On the Big Hartley transform

Abstract

In this paper, we introduce a new type of singular first-order differential-difference operator of Dunkl type on the real line. This operator is obtained as a limiting case from both the first-order Dunkl-type operators corresponding to Bannai-Ito and Big 1-Jacobi orthogonal polynomials. We provide an explicit expression for the eigenfunction of this operator in terms of Bessel functions. The obtained kernel is called the Big Hartley function, which is a generalization of the usual Hartley kernel and the little Hartley function studied in Bouzeffour [The generalized Hartley transform. Integral Transforms Spec Funct. 2014;25(3):230–239]. Additionally, we present a new product formula for the little Hartley function, which is related to the Kingman-Bessel hypergroup and the Rosler-Dunkl signed hypergroup. Finally, we investigate inversion formulae for the transforms of both the little Hartley function and the big Hartley function.

Keywords:

• Generalized differential-difference operator
• Bessel functions
• Hartley transform Plancherel formula

2010 Mathematics Subject Classifications:

• 42A38
• 42B10
• 43A32
• 43A15

Acknowledgments

The first-named author expresses appreciation for the support provided by Researchers Supporting Project Number (RSPD2023R974), King Saud University, Riyadh, Saudi Arabia.

Disclosure statement

The authors declare that she has no conflicts of interest.

Data Availability

No data has been used for producing the result of this paper.