Fourier frames for singular measures and pure type phenomena

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Abstract

Let μ be a positive measure on ℝd. It is known that if the space L2(μ) has a frame of exponentials, then the measure μ must be of “pure type”: it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that μ cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each “pure” component of the measure separately may admit such a frame.

Original languageEnglish
Pages (from-to)2883-2896
Number of pages14
JournalProceedings of the American Mathematical Society
Volume146
Issue number7
DOIs
StatePublished - Jul 2018

Bibliographical note

Publisher Copyright:
© 2018 American Mathematical Society.

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