A Duality Principle for Groups II: Multi-frames Meet Super-Frames
Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda$$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .  more » « less
Award ID(s):
NSF-PAR ID:
10291409
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Journal of Fourier Analysis and Applications
Volume:
26
Issue:
6
ISSN:
1069-5869
Format(s):
Medium: X
1. Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda$$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty$$ λ → ∞ \begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned} ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.
3. Abstract The multihadron decays $${\Lambda}_b^0$$ Λ b 0 → D + pπ−π− and $${\Lambda}_b^0$$ Λ b 0 → D * + pπ−π− are observed in data corresponding to an integrated luminosity of 3 fb − 1 , collected in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV by the LHCb detector. Using the decay $${\Lambda}_b^0$$ Λ b 0 → $${\Lambda}_c^{+}$$ Λ c + π + π − π − as a normalisation channel, the ratio of branching fractions is measured to be $$\frac{\mathcal{B}\left({\Lambda}_b^0\to {D}^{+}p{\pi}^{-}{\pi}^{-}\right)}{\mathcal{B}\left({\Lambda}_b^0\to {\Lambda}_c^0{\pi}^{+}{\pi}^{-}{\pi}^{-}\right)}\times \frac{\mathcal{B}\left({D}^{+}\to {K}^{-}{\pi}^{+}{\pi}^{+}\right)}{\mathcal{B}\left({\Lambda}_c^0\to {pK}^{-}{\pi}^{-}\right)}=\left(5.35\pm 0.21\pm 0.16\right)\%,$$ B Λ b 0 → D + p π − π − B Λ b 0 → Λ c 0 π + π − π − × B D + → K − π + π + B Λ c 0 → pK − π − = 5.35 ± 0.21 ± 0.16 % , where the first uncertainty is statistical and the second systematic. The ratio of branching fractions for the $${\Lambda}_b^0$$ Λ b 0 → D *+ pπ − π − and $${\Lambda}_b^0$$ Λ b 0 → D + pπ − π − decays is found to be $$\frac{\mathcal{B}\left({\Lambda}_b^0\to {D}^{\ast +}p{\pi}^{-}{\pi}^{-}\right)}{\mathcal{B}\left({\Lambda}_b^0\to {D}^{+}p{\pi}^{-}{\pi}^{-}\right)}\times \left(\mathcal{B}\left({D}^{\ast +}\to {D}^{+}{\pi}^0\right)+\mathcal{B}\left({D}^{\ast +}\to {D}^{+}\gamma \right)\right)=\left(61.3\pm 4.3\pm 4.0\right)\%.$$ B Λ b 0 → D ∗ + p π − π − B Λ b 0 → D + p π − π − × B D ∗ + → D + π 0 + B D ∗ + → D + γ = 61.3 ± 4.3 ± 4.0 % .