Shape preserving polynomial curves

We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical.     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

New examples of Osserman metrics with nondiagonalizable Jacobi operators

Explicit examples of Osserman 4-manifolds with exactly two distinct eigenvalues of the Jacobi operators, α and β=4α≠0, are given. The former has multiplicity two and is a double root of the minimal polynomial of the Jacobi operators.     

Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

This is an ultimate completion of our earlier paper [3] where mapping properties of several fundamental harmonic analysis operators in the setting of symmetrized Jacobi trigonometric expansions were investigated under certain restrictions on the underlying parameters of type. In the present article we take advantage of very recent results due to Nowak, Sjögren and Szarek to fully release those restrictions, and also to provide shorter and more transparent proofs of the previous restricted results. Moreover, we also study mapping properties of analogous operators in the parallel context of symmetrized Jacobi function expansions. Furthermore, as a consequence of our main results we conclude some new results related to the classical non-symmetrized Jacobi polynomial and function expansions.     

Domains of exponentiated fractional Jacobi operators: Characterizations,classifications, expansion results

The main topic of this paper is the study of bases of Jacobi polynomials in topological vector spaces of entire functions of slow growth. These topological vector spaces are weightedL ₂-spaces and inductive-projective limits of these. One of the side results is a characterization of the analyticity and entireness domains of general fractional Jacobi operators.     

A control problem with application to integral inequalities

The paper deals with extensions of Carlitz and Cohen et al. of the Brock recurrence relation, which occurs in a sorting problem, and is connected with generating functions of the Jacobi polynomial. New expressions are also presented regarding the positivity of coefficients in the power series expansions of rational functions.     

Differential operators of certain special functions and their applications to linear differential equations

The differential operators associated with Jacobi polynomials, the Languerrer polynomials and the parabolic cylinder (or Weber—Hermite) functions are denned. The corresponding commutator brackets are constructed, and it is shown that in each case the operators defined are non‐commutative. Some applications of these operators to linear differential equations are also considered.     

Some extensions of Carlitz on cyclic sums and generating functions

Recently, the author (Proc. Amer. Math. Soc., 57 1976, 271–275) derived two theorems involving double series, which gave as a consequence new and known generating functions for the Jacobi polynomial. The method of proof differed from that of previous workers. Using an extension of this procedure, we present in this paper two theorems for double and m-dimensional series which generalize our previous work. These formulas also yield new generating functions for the Jacobi polynomial and extend some formulas of Carlitz (Boll. U.M.I. (3), 16 1961, 150–155) and others. A feature of this work is the inclusion of the Jacobi polynomial within the framework of m-dimensional cyclic sums, thus generalizing a main result of Carlitz (SIAM Rev., 6 1964, 20–30).     

Function weighted measures and orthogonal polynomials on Julia sets

LetT(z) be a monic polynomial of degreed ?2 chosen so that its Julia setJ is real. A class of invariant measures supported onJ is constructed and discussed. We then construct the Jacobi matrices associated with these measures and show that they satisfy a renormalization group equation, a special case of which was discovered by Bellissard. Finally, we examine the asymptotic behavior of the orthogonal polynomials associated with these operators. We note that the operators have singular continuous spectra.     

Polynomial sequences associated with the classical linear functionals

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero–Sutherland Type

In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero–Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.     

Some Representations of Translations of the Product of Two Functions for Hankel Transforms and Jacobi Transforms

By introducing a series of operators T_y^(m), m = 0,1,. . . , a representation of the generalized translation of the product of two functions associated to Hankel transforms is given. These operators have certain properties of finite differences. The analogous problem related to Jacobi transforms is also studied.     

On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

Hypercyclic and Chaotic Convolution Associated with the Jacobi–Dunkl Operator

In this paper, we study the Jacobi–Dunkl convolution operators on some distribution spaces. We characterize the Jacobi–Dunkl convolution operators as those ones that commute with the Jacobi–Dunkl translations and with the Jacobi–Dunkl operators. Also we prove that the Jacobi–Dunkl convolution operators are hypercyclic and chaotic on the spaces under consideration and we give a universality property for the generalized heat equation associated with them.