Coorbit spaces and wavelet coefficient decay over general dilation groups

Abstract

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L 2 ( R d ) \textrm {L}^2(\mathbb {R}^d) . It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed L p \textrm {L}^{p} -estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

“…Later it was seen to apply to other settings and groups as well, for example the shearlet groups in dimensions two and higher [7,9,6]. This paper continues work begun in [24,25], which provided explicit criteria for wavelets associated to general dilation groups, ensuring their suitability as analyzing vectors and/or atoms in the coorbit scheme. The results in these sources depend on a number of fairly technical conditions on the dilation groups, defined as ( strong) temperate embeddedness of the associated open dual orbit.…”

“…The following theorem summarizes the chief results of [24,25] regarding membership in A v 0 and B v 0 . Informally, the significance of the theorem can be described as follows: Part (a) shows that there is a large and fairly handy class of nice wavelets around, in the form of bandlimited Schwartz functions.…”

“…For more background on the representation-theoretic aspects we refer to the book [22], and to the previous papers [24,25] for additional information concerning coorbit theory for this setting. We assume that the dilation group H is chosen such that π is an (irreducible) squareintegrable representation, and call such dilation groups irreducibly admissible.…”

“…Ideally, one would wish for transparent criteria similar to the above-mentioned ones for wavelet ONB's in the one-dimensional case, i.e., in terms of smoothness, decay and vanishing moments. The papers [24,25] are chiefly concerned with providing such criteria for general dilation groups. Here it is of key importance to employ a notion of vanishing moments that takes into account the dual orbit in a proper way.…”

“…As a consequence, we will obtain a simple argument that the properties of (strong) temperate embeddedness are invariant under conjugacy. This was mentioned (and used) in [24,25], but since the norm need not be invariant under a given linear change of coordinates, we found it useful to elaborate on this point. The following somewhat technical lemma contains the main estimate.…”