Systems of left translates and oblique duals on the Heisenberg group

Authors : Santi Das, Radha Ramakrishnan, Peter Massopust

Pages : 222-236

Doi:10.33205/cma.1382306

View : 133 | Download : 113

Publication Date : 2023-12-15

Article Type : Research Paper

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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