Related papers: Bernstein's problem on weighted polynomial approxi…
We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.
We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…
In this paper, we consider the problem of determining the density of monic polynomials over $\mathbb{Z}_p$ with squarefree discriminant over various subsets of the set of monic polynomials over $\mathbb{Z}_p$ of fixed degree. We compute the…
We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$…
We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…
The general methods which are powerful for the necessity of bounded commutators are given. As applications, some necessary conditions for bounded commutators are first obtained in certain endpoint cases, and several new characterizations of…
In this paper, we prove the boundedness of the Bergman projection on weighted mixed norm spaces of the upper-half space for some weights that are constructed using the logarithm function and growth functions. Our necessary and sufficient…
Let $\mu$ be a probability measure on $\mathbb{R}$. We give conditions on the Fourier transform of its density for functionals of the form $H(a)=\int_{\mathbb{R}^n}h(\langle a,x\rangle)\mu^n(dx)$ to be Schur monotone. As applications, we…
This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are…
Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…
Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial $f$ if it applies to the number field $K=\mathbb{Q}[x]/(f)$…
We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein operators.
We establish $L^p$-type universal approximation theorems for general and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to…
In the case when the weight and its inverse belong to BMO(T), we prove the asymptotics of the monic orthogonal polynomials in L^p, 2<p<p_0. Immediate applications include the estimates on the uniform norm and asymptotics for the polynomial…
In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…
In this short note we have proved an enhanced version of a theorem of Lorentz [1] and its generalization to the multivariate case which gives a non- uniform estimate of degree of approximation by a polynomial with positive coefficients. The…
We study comparison of Lp norms of polynomials on the sphere with respect to doubling measures. From our description it follows an uncertainty principle for square integrable functions on the sphere. We consider also weighted uniform…
We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…
We consider the pointwise weighted approximation by Bernstein operators with inner singularities. The related weight functions are weights $\bar{w}(x)=|x-\xi|^\alpha(0<\xi<1,\ \alpha>0).$ In this paper we give direct and inverse results of…