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We define a class of spaces $H_{\mu }^p$, 0 < p < ∞, of holomorphic functions on the tube, with a norm of Hardy type: ${\left|\right|F\left|\right|}_{H_{\mu }^p}^p=su{p}_{y\in \Omega }{\int }_{\Omega ̅}{\int }_{ℝⁿ}{\left|F(x+i(y+t))\right|}^{p}dxd\mu \left(t\right)$. We allow μ to be any quasi-invariant measure with respect to a group acting simply transitively on the cone. We show the existence of boundary limits for functions in $H_{\mu }^p$, and when p ≥ 1, characterize the boundary values as the functions in $L_{\mu }^p$ satisfying the tangential CR equations. A careful description of the measures μ when their supports lie on the boundary of the cone is also provided....

A family of holomorphic function spaces can be defined with reproducing kernels $B_{\alpha }\left(z,w\right)$, obtained as real powers of the Cauchy-Szegö kernel. In this paper we study properties of the associated Poisson-like kernels: $P_{\alpha }\left(z,w\right)={\left|B_{\alpha }\left(z,w\right)\right|}^2/B_{\alpha }\left(z,z\right)$. In particular, we show boundedness of associated maximal operators, and obtain formulas for the limit of Poisson integrals in the topological boundary of the cone.

In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1/2 &lt; α ≤ ∞. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence m → φ' = Π m(2 ·), between low-pass filters in H(T) and Fourier transforms of α-localized scaling functions (in H(R)), is actually a homeomorphism of topological spaces. We also show that the space of such filters can be regarded as a connected infinite dimensional...

We give a characterization of biorthogonal wavelets arising from MRA's of multiplicity D entirely in terms of the dimension function. This improves the previous characterization in [8] removing an unnecessary angle condition. Besides we characterize Riesz wavelets arising from MRA's, and present new proofs based on shift-invariant space theory, generalizing the 1-dimensional results appearing in [17].