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5
votes
2answers
586 views

Stone-Weierstrass for monotone functions

Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$. Does it follow that that there exists a real ...
2
votes
0answers
172 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 $U_n \subset L^\infty[-1,1]$ ...
4
votes
2answers
580 views

Polynomial approximation in L^p norms

Hello, I am very new to the field of approximation theory, and since an extended search on the Internet did not provide answers for two rather basic questions, I decided to ask them here. 1) From ...
4
votes
1answer
419 views

Low degree polynomial approximation for the entropy function

Let $X$ be a discrete random variable with possible values $\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of $X$. In addition, denote $p_i=p(x_i)$. The entropy of $X$ is ...
16
votes
1answer
1k views

Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 3

Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many ...
0
votes
1answer
194 views

Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
2
votes
1answer
259 views

Trigonometrical approximation for the characteristic function of an interval

Hello, Denoting $e(x)$ for $e^{2i\pi x}$, set $$E(R):=\left\{f\ \left|\ f(x)=\sum_{r=0}^{R-1}a_re(rx)\mbox{ where }a_r\in\mathbb{C}\ \forall r\mbox{ and }\sum_r|a_r|^2=1\right\}\right.$$ ...
10
votes
0answers
375 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 ...
1
vote
4answers
1k views

Approximation to the ratio of a Gaussian CDF to PDF

Johnstone and Silverman (2005) claimed that for large x $\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$ where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable. I was able ...
5
votes
3answers
751 views

Uniform approximation of $x^n$ by a degree $d$ polynomial: estimating the error

The answer to this question should be well known, but it's a hard question to search for online. Suppose we want to approximate the function $x^n$ by a polynomial of degree $d$ in the $L_\infty$ norm ...
3
votes
1answer
668 views

Least-squares regression and differential geometry

For $k, n \in \mathbb{N}$, let $\mathcal{C}_n \mathbb{R}^k$ denote the configuration space of $n$ distinct points in $\mathbb{R}^k$. (1) Is there a description of the tangent space $T_C ...
10
votes
1answer
710 views

Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2

This is a sequel to the question Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) . Let $Y$ be some projective variety, over $\mathbb{C}$. Let $X\subset Y$ be ...
21
votes
2answers
1k views

Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)

Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the ...
5
votes
0answers
300 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 ...
1
vote
3answers
1k views

Approximation in $L^2$ by piecewise constant functions

Dear all, in order to prove the validity of my Galerkin approach of a certain variational problem, I need to check the so-called approximability property. In my case, it boils down to showing that for ...
1
vote
1answer
385 views

nonnegative series expansion of nonnegative functions

The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
2
votes
1answer
341 views

Regularization of Zygmund functions

Dear community. I would like to derive a "good" estimate on $\frac{d}{dt}f_\epsilon(t)$, where $f_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e. ...
7
votes
3answers
1k views

Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer. ...
5
votes
6answers
605 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
221 views

Cubic spline smooting question

Hello, I came across this link when searching for an algorithm for spline smoothing. Though I understand basically what I have to do I need further clarifications on the formula chosen for curvature ...
4
votes
3answers
372 views

Approximating derivatives between gridpoints

Hi, Suppose we have a grid (possibly irregular) of N function/value pairs, $(x_i, f_i)$, $i=1...N$. The function is differentiable everywhere at least twice (perhaps more). What would be a good way ...
5
votes
2answers
895 views

L1 distance from a trigonometric susbspace

How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in ...
1
vote
1answer
1k views

smooth approximation of the hinge loss function

I came across this paper but the smooth approximation for the hinge loss function is wrong. Can someone guide me to the proper smooth approximation (using polynomials) of the function ...
3
votes
0answers
219 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 ...
2
votes
1answer
1k views

Simple algorithm to generate a Mondrian “Random Grid”

Hello, I was wondering if there is a simple way or algorithm that can generate 2-d grids resembling Mondrian paintings like the boogie woogie grid ( ...
9
votes
1answer
654 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
0answers
628 views

What is the function $\sin(n \omega) / (n \sin \omega)$?

During my work, I encounter the function like $\frac{\sin(n \omega)}{n \sin \omega}$. I'm puzzled and knew nothing about this function before. Given integer $n>1$, my question is how to find a ...
6
votes
2answers
856 views

Non-linear “Fourier analysis”

Call a function of the following form a beep: $e^{-(\frac{x-\alpha}{\beta})^2}\sin(\rho x+\theta)$. Given a real-valued function $f\in L^2(R)$ and a number $n$, I'm interested in the approximating ...
7
votes
4answers
901 views

A senseful meaning of 'approximation of manifolds'?

Any continuous function can be uniformly approximated by smooth functions. I would like to have something similar - in what-ever sense - for continuous manifolds. For example, by Whitney's theorem, ...
5
votes
0answers
212 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} ...
4
votes
1answer
286 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 ...
2
votes
0answers
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' ...
3
votes
1answer
419 views

methods for interpolating a function, holomorphic in the upper halfplane

Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...
3
votes
1answer
626 views

Where does the Chebyshev polynomial notation come from?

The $k$th Chebyshev polynomial is denoted by $T_k$ where $T_k(x) = \cos(k\cos^{-1}(x))$ I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the ...
2
votes
5answers
1k views

Relative error approximation by polynomials

For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define $$ D(f,g) = \sup_{x \in [-1,1]} \left|{\frac{f(x)-g(x)}{f(x)}}\right| $$ (in this context, let's take $0/0$ to be $0$ ...
1
vote
1answer
645 views

Padé approximation - usability in iterative algorithms

Firstly, I have to say that I don't understand Padé approximation well. But I discovered that, it is more precise than Taylor series. I have to create approximation for these functions: Log(x) and ...
2
votes
2answers
810 views

Interpolation Splines of Bounded Curvature

Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Can we have a polynomial spline (e.g natural cubic spline) passing through all these points, such that: ...
3
votes
1answer
472 views

Approximating an integral representation of the Number Partition Problem

One can write out an integral whose solution gives the number of solutions to the NP-Complete Number Partition Problem and I'm wondering if anyone has an suggestions or ideas on who to solve or ...
0
votes
1answer
450 views

Rational solutions of homogeneous equations

Can every solution of a homogeneous linear system be approximated by a solution in rational numbers? In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an ...
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)?
4
votes
2answers
271 views

Can SO_n(R) be approximated arbitrarily well using a discrete subgroup?

Let $G := SO_n(R)$ be equipped with the Euclidean metric on vectors of length $n^2$. Is it true that for any $\epsilon >0$, there is a finite subgroup of $G$ which intersects every metric ball of ...
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 $x \in \mathbb{R}$ $f_n(x)$ ...
4
votes
3answers
518 views

Does NP = “epsilon-P” (PTAS / BPP)?

Some NP-complete optimization problems, like the knapsack problem, have a solution reachable in polynomial time that is guaranteed to be within arbitrary ε of the optimum answer. (aka PTAS - ...
6
votes
2answers
1k views

Efficient approximation of a matrix and its inverse

Assume that $ A $ is a real $ n\times n $ matrix whose rows constitute an orthonormal basis of $ \mathbb R^n $. Informal statement of question: Assume we want to approximate $ A $ by a rational ...
5
votes
4answers
2k views

Approximation by exponential polynomials

Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of $\underline{order}$ $n$. Define $E_n$ to be the collection of all ...
2
votes
1answer
721 views

Probability of system failure in a distributed network

I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ ...
12
votes
5answers
887 views

Does this sequence span $L^2$?

Consider the following sequence of functions in $L^2[0,\infty)$: $$f_n(x)=e^{-x/n}x^n,\;\;n\geq 1$$ Does this sequence span $L^2[0,\infty)$ (that is, is the set of finite linear combinations of these ...
1
vote
1answer
730 views

Fast gradient approximations

I am trying to use the one-sided Simultaneous Perturbation (SP) method to get a gradient approximation for multi-variable function. The equations are very simple: ...
1
vote
1answer
304 views

References regarding unisolvent sets

Let $X = {x_1, ..., x_N}$ be a finite subset of $R^n$ and let $p$ and $q$ be any polynomials of degree $k$ or less. X is called $\underline{P_k-unisolvent}$ if $p(x_j) = q(x_j)$ ($j = 1, ..., N$) ...
7
votes
2answers
3k views

approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...