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### Pade approximation of gaussian distribution to given precision

Apologies if the question is too elementary here. For a certain computational application I need to approximate Gaussian distribution $e^{-x^2}$ with specific absolute precision (within $10^{-7}$ ...
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### Approximation of $e^x$ by rational functions

Define a sequence of polynomials: $p_0(x)=1$, $p_1(x)=2+x$, and, for $n\ge 1$, $p_{n+1}=(4n+2)p_n(x)+x^2p_{n-1}(x)$ so that the first few are $1, x+2, x^2+6x+12, x^2+12x^2+60x+120$. Is there an ...
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### Cubic splines convergence?

I am looking for a basic, classical, result on approximating a smooth function using cubic and linear splines. Is there a reference on the convergence, in some sense, of the splines to the function of ...
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### On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...
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### approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
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### Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...
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### Approximation of a volume-preserving Hölder homeomorphism by diffeomorphisms?

Is it known whether a volume-preserving Hölder homeomorphism of an arbitrary manifold can be approximated by a volume-preserving diffeomorphism? The answer is clearly no if the volume-preserving ...
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### approximation of products of polynomials

I am wondering whether the following can be proved: Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...
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### Interpolation of a series of data points via Chebyshev approximation?

first of all: english is not my native language, so there might be differences between what I meant and what you understood. Sorry for that in advance. As a research project, I try to comprehend and ...
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### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
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### How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...
The Euler--MacLaurin summation formula can be written as $$\sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)... 0answers 28 views ### Uniform convergence of the best L_1 approximations by polynomials Let P_n be the vector space of real multivariate polynomials in d variables of total degree no more than n and let f \in C(X), where X \subset \mathbb{R}^d is compact. Let (p_n)_{n=1}^\... 0answers 189 views ### Constructive approximation of Lipschitz functions There are a number of theorems in classical functional analysis about approximation of Lipschitz functions by smooth functions. I was wondering if there are any similar constructive and explicit ... 1answer 176 views ### Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane Setting: Let \Gamma be a simple smooth(C^\infty) curve in \mathbb{C} parametrized by the injective map \gamma:[0,1] \to \mathbb{C}. Assume f is a function defined on \Gamma s.t. f is ... 0answers 73 views ### Basis functions for approximation of a convex function on unit simplex Consider the unit D-simplex S^D=\left\lbrace (x_0, x_1, \ldots, x_D) \in \mathbb{R}^{D+1} \mid \sum\limits_{i=0}^{D}x_i = 1, x_i \geq 0 \right\rbrace. I have a bounded, convex function f:S^D\to\... 1answer 69 views ### Minimum degree of a nonnegative polynomial uniformly approximating two constant values on two disjoint closed intervals This is a one-dimensional problem over \mathbb{R}. Given y_0, y_1 \ge 0 with y_0 \neq y_1, and closed intervals I_0 and I_1 with I_0 \cap I_1 = \emptyset, define a partial function f(x) = ... 0answers 34 views ### How to treat equation with alternating square of frequency? Let's have equation$$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$Here$$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$and functions A(t), B(t) have ... 0answers 139 views ### Uniform approximation of separately continuous functions on zero-dimensional spaces For topological spaces X,Y,Z а function f:X\times Y\to Z is called separately continuous if for any (x,y)\in X\times Y the restrictions of f to the sets \{x\}\times Y and X\times \{y\} are ... 1answer 83 views ### Approximation theoretic question about operator norm Let \|M\|:=\sup_{u:\|u\|=1}\|Mu\| be the operator norm induced by the Euclidean distance. Suppose A is a k\times k symmetric matrix with A_{ij}>0 for all i,j and \sum_{i,j} A_{ij} = 1.... 1answer 45 views ### Separation of peaks Could you please give any reference to literature on "separation o peaks", i.e. approximation of a numerically given function by a linear combination of two or several Gaussians with unknown ... 2answers 251 views ### A generalization of Chebyshev polynomials What is the monic polynomial p(x) of degree n which minimizes \max_{x \in [-1,1]} |p(x)|? The answer is the Chebyshev polynomial, and its largest value on [-1,1] is 1/2^{n-1}. Now suppose ... 0answers 85 views ### Construct a PDE solution from a net of approximations Consider P a linear partial differential operator in \Bbb R ^n. Consider some boundary condition given in the generic form C(u) = 0, that guarantees a unique solution (if any) of Pu = 0. Let ... 0answers 29 views ### Error bounds for approximation with dyadic sums of polynomials Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables? In the 2-dimensional case the ... 1answer 120 views ### Polynomial interpolation of binary word signal Let consider a binary word x_1 \ldots x_n (finite sequence of elements of \{0,1\}. I want to construct a polynomial P that interpolates the points (i, x_i) for i \in \{1\ldots n\} , such ... 0answers 48 views ### Small open sets around a point intersecting pieces of orbits Let T be an ergodic rotation on a compact Abelian group. Can one always find a point x_0 and a decreasing sequence of open sets O_n \searrow \{x_0\} such that for every n there exists K \geq ... 0answers 53 views ### Are functions whose partial derivatives are simple functions dense in W^{1,\infty}? In a 2D domain, are the functions whose partial derivatives are simple functions dense in W^{1,\infty} ? 1answer 226 views ### Are piecewise linear functions dense in W^{1,\infty}? Are piecewise linear functions dense in W^{1,\infty} ? 1answer 207 views ### Chebyshev Polynomials Given$$-\frac{1}2<a<\alpha<0<\beta<b<+\frac{1}2+\frac{1}2<c<\gamma<1<\delta<d<+\frac{3}2$$I want to find a polynomial f(x)\in\Bbb R[x] such that f([a,b])... 1answer 139 views ### Simultaneous approximation of arbitrary functions in Hölder space and in L^2(\mu) by a smooth function and its derivative Let \mu be a probability measure on the circle S^1=\mathbb{R}/\mathbb{Z} which is singular with respect to the Lebesgue measure \lambda. Consider the functions spaces L^2(\mu) on the one hand, ... 2answers 176 views ### Accuracy of the truncated Hausdorff moment problem For a sequence of real numbers s = (s_i)_{i \in n} let M_s be the collection of functions f:[0,1] \to [0,1] such that$$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i In other words, $M_s$ ...
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...