On a homogeneous group, we characterize the one-parameter groups of dilations whose associated Hardy spaces in the sense of Folland and Stein are the same.

This paper provides a characterization of expansive matrices A∈ GL (d, R) generating the same anisotropic homogeneous Triebel–Lizorkin space F˙p,qα(A) for α∈ R and p, q∈ (0 , ∞] . It is shown that F˙p,qα(A)=F˙p,qα(B) if and only if the homogeneous quasi-norms ρ_A, ρ_B associated to the matrices A, B are equivalent, except for the case F˙p,20=Lp with p∈ (1 , ∞) . The obtained results complement and extend the classification of anisotropic Hardy spaces Hp(A)=F˙p,20(A) , p∈ (0 , 1] , in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).

This paper concerns the overcompleteness of coherent frames for unimodular amenable groups. It is shown that for coherent frames associated with a localized vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces F˙p,qα for the endpoint case of p= ∞ and the full scale of parameters α∈ R and q∈ (0 , ∞]. In particular, a Peetre-type characterization of the anisotropic Besov space B˙∞,∞α=F˙∞,∞α is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in F˙∞,qα.

This paper provides maximal function characterizations of anisotropic Triebel–Lizorkin spaces associated to general expansive matrices for the full range of parameters p∈ (0 , ∞) , q∈ (0 , ∞] and α∈ R. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of mild decay, moment, and smoothness conditions.

Let π_α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space Aα2(Ω) on a bounded symmetric domain Ω , of formal dimension dπα>0. It is shown that if the Bergman kernel kz(α) is a cyclic vector for the restriction π_α| _Γ to a lattice Γ ≤ G (resp. (πα(γ)kz(α))γ∈Γ is a frame for Aα2(Ω)), then vol(G/Γ)dπα≤|Γz|-1. The estimate vol(G/Γ)dπα≥|Γz|-1 holds for kz(α) being a p_z-separating vector (resp. (πα(γ)kz(α))γ∈Γ/Γz being a Riesz sequence in Aα2(Ω)). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G= PSU (1 , 1).

This note provides new criteria on a unimodular group G and a discrete series representation (π, H_π) of formal degree d_π> 0 under which any lattice Γ ≤ G with vol(G/Γ)dπ≤1 (resp. vol(G/Γ)dπ≥1) admits g∈ H_π such that π(Γ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

Let G = N ⋉ A, where N is a graded Lie group and A = R⁺ acts on N via homogeneous dilations. The quasi-regular representation π = ind^G_A(1) of G can be realised to act on L²(N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L²(N) into L²(G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.

We derive an extension of the Walnut–Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that L² frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the L² canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces.

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.

Let G be a nilpotent Lie group and let π be a coherent state representation of G. The interplay between the cyclicity of the restriction πjΓ to a lattice ≤ G and the completeness of subsystems of coherent states based on a homogeneous G-space is considered. In particular, it is shown that necessary density conditions for Perelomov's completeness problem can be obtained via density conditions for the cyclicity of πjΓ.

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian–Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C^⁎-module. An important ingredient in the approach is that twisted group C^⁎-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C^⁎-algebra has strict comparison of projections.

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood—Paley-type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.