Summability of Multi-Dimensional Fourier Series and Hardy Spaces / Edition 1

by Ferenc Weisz

Overview

The history of martingale theory goes back to the early fifties when Doob [57] pointed out the connection between martingales and analytic functions. On the basis of Burkholder's scientific achievements the mar­ tingale theory can perfectly well be applied in complex analysis and in the theory of classical Hardy spaces. This connection is the main point of Durrett's book [60]. The martingale theory can also be well applied in shastics and mathematical finance. The theories of the one-parameter martingale and the classical Hardy spaces are discussed exhaustively in the literature (see Garsia [83], Neveu [138], Dellacherie and Meyer [54, 55], Long [124], Weisz [216] and Duren [59], Stein [193, 194], Stein and Weiss [192], Lu [125], Uchiyama [205]). The theory of more-parameter martingales and martingale Hardy spaces is investigated in Imkeller [107] and Weisz [216]. This is the first monograph which considers the theory of more-parameter classical Hardy spaces. The methods of proofs for one and several parameters are en­ tirely different; in most cases the theorems stated for several parameters are much more difficult to verify. The so-called atomic decomposition method that can be applied both in the one-and more-parameter cases, was considered for martingales by the author in [216].

Product Details

ISBN-13:	9781402005640
Publisher:	Springer Netherlands
Publication date:	03/31/2002
Series:	Mathematics and Its Applications , #541
Edition description:	2002
Pages:	332
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

1. Multi-Dimensional Dyadic Hardy Spaces.- 2. Multi-Dimensional Classical Hardy Spaces.- 3. Summability of D-Dimensional Walsh-Fourier Series.- 4. The D-Dimensional Dyadic Derivative.- 5. Summability of D-Dimensional Trigonometric-Fourier Series.- 6. Summability of D-Dimensional Fourier Transforms.- 7. spline and Ciesielski Systems.- References.