The theory of general orthogonal series originated at the turn of the century as a natural generalization, based on Lebesgue integration, of the theory of trigonometric series. However, the most active developments have occurred in the past 25 years, when it has become clear that many propositions about properties of the trigonometric system remain valid for a broad class of orthonormal systems.

Focusing on the fundamental methods of the theory of orthogonal series, this book presents a study of general orthonormal systems as well as specific systems such as the Haar and Franklin systems, covering both classical and recent results. The authors prove a number of results that have appeared in the literature but have not been gathered together in a monograph, so this book will be of interest to specialists in the field. However, the book is primarily oriented toward beginners in this area, and many of the fundamental theorems are given full proofs. The required background includes a familiarity with functional analysis and with the basic theory of functions of a complex variable; some background material on the theory of functions and functional analysis is presented in the appendices.

Table of Contents

• Introductory concepts and some general results
• Independent functions and their first applications
• The Haar system
• 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