The approximation-theory tag has no wiki summary.

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### 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 $n \geq 1$ ...

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### 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 ...

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### 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, ...

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213 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 $C$ ...

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**1**answer

278 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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437 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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704 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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647 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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308 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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### 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 ...

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**1**answer

153 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, ...

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### 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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114 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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### 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 ...

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142 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 ...

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344 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 ...

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423 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 ...

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**1**answer

538 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 ...

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### Bound on trigonometric sum

I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have
$\sum_{k=1}^n -1+\cos(k\theta) \leq C(n)$.
If we rewrite this ...

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213 views

### Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact}

Given a function F, how to find polynom which is best/good approximate with respect supreremum-norm, i.e. minimize over P_{approx} sup|F-P_{approx}| ?
I am intersted in polynoms in two variables of ...

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### A differentiable approximation to the minimum function over a vector of reals

In A differentiable approximation to the minimum function, a differentiable approximation of the minimum function is given, but it seems it only works for positive reals.
Is there an ...

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153 views

### Approximating $\prod_{i=1}^{n-1} (1-ai)$ for large $n$

I have a function of the form:
$f(n) = \prod_{i=1}^{n-1} (1-ai)$
Here, $a \geq 0$ and $(a*i) < 1$. For $n > 10^5$ or $10^6$, what is the best possible analytic approximation for $f(n)$ that ...

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193 views

### Algebraic curve approximation

I am wondering wether it exists a theorem that any continuous path on the plane one can
approximate with algebraic curve $P(x,y)=0$ ($P$- is a polynom)?

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### are p-limits scales dense in the infinite musical scale of all rational frequencies?

In the wiki section on prime limit tuning, one reads:
...

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142 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 ...

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387 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 ...

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### 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 ...

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### 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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313 views

### A question about the Beurling-Selberg majorant

Beurling's majorant is defined as the unique entire function $B(z)$ such that the Fourier transform of $B(x)$ is compactly supported in the interval $[-1;1]$, $B(x) \geq \text{sgn}(x)$ and $B(x)$ ...

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### approximation of holomorphic functions on a halfplane.

Let $\mathbb {C} _ + $ denote the right halfplane and $A$ the algebra
$$
A = \{ f \in H^\infty({\mathbb C} _ +) \cap C(\overline{{\mathbb C} _ +}): \;
|f(z)| \le M (1+|z|)^{-\epsilon} \text{ ...

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### 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 ...

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555 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 ...

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362 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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### 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 ...

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557 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 ...

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169 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 $U_n \subset L^\infty[-1,1]$ ...

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551 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 ...

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**1**answer

404 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 ...

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**1**answer

967 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 ...

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### 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 ...

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**1**answer

256 views

### Trigonometrical approximation for the characteristic function of an interval

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

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### 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 ...

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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 ...

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### 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 ...

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654 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 ...

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### 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 ...

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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 ...

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295 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 ...

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### 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 ...

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372 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 ...