The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

A hilbert space problem book

The Convolution Transform. By Hirschmann & Widder . Pp. x + 268. 45s. 1955. (Princeton University Press. London : Cumberlege)

Polynomials orthogonal on the unit circle

An introduction to orthogonal polynomials, by Theodore S. Chihara. Pp xii, 249. £26·30. 1978. SBN 0 677 04150 0 (Gordon and Breach)

The q-jα Bessel Function

Nonsymmetric Askey–Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl–Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey–Wilson and of the little q-Jacobi case.

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Nonsymmetric Askey–Wilson polynomials as vector-valued polynomials

Elements of harmonic analysis related to the third basic zero order Bessel function

Heisenberg uncertainty principle for a fractional power of the Dunkl transform on the real line

We solve the problem of constructing entire functions where ln M (r; f) grows like ln2 r from their values at q−n, for 0 < q < 1. As application we give a product formula for the basic sine function.

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Fethi Bouzeffour

Papers

The q-jα Bessel Function

Nonsymmetric Askey–Wilson polynomials as vector-valued polynomials

Elements of harmonic analysis related to the third basic zero order Bessel function

Heisenberg uncertainty principle for a fractional power of the Dunkl transform on the real line

Interpolation of entire functions, product formula for basic sine function