Almost-convergent sequences with respect to polynomial hypergroups

Each polynomial hypergroup on ?₀ generates a family of generalized translation operators?T _m on sequence spaces. We introduce the concept of almost convergence for polynomial hypergroups (determined by the operators?T _m ), extending the notion of almost convergence introduced by Lorentz. Our investigations lead to two theorems characterizing almost convergent sequences on polynomial hypergroups.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

Positive additive functions with respect to interval filling sequences and some sequences of integers with gap conditions

This work was financially supported by the Hungarian Research Fund, No. 907.     

Basis of splines associated with some singular differential operators

Functions being piecewise in Ker (D ^k DpD) are a special case of Chebyshev splines having one nontrivial weight and also a special case of singular splines. An algorithm is designed which enables calculating with related B-splines and their derivatives. Ifp(t) is approximated by a piecewise constant, an interesting recurrence for calculating with polynomial B-splines is obtained.     

A polynomial generalization of some associated sequences related to set partitions

Periodica Mathematica Hungarica - In this paper, we consider a common polynomial generalization, denoted by $$w_m(n,k)=w_m^{a,b,c,d}(n,k)$$ , of several types of associated sequences. When $$a=0$$...     

Nuclear multilinear operators with respect to a partition

We introduce the notion of multilinear nuclear operator with respect to a partition and extend Grothendieck??s characterizations of nuclear linear operators into $l_{1},$ resp. $L_{1}( \mu ),$ in this new setting. We give the necessary and sufficient conditions for a natural operator to be nuclear with respect to a partition.     

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

In this paper we study the class of differential operators with polynomial coefficients Q_j in one complex variable satisfying the condition degQ_jj with equality for at least one j. We show that if degQ_k<k then the root with the largest modulus of the nth degree eigenpolynomial p_n of T tends to infinity when n→∞, as opposed to the case when degQ_k=k, which we have treated previously in [T. Bergkvist, H. RullgÅrd, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153–171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of p_n. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.     

Convexity with respect to a differential equation

The concepts of convexity of a set, convexity of a function and monotonicity of an operator with respect to a second-order ordinary differential equation are introduced in this paper. Several well-known properties of usual convexity are derived in this context, in particular, a characterization of convexity of function and monotonicity of an operator. A sufficient optimality condition for a optimization problem is obtained as an application. A number of examples of convex sets, convex functions and monotone operators with respect to a differential equation are presented.