Haar Series and Linear Operators

Description

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 multipliers with respect to the Haar system; subspaces generated by subsequences of the Haar system; the criterion of equivalence of the Haar and Franklin systems. Audience: This book will be of interest to graduate students and researchers whose work involves functional analysis and operator theory. Front Matter....Pages i-xv Preliminaries....Pages 1-13 Definition and Main Properties of the Haar System....Pages 15-18 Convergence of Haar Series....Pages 19-24 Basis Properties of the Haar System....Pages 25-31 The Unconditionality of the Haar system....Pages 33-39 The Paley Function....Pages 41-50 Fourier-Haar Coefficients....Pages 51-72 The Haar system and martingales....Pages 73-82 Reproducibility of the Haar system....Pages 83-87 Generalized Haar Systems and Monotone Bases....Pages 89-107 Haar System Rearrangements....Pages 109-125 Fourier-Haar Multipliers....Pages 127-131 Pointwise Estimates of Multipliers....Pages 133-142 Estimates of Multipliers in L 1 ....Pages 143-149 Subsequences of the Haar system....Pages 151-167 Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces....Pages 169-189 Olevskii System....Pages 191-193 Back Matter....Pages 195-224