We prove that transplantations for Jacobi polynomials can be derived from representation of a special integral operator as fractional Weyl’s integral. Furthermore, we show that, in a sense, Jacobi transplantation can be reduced to transplantations for ultraspherical polynomials. As an application of these results, we obtain transplantation theorems for Jacobi polynomials in ReH1 and BMO. The pa...

In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld [BD] have recently given a notion of “chiral algebra” in terms of D-modules on algebraic curves. This definition consists of a “skew-symmetry” relation and a “Jacobi identity” relation in a categorical setting, and it leads to the operator product expansion...

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As a special case we obtain a generalization of Kenser’s criterion for Sturm-Liouville operators to Jacobi operators.

We determine the matrix of the balanced metric of the Siegel–Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace–Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel–Jacobi ball.

A four-parameter family of multivariable big q-Jacobi polynomials and a threeparameter family of multivariable little q-Jacobi polynomials are introduced. For both families, full orthogonality is proved with the help of a second-order q-difference operator which is diagonalized by the multivariable polynomials. A link is made between the orthogonality measures and R. Askey’s q-extensions of Sel...

The notions of N = 1 Neveu-Schwarz vertex operator superal-gebra over a Grassmann algebra and with odd formal variables and of N = 1 Neveu-Schwarz vertex operator superalgebra over a Grass-mann algebra and without odd formal variables are introduced, and we show that the respective categories of such objects are isomorphic. The weak supercommutativity and weak associativity properties for an N ...

In 1967 Durrmeyer introduced a modiication of the Bernstein polynomials as a selfadjoint polynomial operator on L 2 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 modiication, and identiied these operators as de la Vall ee{Poussin means with...

First we give a compact treatment of the Jacobi polynomials on a simplex in IR which exploits and emphasizes the symmetries that exist. This includes the various ways that they can be defined: via orthogonality conditions, as a hypergeometric series, as eigenfunctions of an elliptic pde, as eigenfunctions of a positive linear operator, and through conditions on the Bernstein–Bézier form. We the...

Without the factor of 1 2 , this would be the semidirect product Lie algebra for the usual action of gl(n,R) on R. With the factor of 1 2 , the bracket does not satisfy the Jacobi identity. Nevertheless, it does satisfy the Jacobi identity on many subspaces which are closed under the bracket. In fact, we will see that any Lie algebra structure on R is realized on such a subspace. If B is any bi...

We establish a rate of convergence for a semi-discrete operator splitting method applied to Hamilton-Jacobi equations with source terms. The method is based on sequentially solving a Hamilton-Jacobi equation and an ordinary diierential equation. The Hamilton-Jacobi equation is solved exactly while the ordinary diierential equation is solved exactly or by an explicit Euler method. We prove that ...