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: \begin{equation} \nu = \frac …

Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are unifo …

we have a function $f(x)\in C^{K}[-1,1]^{d}$, whether there exists a multivariate B-spline function $s(x)$, such that $|s(x)-f(x)|_{\infty}=O(h^{K})$? where $h$ is the width of th …

I am interested in approximating the sum of the squares of the multinomial coefficients, i.e. $a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$ or mo …

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0 …

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\ …

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim …

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 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^{ …

Let $T_{N}$ be the (Hilbert) matrix defined by $T_{N}(m,n)=\frac{1}{m-n}$ if $1\leq m,n \leq N$ and $m\neq n$ , and $ T_{N}(n,n)=0$ if $1\leq n \leq N$ . It's well known that $\Ver …

Hi everybody, this is my first question.$L^2$,$H^p$ are the standard Lebesgue,Sobolev spaces here, and I am deliberately omitting the domains because I'll accept an answer if it's …

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following: $$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k …

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 $n$ bo …

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' …

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 …