相关论文: Bohr--Favard Inequality for differences and consta…

We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an $m$-dimensional cube and ball and the corresponding constants for entire functions of exponential type.

We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…

The Bohnenblust-Hille inequality for $m$-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as $m$ grows) is unknown, but significant for…

We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials…

We argue that a classical inequality due to Fan, Taussky and Todd (1955) is equivalent to the dissipativity of a Jordan block. As the latter can be characterised via the zeros of Chebyshev polynomials, we obtain a short new proof of the…

The main goal of this paper is to prove the Schmidt--Kolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next,…

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{P}\Delta(0;1_n)$. We also prove two other sharp versions of the Bohr inequality in…

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_p^2$ spaces, $2<p\leq\infty$…

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets,…

We study the Fourier approximation $\mathcal{F}_N$ of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast…

Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, so called Jackson-type operator…

We consider the variance of a function of $n$ independent random variables and provide new inequalities which, in particular, extend previous results obtained for symmetric functions in the i.i.d.~setting. For instance, we obtain various…

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials…

We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained…

Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate…

We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals…

We improve constants in the Rademacher-Menchov inequality.

Since its original formulation, Jensen's inequality has played a fundamental role across mathematics, statistics, and machine learning, with its probabilistic version highlighting the nonnegativity of the so-called Jensen's gap, i.e., the…

A relation between Fick-Jacobs and Schr\"{o}dinger equation is shown. When the diffusion coefficient is constant, exact solutions for Fick-Jacobs equation are obtained. Using a change of variable the general case is studied.

We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Ra\c{s}a (2017). The methods of our proof…