Summability of Multi-Dimensional Fourier Series and Hardy Spaces

ISBN-13: 9789048159925 / Angielski / Miękka / 2010 / 332 str.

Ferenc Weisz

Opis

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 stochastics 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 mono graph 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]."

Spis treści

Preface. Acknowledgments. 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. Index.