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Full 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.
Contents
1. Preliminaries.- 2. Definition and Main Properties of the Haar System.- 3. Convergence of Haar Series.- 4. Basis Properties of the Haar System.- 5. The Unconditionality of the Haar System.- 6. The Paley Function.- 7. Fourier-Haar Coefficients.- 8. The Haar System and Martingales.- 9. Reproducibility of the Haar System.- 10. Generalized Haar Systems and Monotone Bases.- 11. Haar System Rearrangements.- 12. Fourier-Haar Multipliers.- 13. Pointwise Estimates of Multipliers.- 14. Estimates of Multipliers in L1.- 15. Subsequences of the Haar System.- 16. Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces.- 17. Olevskii Systems.- References.
