Mock Fourier series and transforms associated with certain Cantor measures

Abstract

We extend the construction, originally due to Jorgensen and Pedersen, of spectral pairs {μ, Λ}, consisting of Cantor measures μ on ℝⁿ and discrete sets Λ such that the exponentials with frequency in Λ form an orthonormal basis forL ²(μ). We give conditions under which these mock Fourier series expansions ofL ¹(μ) functions converge in a weak sense, and for a dense set of continuous functions the convergence is uniform. We show how to construct spectral pairs (²(μ) of infinite Cantor measures with unbounded support such that$\hat f(\lambda ) = \smallint e( - x \cdot \lambda )f(x)d\tilde \mu (x),$ defined for a dense subset ofL ²(μ), extends to an isometry fromL ²(μ) ontoL ²(μ'), a kind of mock Fourier transform. Our constructions do not require self-similarity, but only a compatible product structure for the pairs. We also give an analogue of the Shannon Sampling Theorem to reconstruct a function whose Fourier transform is supported in the Cantor set associated with μ from its values on Λ.