Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)
Volume 87 / 2009
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.
