CONTINUOUS AND DISCRETE FOURIER FRAMES FOR FRACTAL MEASURES

    Authors

    D. E. Dutkay; D. G. Han;E. Weber

    Comments

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    Abbreviated Journal Title

    Trans. Am. Math. Soc.

    Keywords

    Plancherel theorem; frame; Bessel; Fourier series; Hilbert space; fractal; self-similar; iterated function system; SELF-SIMILAR MEASURES; BEURLING DIMENSION; TRANSFORMS; SUBSPACES; SYSTEMS; SERIES; SPACES; Mathematics

    Abstract

    Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on R-d, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure, then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular, we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure, then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

    Journal Title

    Transactions of the American Mathematical Society

    Volume

    366

    Issue/Number

    3

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    1213

    Last Page

    1235

    WOS Identifier

    WOS:000329123600005

    ISSN

    0002-9947

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