Related papers: On operators preserving inequalities between polyn…
In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to…
The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
The properties of the Pastro polynomials on the real line are studied with the help of a triplet of $q$-difference operators. The $q$-difference equation and recurrence relation these polynomials obey are shown to arise as generalized…
We obtain strong converse inequalities for the Bernstein operators with explicit constants. One of the main ingredients in our approach is the representation of the derivatives of the Bernstein operators in terms of the orthogonal…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…
We show that linear differential operators with polynomial coefficients over a field of characteristic zero can be multiplied in quasi-optimal time. This answers an open question raised by van der Hoeven.
The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Our approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine…
We study non-linear surjective mappings on subsets of ${\cal M}_n(F)$, which preserve the zeros of some fixed polynomials in noncommuting variables. Keywords: Matrix algebra, Multilinear polynomials, Preservers.
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…
Higher order Bernstein- and Markov-type inequalities are established for trigonometric polynomials on compact subsets of the real line and algebraic polynomials on compact subsets of the unit circle. In the case of Markov-type inequalities…
In this note we attempt to develop an analog of P\'olya-Schur theory describing the class of univariate hyperbolicity preservers in the setting of linear finite difference operators. We study the class of linear finite difference operators…
This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated…
In this short paper, the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anticommutators. The formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The role of…
The characterization mentioned in the title is found.
Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without…
A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real…
Real linear operators emerge in a range of mathematical physics applications. In this paper spectral questions of compact real linear operators are addressed. A Lomonosov-type invariant subspace theorem for antilinear compact operators is…