A phase space localization operator in negative binomial states

Our work is inspired by the seminal paper by I. Daubechies, "Time-frequency localization operators: a geometric phase space approach" (IEEE Trans. Inf. Theory, 1988), which formalized a fundamental problem in signal analysis: a signal can only be measured within a finite time-frequency region. Daubechies transformed this practical constraint into a rigorous mathematical framework using the coherent states (CS) of the harmonic oscillator. Building directly upon this foundation, our presentation will introduce and analyze a hyperbolic geometric analogue of Daubechies' localization operators. The discussion will, naturally, lead to a possible generalization of the Bergman space on the disc (a functional space) but one has to solve a non-trivial reconstruction problem in the context of Reproducing Kernel Hilbert Spaces.

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