Boundedness and reproducing kernels of multiplier operators in the Fourier-Laguerre setting

Abstract

This paper investigates the Laguerre multiplier operator $M_{u,\beta }^{\alpha }$ within the Fourier-Laguerre transform framework. We begin by deriving Calderón reproducing formulas, including an inversion formula and a method for approximating square-integrable functions. We then establish uncertainty principles for $M_{u,\beta }^{\alpha }$, extending the Pauli-Weyl inequality and introducing a concentration-type inequality. These results demonstrate the enhanced capabilities of the multiplier operator compared to the classical Fourier transform, offering improved uncertainty bounds and advanced tools for harmonic analysis in the Fourier-Laguerre context.