The approximation-theory tag has no usage guidance.

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votes

**1**answer

318 views

### Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation*}
My ...

**3**

votes

**2**answers

290 views

### Approximation of curves

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...

**1**

vote

**3**answers

699 views

### State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 (http://www.ams.org/journals/mcom/...

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votes

**1**answer

108 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?

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votes

**0**answers

84 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

**2**answers

514 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

**1**answer

405 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 ...

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vote

**2**answers

186 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

**1**answer

126 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

**0**answers

686 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

**2**answers

83 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

**2**answers

139 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

**1**answer

194 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 } f_{n-1}\left(\frac{x}{t}...

**1**

vote

**1**answer

379 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

**1**answer

320 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", ...

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votes

**0**answers

91 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

**1**answer

144 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 $f$...

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votes

**0**answers

203 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 $C^{1}$?

**7**

votes

**0**answers

125 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

**1**answer

100 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 ...

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vote

**1**answer

190 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 $\mathbf{r},\...

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vote

**1**answer

260 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

**1**answer

569 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

**0**answers

85 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}}.
\end{...

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**0**answers

167 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

**2**answers

149 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

**1**answer

186 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 ...

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votes

**0**answers

113 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 ...

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votes

**3**answers

337 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\|}
\end{...

**4**

votes

**1**answer

719 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

**1**answer

743 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,
$a_\...

**1**

vote

**1**answer

104 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 ...

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**0**answers

71 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 ...

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**0**answers

134 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

**1**answer

200 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 $X_i\sim \text{Binom}(n,p_i)$...

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109 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 \{0,\dots,n\}^m}...

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**0**answers

279 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 $n \geq 1$ ...

**22**

votes

**2**answers

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 ...

**23**

votes

**1**answer

538 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, ...

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votes

**1**answer

266 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

**1**answer

417 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 ...

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votes

**2**answers

458 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 ...

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votes

**2**answers

1k 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 ...

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**4**answers

1k 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 ...

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votes

**2**answers

564 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 ...

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votes

**4**answers

1k 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

**1**answer

315 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

**0**answers

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 $...

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vote

**0**answers

229 views

### Decay rate of the singular values of functions

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

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136 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 ...