymptotic behavior, for large real $t$, of the following sum \begin{align} Q(t)=\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{1+n}L_n^{(1)}(t-n) \end{align} where $L_n^{(\alpha)}$ is ...
1
vote
0answers
101 views

Decay rate of the singular values of functions

Suppose a function $f:[-1,1]^2\rightarrow \mathbb{C}$ has a singular value decomposition: $$ f(x,y) = \sum_{k=1}^\infty \sigma_k u_k(y) v_k(x), \qquad \sum_{k=1}^\infty \sigma_k^2 <\infty, $$ ...
2
votes
2answers
141 views

Non-global oscillation of banded Fourier transform

Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$ $$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$ Of ...
1
vote
4answers
1k views

A differentiable approximation to the minimum function

Suppose we have a function $f : \Re^N \rightarrow \Re$ which, given a vector, returns the value of its smallest element. How can I approximate $f$ with a differentiable function(s)?
2
votes
1answer
135 views

Approximation Runge's Theorem

Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...
2
votes
0answers
96 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
1answer
213 views

Multivariate polynomial approximation of smooth functions

Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are uniformly upper bounded ...
0
votes
0answers
40 views

multivariate B-spline approximation

we have a function $f(x)\in C^{K}[-1,1]^{d}$, whether there exists a multivariate B-spline function $s(x)$, such that $|s(x)-f(x)|_{\infty}=O(h^{K})$? where $h$ is the width of the spline.
0
votes
0answers
46 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let ...
1
vote
1answer
308 views

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, ...
1
vote
1answer
86 views

Approximating rational generating functions

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0}^\infty x_iX^i$ is ...
3
votes
1answer
168 views

Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...
1
vote
0answers
61 views

Jackson inequality for a nonpolynomial basis

Hi everybody, this is my first question.$L^2$,$H^p$ are the standard Lebesgue,Sobolev spaces here, and I am deliberately omitting the domains because I'll accept an answer if it's on an interval or a ...
1
vote
0answers
91 views

Spectral norm for a truncated Hilbert matrix

Let $T_{N}$ be the (Hilbert) matrix defined by $T_{N}(m,n)=\frac{1}{m-n}$ if $1\leq m,n \leq N$ and $m\neq n$ , and $ T_{N}(n,n)=0$ if $1\leq n \leq N$ . It's well known that $\Vert T_{N}\Vert < ...