Felix Voigtlaender
Coorbit dual molecules and convolution dominated operators
Abstract: Many continuous frames—including Gabor frames, wavelets, and certain types of shearlets—are induced by group representations. An important question is whether one can obtain a discrete frame by sampling such a localized continuous system. Preferably, the discrete dual frame should be well-localized as well, ensuring that the frame expansion extends to function spaces beyond the Hilbert space setting.
One of the main results of coorbit theory shows that this is indeed possible in many cases. In this talk, we present a novel proof of this fact. This proof is more accessible than the traditional one and yields a stronger conclusion, showing that the dual frame forms a family of coorbit molecules. The main proof ingredient is a novel local holomorphic calculus for convolution-dominated operators on general locally compact groups, which might be of independent interest.
In more technical detail, let G be a locally compact group and let π be an irreducible representation of G on a Hilbert space ℋπ. The voice transform of f with respect to g is Vgf(x)=⟨f, π(x)g⟩. We assume that Vgg belongs to the Wiener amalgam space W(L∞, Lw1) and is normalized so that
f = ∫GVgf(x) π(x)g dμ(x) ∀ f ∈ ℋπ.
We show that for every discrete, sufficiently dense set Λ ⊂ G, the family (π(λ)g)λ ∈ Λ is a frame for ℋπ, and there exists a dual frame (hλ)λ ∈ Λ that forms a family of coorbit-molecules, in the sense that
|Vghλ(x)| ≤ Θ(λ−1x)
for all λ ∈ Λ, with a suitable envelope Θ ∈ W(L∞, Lw1).
This molecule condition implies that the size of the coefficients (⟨f, hλ⟩)λ ∈ Λ fully reflects the size profile of the voice transform Vgf. It also ensures that the frame expansion
f = ∑λ ∈ λ⟨f, hλ⟩ π(λ)g = ∑λ ∈ λ⟨f, π(λ)g⟩ hλ
extends to the associated coorbit spaces.
Finally, for very regular sets Λ (in particular for quasi-lattices), we show that the canonical dual frame forms a family of coorbit-molecules.