127 research outputs found
Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series
We study the analog of power series expansions on the Sierpinski gasket, for
analysis based on the Kigami Laplacian. The analog of polynomials are
multiharmonic functions, which have previously been studied in connection with
Taylor approximations and splines. Here the main technical result is an
estimate of the size of the monomials analogous to x^n/n!. We propose a
definition of entire analytic functions as functions represented by power
series whose coefficients satisfy exponential growth conditions that are
stronger than what is required to guarantee uniform convergence. We present a
characterization of these functions in terms of exponential growth conditions
on powers of the Laplacian of the function. These entire analytic functions
enjoy properties, such as rearrangement and unique determination by infinite
jets, that one would expect. However, not all exponential functions
(eigenfunctions of the Laplacian) are entire analytic, and also many other
natural candidates, such as the heat kernel, do not belong to this class.
Nevertheless, we are able to use spectral decimation to study exponentials, and
in particular to create exponentially decaying functions for negative
eigenvalues
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