Mock Fourier series and transforms associated with certain Cantor measures (Q1591324)

This paper extends a construction, originally introduced by \textit{P. E. T. Jorgensen} and \textit{S. Pedersen} [J. Anal. Math. 75, 185-228 (1998; Zbl 0959.28008)], of spectral pairs \(\{\mu, \Lambda \}\) consisting of Cantor measures \(\mu\) on \(R^n\) and discrete sets \(\Lambda\) such that the exponentials with frequency in \(\Lambda\) form an orthonormal basis for \(L^2(\mu)\). The author gives conditions under which these mock Fourier series expansions of \(L^1(\mu)\) functions converge in a weak sense, and shows that for a dense set of continuous functions the convergence is uniform. He also shows how to construct spectral pairs \((\widetilde{\mu}, \widetilde{\mu}')\) of infinite Cantor measures with unbounded support such that \[ \widehat{f}(\lambda) := \int e(-x \cdot \lambda) f(x) d\widetilde{\mu}(x), \] defined for a dense subset of \(L^2(\widetilde{\mu})\), extends to an isometry from \(L^2(\widetilde{\mu})\) to \(L^2(\widetilde{\mu}')\), a kind of mock Fourier transform. This construction does not require self--similarity, but only a compatible product structure for the pairs. He also gives an analogue of the Shannon Sampling Theorem to reconstruct a function whose Fourier transform is supported in the Cantor set associated with \(\mu\) from its values on \(\Lambda\).