New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms

Abstract

This paper establishes explicit inversion formulae for the Widder–Lambert transform and the Stieltjes–Poisson transform, extending their applicability to function spaces and compactly supported distributions. Utilizing a generalized Lambert transform, the study provides a unified framework for inversion, enhancing the theoretical understanding of these integral transforms. Additionally, a specific function class satisfying Salem’s equivalence to the Riemann hypothesis is identified, further broadening the analytical scope of the results within generalized function spaces.

Keywords

• Widder–Lambert transform
• Stieltjes–Poisson transform
• inversion formulae
• generalized Lambert transform
• distributions of compact support
• Salem’s equivalence