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Orthogonal polynomials and moment problems. (Italian) Zbl 0598.44007
Pubbl., Ser. III, Ist. Appl. Calcolo 202, 139 p (1981).
The booklet (139 pages) presents in a very attractive way the theory of orthogonal polynomials and that of the moment problem based on the well known results of M. Riesz [Arkiv Mat., Astr. Fys. 16, No.12, 23 p. and No.19, 21 p. (1922; JFM 48.1227.01) and 17, No.16 (1923; JFM 49.0195.01)]. The book contains five chapters. Ch. 1. contains some short preliminaries mostly without proofs. Ch. 2. deals with the properties of orthogonal and quasi-orthogonal polynomials following the idea of M. Riesz, i.e. developing this theory without using integrals. In Ch. 3. the moment problem is discussed. Conditions for the solvability and uniqueness conditions are given. Ch. 4. has the title: ”Conditions under which the moment problem is determined”. The last Ch. 5. investigates the problem of completeness of an orthogonal system of polynomials and its relations to the moment problem. Also this treatement is based on the ideas of M. Riesz [Acta Litt. ac Scient. Univ. Hung. 1, 209-225 (1923; JFM 49.0708.02)]. At the end of the book the reader finds a good bibliography containing 25 items; the end of every chapter gives detailed indications to the bibliography. The presentation is precise and easy to understand.
Reviewer: St.Feny√∂
44A60 Moment problems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)