# Tagged Questions

190 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 ...
74 views

### The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion: \begin{equation*} B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}. ...
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### Bound on a sum of Laguerre polynomials

I am trying to find an asymptotic behavior, for large real $t$, of the following sum \begin{align} Q(t)=\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{1+n}L_n^{(1)}(t-n) \end{align} where $L_n^{(\alpha)}$ is ...
231 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: \nu = \frac{s}{\|s\|} ...
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 ...
112 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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### 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 ...
550 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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### 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 $U_n \subset L^\infty[-1,1]$ ...
333 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. ...
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. ...
566 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
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### 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} ...
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### 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 $x \in \mathbb{R}$ $f_n(x)$ ...
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### 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 ...
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 $X$ ...