AFFINE FRACTALS AS BOUNDARIES AND THEIR HARMONIC ANALYSIS
Abbreviated Journal Title
Proc. Amer. Math. Soc.
Affine fractal; Cantor set; Cantor measure; iterated function system; Hilbert space; Fourier bases; CANTOR MEASURES; SPACES; Mathematics, Applied; Mathematics
We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these fractals. We prove that as sets these fractals arise as boundaries of functions in closed subspaces of the Hardy space H-2. By this we mean that there are lacunary subsets Gamma of the non-negative integers and associated closed Gamma-subspace in the Hardy space H-2(D), D denoting the disk, such that for every function f in H-2 (Gamma) and for every point z in D, f (z) admits a boundary integral represented by an associated measure mu, with integration over supp(mu) placed as a Cantor subset on the circle T := bd(D). We study families of pairs: measures mu and sets Gamma of lacunary form, admitting lacunary Fourier series in L-2(mu); i.e., configurations Gamma arranged with a geometric progression of empty spacing, missing parts, or gaps. Given Gamma, we find corresponding generalized Szego kernels G(Gamma), and we compare them to the classical Szego kernel for D. Rather than the more traditional approach of starting with mu and then asking for possibilities for sets Gamma, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset Gamma and within a new duality framework, we study the possibilities for choices of measures mu.
Proceedings of the American Mathematical Society
"AFFINE FRACTALS AS BOUNDARIES AND THEIR HARMONIC ANALYSIS" (2011). Faculty Bibliography 2010s. 1270.