The approximation-theory tag has no wiki summary.

**20**

votes

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### 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 ...

**16**

votes

**1**answer

923 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 ...

**12**

votes

**5**answers

871 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 ...

**12**

votes

**1**answer

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 ...

**11**

votes

**6**answers

3k views

### Solving NP problems in (usually) Polynomial time?

Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.
The best example of this is probably the traveling salesman problem, for which extraordinarily large instances ...

**10**

votes

**4**answers

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 ...

**10**

votes

**4**answers

487 views

### Finite interpolation by a nondecreasing polynomial

Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences
of $n$ real numbers. It is well known that there are polynomials that "interpolate"
in that ...

**10**

votes

**1**answer

674 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 ...

**10**

votes

**0**answers

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, ...

**10**

votes

**0**answers

361 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 ...

**8**

votes

**3**answers

632 views

### Approximating with translated Gaussians and low-frequency trig functions

Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any ...

**8**

votes

**1**answer

554 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.
...

**8**

votes

**1**answer

496 views

### Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...

**8**

votes

**0**answers

334 views

### Padé approximations of $e$

The following question came up in the analysis of some algorithm.
Let $R_{s,t}(z)$ be the PadÃ© approximants of $e^z$, and define $r_{s,t} = R_{s,t}(1)$. Using the explicit expression for the error ...

**7**

votes

**3**answers

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.
...

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

**7**

votes

**4**answers

855 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, ...

**7**

votes

**2**answers

331 views

### polynomials with minimal $L_\infty$ norm on multiple disjoint intervals

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...

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votes

**2**answers

651 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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votes

**2**answers

519 views

### Approximation by polynomials

Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x_0, ..., x_m$ be different numbers from $[a,b]$.
Does for each $\varepsilon >0$ there exist a polynom $P$ such that ...

**6**

votes

**1**answer

439 views

### Stone-Weierstrass analogue for $L^p$

Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...

**6**

votes

**2**answers

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 ...

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votes

**2**answers

2k 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 $X$ ...

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votes

**2**answers

772 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 ...

**5**

votes

**5**answers

1k views

### Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...

**5**

votes

**6**answers

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

**5**

votes

**4**answers

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 ...

**5**

votes

**1**answer

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
$$
|f(x+2h)âˆ’2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
...

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votes

**2**answers

534 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 ...

**5**

votes

**1**answer

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 } ...

**5**

votes

**1**answer

293 views

### Sparse approximate representation of a collection of vectors

Suppose I have a collection of $n$ vectors $C \subset \mathbb{F}_2^n$. They are of course spanned by the canonical set of $n$ basis vectors.
What I would like to find is a much smaller (~ $\log n$) ...

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votes

**2**answers

887 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 ...

**5**

votes

**0**answers

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 $|p(x)| \leq (1+|x|)^d$. ...

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votes

**0**answers

232 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 $n \geq 1$ ...

**5**

votes

**0**answers

140 views

### what books to read to quickly understand adiabatic approximation

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...

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votes

**0**answers

283 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 ...

**5**

votes

**0**answers

207 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} ...

**5**

votes

**0**answers

405 views

### convergence rate in Wiener's approximation theorem

Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in ...

**4**

votes

**3**answers

513 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 - ...

**4**

votes

**2**answers

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 ...

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votes

**3**answers

366 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 ...

**4**

votes

**3**answers

377 views

### Approximation theory reference for a bounded polynomial having bounded coefficients

Let $P(x)$ be a real polynomial of degree at most $d$. Assume $|P(x)| \leq 1$ for $|x| \leq 1$. I would like a bound saying that each coefficient of $P(x)$ is at most $C^d$ in magnitude, for some ...

**4**

votes

**4**answers

724 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 ...

**4**

votes

**1**answer

376 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 ...

**4**

votes

**1**answer

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 ...

**4**

votes

**2**answers

532 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

**1**answer

406 views

### Interpolation by a function whose second derivative is bounded

I don't know if this is an easy question for specialists in the field. Consider
the following interpolation problem : let $\varepsilon >0$, $X$ be a finite
set of real numbers and $g$ be a ...

**4**

votes

**1**answer

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 ...

**3**

votes

**1**answer

600 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 ...

**3**

votes

**2**answers

831 views

### Variant of Fermat's last theorem

By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions
in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$
the distance between $x$ and the nearest ...