Systems of left translates and oblique duals on the Heisenberg group

Abstract

In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\lambda$. Here, $\mathbb{H}$ denotes the Heisenberg group and $g^\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ on $\mathbb{H}$. This result is also illustrated with an example.

Keywords

• B-splines
• Heisenberg group
• Gramian
• Hilbert-Schmidt operator
• Riesz sequence
• moment problem
• oblique dual
• Weyl transform

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