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

Julian ŁawrynowiczTatsuro OgataOsamu Suzuki — 2009

Banach Center Publications

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)....

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