Existence and exactness of exponential Riesz sequences and frames for fractal measures. (English) Zbl 1475.28007
The purpose of this paper is the construction of exponential frames and Riesz sequences for a class of fractal measures on \(\mathbb{R}^d\) which are generated by infinite convolutions of discrete measures. To this end, frame towers and Riesz-sequence towers are employed. The authors completely characterize the exactness and over-completeness of the constructed exponential frames or Riesz sequences in terms of the cardinality at each level of the tower. Moreover, they show, using a version of the solution of the Kadison-Singer problem, known as the \(R_\epsilon\)-conjecture, that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, it is shown that there are always exponential Riesz sequences of maximal possible Beurling dimension provided that the measure is the middle-third Cantor measure, or, more generally, a self-similar measure satisfying the no-overlap condition.
Reviewer: Peter Massopust (München)
MSC:
| 28A80 | Fractals |
| 42C15 | General harmonic expansions, frames |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
Keywords:
exponential frame; exponential Riesz sequence; fractal measure; spectral set; frame tower; Riesz sequence towerReferences:
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