Novikov I., Semenov E. Haar Series and Linear Operators

Dordrecht: Kluwer Academic Publishers. – 1977. – 228 p. (Mathematics and Its Applications: Volume 367) In 1909 Alfred Haar introduced into analysis a remarkable system which bears his name. The Haar system is a complete orthonormal system on [0,1] and the Fourier-Haar series for arbitrary continuous function converges uniformly to this function. This volume is devoted to the investigation of the Haar system from the operator theory point of view. The main subjects treated are: classical results on unconditional convergence of the Haar series in modern presentation; Fourier-Haar coefficients; reproducibility; martingales; monotone bases in rearrangement invariant spaces; rearrangements and multiplies with respect to the Haar system; subspaces generated by subsequences of the Haar system; the criterion of equivalence of the Haar and Franklin systems. This book will be of interest to graduate students and researchers whose work involves functional analysis and operator theory.Acknowledgements
Remarks
Preliminaries
Definition and Main Properties of the Haar System
Convergence of Haar Series
Basis Properties of the Haar System
The Unconditionality of the Haar System
The Paley Function
Fourier-Haar Coefficients
The Haar System and Martingales
Reproducibility of the Haar System
Generalized Haar Systems and Monotone Bases
Haar System Rearrangements
Fourier-Haar Multipliers
Pointwise Estimates of Multipliers
Estimates of Multipliers in L,
Subsequences of the Haar System
Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces
Olevskii Systems