The approximation-theory tag has no wiki summary.

**0**

votes

**0**answers

51 views

### How to prove $\lim _{ \delta\rightarrow {0}^{+}}\int_{a}^{b} F_{\delta}(x)dx=0 ? $ [on hold]

This question comes from http://math.stackexchange.com/questions/982231/a-function-fx-that-riemann-integrable-on-a-b
Define a function $f(x)$ that Riemann integrable on $[a,b]$.
Let
...

**0**

votes

**0**answers

47 views

### Guess A Property Of The Integral Average Value Function [on hold]

Let $f$ be a function that is defined on $[a,b]$ and Riemann integrable on $[a,b]$.
Def 1.
$$\hat f(x)=\begin{cases}
f(x),& \text{if }x\in[a,b], \\
f(a),& \text{if }x<a, \\
...

**1**

vote

**1**answer

87 views

### How to formulate approximation from above?

(This is perhaps a stupid question. If so, please give me a hint and a down vote.)
I have a sequence of Banach spaces $X_{1}\supset X_2\supset ...$ and a sequence of elements $x_j\in X_j$ ...

**3**

votes

**1**answer

47 views

### Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...

**0**

votes

**0**answers

42 views

### About the maximum degree of multivariate polynomial interpolation

It is well known that in the univariate case, to interpolate $k$ points in $\mathbb{R}$, we need to use a polynomial of degree $k-1$.
My question is about multivariate polynomial interpolation in ...

**1**

vote

**1**answer

39 views

### Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...

**0**

votes

**0**answers

61 views

### on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...

**3**

votes

**1**answer

58 views

### Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...

**1**

vote

**1**answer

110 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 ...

**11**

votes

**0**answers

178 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

**1**answer

83 views

### using polynomials as lower / upper bound?

I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials ...

**1**

vote

**1**answer

34 views

### Error estimates for $L^2$ approximation of multivariate $C^k$ functions in terms of norms on derivatives

Let $\Omega \subset \mathbb{R}^d$ be compact and convex, and let $f \in C^{k+1}(\Omega)$. Let $P_k$ be the set of multivariate polynomials up to degree $k$ on $\Omega$. I am looking for any results in ...

**1**

vote

**0**answers

87 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 ...

**2**

votes

**2**answers

250 views

### Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$

Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z_1, z_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by ...

**1**

vote

**1**answer

165 views

### Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions.
Are there positive integers $m<n \in \mathbb{N}$, such that for ...

**5**

votes

**0**answers

216 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 ...

**0**

votes

**0**answers

28 views

### Inapproximability of logarithmic factor of independent set

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...

**1**

vote

**0**answers

33 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 $k$ ...

**4**

votes

**2**answers

110 views

### Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not ...

**1**

vote

**1**answer

90 views

### a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} ...

**4**

votes

**0**answers

111 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 ...

**0**

votes

**2**answers

122 views

### Markov-Bernstein like inequalities for monotone polynomials

Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $Pâ€²(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...

**9**

votes

**1**answer

652 views

### The closures in $C^0(\mathbb{R}, \mathbb{R})$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients

This is a follow up of an interesting recent question on the topic. The answer given there by fedia shows that the matter is rich and complicated, and I can't resist to submit here a further question.
...

**1**

vote

**0**answers

45 views

### interpolation and approximation [closed]

Given a function $f$ in C^k[a,b], can we always construct function $g \neq f$ such that $g(x) \ge f(x)$ for all $x \in [a,b]$, $f^{(m)}(a)=g^{(m)}(a)$ and $f^{(m)}(b)=g^{(m)}(b)$ for $m=0,1,\dots, k$ ...

**0**

votes

**0**answers

31 views

### Approximation with Predefined Topology of Niveau Sets

Problem
given are
a finite, connected, undirected and, cycle-free graph (i.e. a "tree") $T(V,E)$, of which one of the vertices (w.l.o.g. $v_0$) is defined to be the root.
a planar imbedding ...

**3**

votes

**1**answer

197 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 ...

**6**

votes

**1**answer

146 views

### $W^{1,1}$ simplicial approximation

Let $f$ be a continuous real-valued function defined on an $n$ dimesional simplex $\Sigma\subset \mathbb{R}^n $. The classical simplicial approximation scheme provides a sequence $f_k$ of piecewise ...

**4**

votes

**0**answers

244 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 ...

**1**

vote

**3**answers

233 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 ...

**1**

vote

**1**answer

166 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 ...

**4**

votes

**2**answers

227 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**

votes

**1**answer

96 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

**0**answers

59 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

175 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

**2**answers

440 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

**1**answer

82 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

179 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

71 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

**0**answers

31 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

**2**answers

72 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

**3**answers

239 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\|}
...

**13**

votes

**1**answer

2k 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

**1**answer

160 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 ...

**6**

votes

**1**answer

186 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

**1**answer

220 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

**0**answers

66 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

186 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

**1**answer

115 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

**0**answers

80 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

**4**answers

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 $x \in \mathbb{R}$ $f_n(x)$ ...