Summability of multi-dimensional fourier series and hardy spaces


Summability of multi-dimensional fourier series and hardy spaces

by Ferenc Weisz

(Mathematics and its applications, Vol.541)

Kluwer Academic, c2002

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Includes reference (p.315-330) and index



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].


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.

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  • ISBN
    • 1402005644
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  • ページ数/冊数
    xv, 332 p.
  • 大きさ
    25 cm
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