This survey paper is devoted to the theory of orthogonal polynomial systems with respect to different inner product spaces. The author also concentrates on some theoretical aspects that are needed in the applications. The paper is divided in seven sections, including a very short introductory section with the definitions of inner product space and orthogonal system of elements of an inner product space. Orthogonality on the real line is considered in detail in Section 2, with the general properties, the classical orthogonal polynomials, the non-classical orthogonal polynomials including the ones which are orthogonal with respect to the generalized Gegenbauer weight \(w(t)= | t|^\mu (1-t^2)^\alpha\), \(\mu,\alpha >-1\), on \((-1,1)\), the hyperbolic weight \(w(t)=1/ \cosh t\) on \((-\infty,+\infty)\), and the logistic weight \(w(t)= e^{-t}/(1+e^{-t})^2\) on \((-\infty,+\infty)\). A short account of the \(s\)-orthogonal polynomials and Sobolev type orthogonal polynomials is given in the same Section 2. Section 3 is devoted to important applications, as Gaussian-type quadrature formulas, moment-preserving spline approximation and summation of slowly convergent series. Gaussian quadratures for integrals involving non-classical measures for which the coefficients of the recursion formula have been tabulated are included. Section 4 is concerned with the polynomials orthogonal on the unit circle and their properties, in particular their extremal properties and results for the zeros are recalled. The Section 5 presents some applications of polynomials orthogonal on the semicircle and a circular arc, including Geronimus’ version of orthogonality on a contour and an orthogonality on a growing semicicle with radius that tends to infinity. Finally, in Section 7 the author considers orthogonal polynomials on the radial rays in the complex plane. This well-written paper is a valuable resource for all people interested in the modern theory of orthogonal polynomials and their applications. A long list of references directs the reader to more detailed information on topics of particular interest and shows the big amount of the author’s work.
For the entire collection see [Zbl 0882.00018].