Related papers: On weighted polynomial approximation

We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all…

It is shown by the author in [J. Lie Theory 29:4, 1045-1070, 2019] that for every connected linear complex Lie group the algebra of polynomials (regular functions) is dense in the algebra of holomorphic functions of exponential type.…

In 1924 S.Bernstein asked for conditions on a uniformly bounded on $\mathbb{R}$ Borel function (weight) $w: \mathbb{R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials ${\mathcal{P} }$ in the seminormed space $…

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open…

We prove that if $f:(a,b)\to\mathbb{R}$ is convex, then for any $\varepsilon>0$ there is a convex function $g\in C^2(a,b)$ such that $|\{f\neq g\}|<\varepsilon$ and $\Vert f-g\Vert_\infty<\varepsilon$.

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…

We consider approximating of the median of absolutely continuous distribution given by a probability density function $f$. We assume that $f$ has $r$ continuous derivatives, with derivative of order $r$ being H\"older continuous with the…

Let $x \in \mathbb{R}$ be arbitrary and consider the `greedy' approximation of $x$ by signed harmonic sums: given $a_n = \sum_{k \leq n} \varepsilon_k/k$ with $\varepsilon_k \in \left\{-1,1\right\}$, we set $\varepsilon_{n+1} = 1$ if $a_n…

We give an abstract approach to approximations with a wide range of regularity classes $X$ in spaces of pseudocontinuable functions $K^p_\vartheta$, where $\vartheta$ is an inner function and $p>0$. More precisely, we demonstrate a general…

We establish a relation between the approximation in $L^2[-\pi,\pi]$ by exponentials with the set of frequencies of Beurling--Malliavin density less than $1$ and the meromorphic interpolation at $\mathbb Z$. Furthermore, we show that…

Renyi's result on the density of integers whose prime factorizations have excess multiplicity has an analogue for polynomials over a finite field.

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$ be a function of class $C^1$, with $\sup_{\bf R}f'>0$, such that $$\lim_{|\xi|\to +\infty}{{f(\xi)}\over {\xi}}=0\ .$$ Then, for each $\lambda>{{\pi^2}\over…

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each…

Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as…

For any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P $\in$ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.

Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \Psi(t) be a positive continuous function such that \|\Psi f^{\beta}\|_{\infty}<\infty for some 0<\beta<1/2. Under natural…

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if there…

Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$. This requires…

We study multiplicities $a^{d\lambda}_{\mu,(dk)}$ of highest weight representations $\mathbb S_{d\lambda}(\mathbb C^n)$, $\lambda\vdash pk$, of length at most $p$, in $\mathbb{S}_{\mu}(S^{dk}(\mathbb C^n))$, $\mu\vdash p$, so called…