Title

Affine Fractals As Boundaries And Their Harmonic Analysis

Keywords

Affine fractal; Cantor measure; Cantor set; Fourier bases; Hilbert space; Iterated function system

Abstract

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 H2. By this we mean that there are lacunary subsets Γ of the non-negative integers and associated closed Γ-subspace in the Hardy space H2(D), D; denoting the disk, such that for every function f in H2(Γ) and for every point z in D, f(z) admits a boundary integral represented by an associated measure μ, with integration over supp(μ) placed as a Cantor subset on the circle T:= bd(D). We study families of pairs: measures μ and sets Γ of lacunary form, admitting lacunary Fourier series in L2(μ); i.e., configurations Γ arranged with a geometric progression of empty spacing, missing parts, or gaps. Given Γ, we find corresponding generalized Szegö kernels GΓ, and we compare them to the classical Szegö kernel for D. Rather than the more traditional approach of starting with μ and then asking for possibilities for sets Γ, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset Γ and within a new duality framework, we study the possibilities for choices of measures μ. © 2011 American Mathematical Society.

Publication Date

9-1-2011

Publication Title

Proceedings of the American Mathematical Society

Volume

139

Issue

9

Number of Pages

3291-3305

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/S0002-9939-2011-10752-4

Socpus ID

79959300593 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/79959300593

This document is currently not available here.

Share

COinS