Let G be a Lie group of polynomial growth. We prove that the second-order Riesz transforms on L 2 (G ; dg) are bounded if, and only if, the group is a local direct product of a compact group and a nilpotent group, in which case the transforms of all orders are bounded.

Abstract In this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish characterizations for translates a single function to form Riesz bases their closed linear span. provide complete characterization biorthogonality scaling functions two multiresolution analysis and associated wavelet families in $$L^2({\mathbb {R}}^+)$$ <mml:math xmlns...

We present a novel framework for the reconstruction of 1D composite signals assumed to be mixture two additive components, one sparse and other smooth, given finite number linear measurements. formulate problem as continuous-domain regularized inverse with multiple penalties. prove that these penalties induce reconstructed indeed take desired form sum smooth component. then discretize this usin...

In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases. Then we construct Bessel sequences frames and Riesz bases for the class of Hilbert Schmidt operators using the tensor product of such sequences in the original...

The key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their Weyl symbols. This observation has two implications in time-frequency analysis. First, it implies the stability of general Gabor frames under small dilations of the time-frequency set, previously known only for the ...

We study Weyl-Heisenberg (=Gabor) expansions for either L 2 (IR d) or a subspace of it. These are expansions in terms of the spanning set where K and L are some discrete lattices in IR d , L 2 (IR d) is nite, E is the translation operator, and M is the modulation operator. Such sets X are known as WH systems. The analysis of the \basis" properties of WH systems (e.g. being a frame or a Riesz ba...

In this paper we establish an optimal Lorentz estimate for the Riesz potential in L1 regime setting of a stratified group G: Let Q?2 be homogeneous dimension G and I? denote order ? on G. Then, every ??(0,Q), there exists constant C=C(?,Q)>0 such that (0.1)?I?f?LQ?(Q??),1(G)?C?XI1f?L1(G)for all f?Cc?(G) XI1f?L1(G), where X denotes horizontal gradient.