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-1
votes
1answer
105 views

How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)

How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
0
votes
0answers
61 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 ...
1
vote
2answers
257 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
vote
1answer
199 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 ...
1
vote
2answers
151 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
vote
1answer
90 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
337 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
73 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 ...
1
vote
2answers
90 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. ...
6
votes
1answer
190 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 } ...
0
votes
1answer
198 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 ...
2
votes
1answer
244 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", ...
3
votes
0answers
76 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 ...
4
votes
1answer
125 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 ...
2
votes
0answers
191 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 $C^{1}$?
6
votes
0answers
118 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 $|p(x)| \leq (1+|x|)^d$. ...
4
votes
1answer
54 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 ...
1
vote
1answer
179 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 ...
1
vote
1answer
225 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 ...
4
votes
1answer
456 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
79 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
0answers
135 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
2answers
146 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 ...
2
votes
1answer
169 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
106 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 ...
0
votes
3answers
255 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\|} ...
4
votes
1answer
339 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 ...
3
votes
1answer
553 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
98 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 ...
1
vote
0answers
66 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
119 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 < ...
3
votes
1answer
190 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 ...
0
votes
0answers
97 views

Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following: $$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...
5
votes
0answers
259 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 $n \geq 1$ ...
15
votes
1answer
3k 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 ...
22
votes
1answer
426 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 1894, ...
2
votes
1answer
227 views

Approximation theory under $L_1$-error

Is there a reference for results in approximation theory of bounded functions of one (and multiple) variables under $L_1$-error? Formal definitions for functions of one variable are below. Let $C$ ...
3
votes
1answer
338 views

Lebesgue constant as condition number of polynomial interpolation

Let $T = \{ x_0,\ldots,x_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X_T$ be the associated interpolation operator on $C[a,b]$: it takes a function $f \in C[a,b]$ into ...
-1
votes
2answers
441 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 ...
6
votes
2answers
843 views

Approximating erf by tanh

It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
11
votes
4answers
751 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 ...
2
votes
2answers
377 views

Convex upper bound on a linear-fractional function

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ...
4
votes
4answers
969 views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
3
votes
1answer
170 views

Approximation power of wavelets

The Wikipedia article on Wavelet Transform states that: Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, ...
0
votes
0answers
66 views

approximation in Lie algebras

Let $x_{1}$, $x_{2}$, $x_{3}$ three disctinct closed points of a curve $X$ over an algebraically closed field k. Let G a connected reductive group and $\mathfrak{g}$ his Lie algebra. I fix a Borel ...
1
vote
0answers
175 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
0answers
123 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
votes
0answers
152 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
2answers
413 views

Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N ...
2
votes
2answers
505 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 ...