We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

Random-matrix theories in quantum physics : common concepts

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke

Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Foreword Preface 1. Basic hypergeometric series 2. Summation, transformation, and expansion formulas 3. Additional summation, transformation, and expansion formulas 4. Basic contour integrals 5. Bilateral basic hypergeometric series 6. The Askey-Wilson q-beta integral and some associated formulas 7. Applications to orthogonal polynomials 8. Further applications 9. Linear and bilinear generating functions for basic orthogonal polynomials 10. q-series in two or more variables 11. Elliptic, modular, and theta hypergeometric series Appendices References Author index Subject index Symbol index.

Basic Hypergeometric Series

Banach Algebras for Jacobi Series and Positivity of a Kernel

Let Pn (α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by [9, p. 67], and let Rn (α,β)(x) = Pn (α,β)(x)/Pn(αβ)(1). Then for n ≧ m, where Since Rn (α, β)(l) = 1, it follows that (1) It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.

Linearization of the Product of Jacobi Polynomials. I

Positivity and the Convolution Structure for Jacobi Series

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.

George Gasper

Papers

Basic Hypergeometric Series

Banach Algebras for Jacobi Series and Positivity of a Kernel

Linearization of the Product of Jacobi Polynomials. I

Positivity and the Convolution Structure for Jacobi Series

Some systems of multivariable orthogonal q-Racah polynomials