Orthogonal series. Transl. from the Russian by Ralph P. Boas. Transl. ed. by Ben Silver. (English) Zbl 0668.42011

Kashin, B. S.; Saakyan, A. A.

Translations of Mathematical Monographs, 75. Providence, RI: American Mathematical Society (AMS). xii, 451 p. $ 146.00 (1989).

For a review of the Russian original (1984) see Zbl 0632.42017.
In this book we present the fundamental methods of the theory of orthogonal series. We study general orthonormal systems as well as specific systems (for example, the Haar and Franklin systems). We present both, classical results and those obtained more recently. All propositions that fall outside the scope of university courses are provided with full proofs.
Contents: Introductory concepts and some general results. Independent functions and their first applications. The Haar systems. Some results on the trigonometric and Walsh systems. The Hilbert transform and some function spaces. The Faber-Schauder and Franklin systems. Orthogonalization and factorization theorems. Theorems on the convergence of general orthogonal series. General theorems on the divergence of orthogonal series. Some theorems on the representation of functions by orthogonal series

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A20	Convergence and absolute convergence of Fourier and trigonometric series

Keywords:

orthogonal series; Haar system; Fourier series; Franklin systems