Best approximate inversion formulas for the Hartley-Bessel-Stockwell transform

The primary objective of this paper is to introduce a novel integral transform, referred to as the Hartley-Bessel-Stockwell transform, and to establish several fundamental results associated with it. Specifically, we derive generalized versions of Parseval’s identity, Plancherel’s theorem, the inversion formula, and Calderon’s reproducing formula for this transform. Furthermore, we investigate the concentration properties of the Hartley-Bessel-Stockwell transform on sets of finite measure and present an uncertainty principle for orthonormal sequences. Finally, leveraging the theory of reproducing kernels and best approximation methods, we examine the extremal functions associated with this transform. We provide their integral representations and derive optimal estimates for these functions within weighted Sobolev spaces.