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

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Ławrynowicz J., Ogata T., Suzuki O.

Banach Center Publications

2009

tom 87

nr 1

115-140

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 L²-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.

Supercomplex structures, surface soliton equations, and quasiconformal mappings

100%

Ławrynowicz J., Kędzia K., Suzuki O.

Annales Polonici Mathematici

1991

tom 55

nr 1

245-268

Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. The following results are obtained: (i) A field operator of Dirac type, which is called a Hurwitz operator, is introduced by use of a Hurwitz pair and its characterization is given (Theorem 1). (ii) A field equation of the elliptic Neveu-Schwarz model of superstring theory is obtained from the Hurwitz pair (𝔼⁴,𝔼³) (Theorem 2), and its counterpart connected with the Hurwitz triple $(𝔼^{11},𝔼^{11},𝔼^{26})$ is mentioned. (iii) Isospectral deformations of the Hurwitz operator of the Hurwitz pair (𝔼²,𝔼²) induce various soliton equations (Theorem 3). (iv) A special complex structure, which is called a supercomplex structure, is introduced on separable Hilbert spaces (Definition 10). A correspondence between such structures and reduction solutions of Sato's version of Kadomtsev-Petviashvili system is established (Theorem 4). (v) The general class of quasiconformal mappings in the plane is obtained from generalized Hurwitz pairs (Theorem 5). From these results we conclude that Hurwitz pairs and triples give rise to several interesting applications.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Banach Center Publications

2 Suzuki O.

2 Ławrynowicz J.

1 Kędzia K.

1 Ogata T.

1 2009

1 1991

Wyniki wyszukiwania

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

Supercomplex structures, surface soliton equations, and quasiconformal mappings