Uniformity of measures with Fourier frames

Frames of multi-windowed exponentials on subsets ofRd

“…(4.2) In this situation, μ is called a tight frame measure for L 2 (Ω) (see [4,6]). It is clear that if μ is the Lebesgue measure on R d , then, by the Plancherel theorem, μ is a tight frame measure for L 2 (Ω) for any Ω.…”

“…This characterization is motivated by the recent work of the second named author on Fourier frames of absolutely continuous measures [20,6]. In fact, we will see that Fourier frames of absolutely continuous measures are equivalent to frames of windowed exponentials generated by a single window (Proposition 5.2).…”

Fourier Series for Singular Measures

“…However, by a remark in [10], since {e n } ∞ n=0 is pseudo-dual to the (in this case Parseval) frame {g n } ∞ n=0 , the upper frame bound for {g n } ∞ n=0 implies a lower frame bound for {e n } ∞ n=0 . Moreover, some of the examples in [3,4] of measures that do not possess an exponential frame are singular, and hence if we normalize them to be probability measures, Theorem 1 applies.…”

“…It is known that there exist singular measures without exponential frames. In fact, Dutkay and Lai [3,4] showed that self-affine measures induced by iterated function systems with no overlap cannot possess exponential frames if the probability weights are not equal.…”

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

“…It has been an interesting question to produce singular measures with Fourier frames but not exponential orthonormal bases. By now we only know we can produce such measures by considering measures which are absolutely continuous with respect to a spectral measure with density bounded above and away from 0 or convolving a spectral measure with some discrete measures [HLL,DL1]. These methods are rather restrictive.…”