Gabor System based on the unitary dual of the Heisenberg group

Abstract In this paper, the Gabor system based on the unitary dual of the Heisenberg group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">H</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{H}^{n} is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>,</m:mo> <m:msub> <m:mi mathvariant="script">B</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>;</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>κ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L^{2}(\mathbb{R}^{*},{\mathcal{B}_{2}};d\kappa) using the Schrödinger representation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">H</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{H}^{n} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">B</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathcal{B}_{2} denotes the class of Hilbert–Schmidt operators on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L^{2}(\mathbb{R}^{n}) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>κ</m:mi> </m:mrow> </m:math> d\kappa denotes the Haar measure on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> \mathbb{R}^{*} . Further, a necessary and sufficient condition is provided for the Gabor system to be an orthonormal system, a Parseval frame sequence, a frame sequence and a Riesz sequence.