Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

Abstract

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

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

“…fractals we use our result on the existence of bump functions to prove a Borel-type theorem, showing that there are compactly supported smooth functions with prescribed jet at a junction point (Theorem 4.3). This gives a very general answer to a question raised in [19,5], and previously solved only for the Sierpinski Gasket [20]. We remark, however, that even in this special case the results of [20] neither contain nor are contained in the theorem proven here, as the functions in [20] do not have compact support, while those here do not deal with the tangential derivatives at a junction point.…”

“…In this theory, either a Dirichlet energy form or a diffusion on the fractal is used to construct a weak Laplacian with respect to an appropriate measure, and thereby to define smooth functions. As a result the Laplacian eigenfunctions are well understood, but we have little knowledge of other basic smooth functions except in the case where the fractal is the Sierpinski Gasket [19,5,20]. At the same time the existence of a rich collection of smooth functions is crucial to several aspects of classical analysis, where tools such as smooth partitions of unity, test functions and mollifications are frequently used.…”

Orthogonal Polynomials on the Sierpinski Gasket

“…Our construction of OP on SG will be based on another basis for the space H j introduced in [11]. Functions in this basis are called monomials and are essentially the fractal analogs of x j j!…”

Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket

“…In this paper we will study the local behavior of functions in the domain of powers of ν , analogous to the theory of Taylor approximations on the real line. For the standard Laplacian, the theory is well understood at boundary points and junctions points [16,17,13], and to a lesser extent at other points [4,1,14]. Here we will deal mostly with boundary points.…”

“…Note that these are different polynomials than those arise if we use the standard Laplacian, and the decay rates are different. In the case of the standard Laplacian, the global size of polynomials was analyzed in [13], and as a result a theory of power series and analytic functions was developed.…”