A+ CATEGORY SCIENTIFIC UNIT

Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

Volume 87 / 2009

Julian Ławrynowicz, Tatsuro Ogata, Osamu Suzuki Banach Center Publications 87 (2009), 115-140 MSC: Primary 37F45; Secondary 28A80. DOI: 10.4064/bc87-0-11

Abstract

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the $L^{2}$-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III). (4) We give the Taylor expansion theorem on a fractal set (Theorem IV and IV'). (5) By use of the Taylor expansion for wavelet functions, we introduce basic functions, for example, exponential and trigonometrical functions, and discuss the relationship between the usual and introduced corresponding special functions (Theorem V). (6) Finally we discuss the relationship between the wavelet functions and the generating functions of the dynamical systems on a fractal set and show that our wavelet expansions can describe several fluctuations observed in nature.

Authors

  • Julian ŁawrynowiczInstitute of Physics
    University of Łódź
    Pomorska 149/53
    PL-90-236 Łódź
    and
    Institute of Mathematics
    Polish Academy of Sciences
    Łódź Branch
    Banacha 22
    PL-90-238 Łódź, Poland
    e-mail
  • Tatsuro OgataDepartment of Computer and System Analysis
    College of Humanities and Sciences
    Nihon University
    Sakurajosui 3-25-40
    156-8550 Setagaya-ku,Tokyo, Japan
    e-mail
  • Osamu SuzukiDepartment of Computer and System Analysis
    College of Humanities and Sciences
    Nihon University
    Sakurajosui 3-25-40
    156-8550 Setagaya-ku,Tokyo, Japan
    e-mail

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