New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms
Abstract
:1. Introduction and Preliminaries
2. The Widder–Lambert Transform
2.1. An Inversion Formula for the Widder–Lambert Transform over a Class of Functions
2.2. An Inversion Formula for the Widder–Lambert Transform on
2.3. Regular Distributions of Compact Support Versus the Salem Equivalence to the Riemann Hypothesis
3. The Stieltjes–Poisson Transform
3.1. An Inversion Formula for the Stieltjes–Poisson Transform over Conventional Functions
3.2. Concerning an Inversion Formula for the Stieltjes–Poisson Transform over Distributions of Compact Support
4. Final Observations and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Widder, D.V. An inversion of the Lambert transform. Math. Mag. 1950, 23, 171–182. [Google Scholar] [CrossRef]
- Carrasco, J.R. The Lambert transform and an inversion formula. Rev. Acad. Sci. 1959, 53, 727–736. [Google Scholar]
- Pennington, W.B. Widder’s inversion formula for the Lambert transform. Duke Math. J. 1960, 27, 561–568. [Google Scholar] [CrossRef]
- Goldberg, R.R. Inversions of generalized Lambert transforms. Duke Math. J. 1958, 25, 459–476. [Google Scholar] [CrossRef]
- Miller, E.L. Convergence properties and an inversion formula for the Lambert transform. Rev. Tecn. Fac. Ingr. Univ. 1979, 2, 77–94. [Google Scholar]
- Ferreira, C.; López, J. The Lambert transform for small and large values of the transformation parameter. Q. Appl. Math. 2006, 64, 515–527. [Google Scholar] [CrossRef]
- Raina, R.K.; Nahar, T.S. A note on a certain class of functions related to Hurwitz zeta function and Lambert transform. Tamkang J. Math. 2000, 31, 49–56. [Google Scholar] [CrossRef]
- Goyal, S.P.; Laddha, R.K. On the generalized Riemann zeta functions and the generalized Lambert transform. Gadnita Sandesh 1997, 11, 99–108. [Google Scholar]
- Raina, R.K.; Srivastava, H.M. Certain results associated with the generalized Riemann zeta functions. Rev. Tecn. Fac. Ingr. Univ. 1995, 18, 301–304. [Google Scholar]
- Negrín, E.R.; Roopkumar, R. Generalized Lambert-type transforms over integrable Boehmians. Integral Transf. Spec. Funct. 2022, 33, 698–710. [Google Scholar] [CrossRef]
- Maan, J.; Negrín, E.R.; González, B.J. On the Generalized Lambert Transform over Lebesgue and Boehmian Spaces. Int. J. Appl. Comput. Math. 2023, 9, 103. [Google Scholar] [CrossRef]
- González, B.J.; Negrín, E.R. Inversion formulae for a Lambert-type transform and Salem’s equivalence to the Riemann hypothesis. Integral Transf. Spec. Funct. 2023, 34, 614–618. [Google Scholar] [CrossRef]
- González, B.J.; Negrín, E.R. Approaching the Riemann hypothesis using Salem’s equivalence and inversion formulae of a Widder-Lambert-type transform. Integral Transf. Spec. Funct. 2024, 35, 291–297. [Google Scholar] [CrossRef]
- Negrín, E.R.; Maan, J.; González, B.J. Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis. Axioms 2025, 14, 129. [Google Scholar] [CrossRef]
- Yakubovich, S. Integral and series transformations via Ramanujan’s identities and Salem’s type equivalences to the Riemann hypothesis. Integral Transf. Spec. Funct. 2014, 25, 255–271. [Google Scholar] [CrossRef]
- Broughan, K. Equivalents of the Riemann Hypothesis; Cambridge University Press: Cambridge, UK, 2017; Volume 2. [Google Scholar]
- Salem, R. Sur une proposition équivalente à l’hypothèse de Riemann. CR Acad. Sci. 1953, 236, 1127–1128. [Google Scholar]
- Patkowski, A.E. On Salem’s integral equation and related criteria. Tsukuba J. Math. 2023, 47, 207–213. [Google Scholar] [CrossRef]
- Sumner, D.B. An Inversion Formula for the Generalized Stieltjes Transform. Bull. Amer. Math. Soc. 1949, 55, 174–183. [Google Scholar] [CrossRef]
- Maan, J.; Negrín, E.R. On the generalized Stieltjes transform over weighted Lebesgue spaces and distributions of compact support. Rend. Circ. Mat. Palermo II Ser. 2023, 72, 3551–3561. [Google Scholar] [CrossRef]
- Maan, J.; Negrín, E.R. A comprehensive study of generalized Lambert, generalized Stieltjes, and Stieltjes-Poisson transforms. Axioms 2024, 13, 283. [Google Scholar] [CrossRef]
- Love, E.R.; Byrne, A. Real Inversion Theorems for Generalized Stieltjes Transforms. J. Lond. Math. Soc. 1980, 22, 285–309. [Google Scholar] [CrossRef]
- Pathak, R.S. Integral Transforms of Generalized Functions and Their Applications; Gordon and Breach Science Publishers: Amsterdam, The Netherlands, 1997. [Google Scholar]
- Srivastava, H.M.; Yürekli, O. A theorem on a Stieltjes-type integral transform and its applications. Complex Var. Elliptic Equ. 1995, 28, 159–168. [Google Scholar] [CrossRef]
- Goldberg, R.R. An inversion of the Stieltjes transform. Pac. J. Math. 1958, 8, 213–217. [Google Scholar] [CrossRef]
- González, B.J.; Negrín, E.R. A distributional inversion formula for a generalization of the Stieltjes and Poisson transforms. Integral Transf. Spec. Funct. 2009, 20, 897–903. [Google Scholar] [CrossRef]
- Zemanian, A.H. Distribution Theory and Transform Analysis: An Introduction to Generalized Functions with Applications; McGraw-Hill Book Co.: New York, NY, USA, 1965. [Google Scholar]
- Zemanian, A.H. Generalized Integral Transformations; Interscience Publishers, John Wiley and Sons: New York, NY, USA, 1968; Volume 18. [Google Scholar]
- Sidorov, N.A.; Sidorov, D.N. Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind. Differ. Equ. 2006, 42, 1312–1316. [Google Scholar] [CrossRef]
- Hayek, N.; González, B.J.; Negrín, E.R. The generalized Lambert transform on distributions of compact support. J. Math. Anal. Appl. 2002, 275, 938–944. [Google Scholar] [CrossRef]
- Maan, J.; Negrín, E.R. The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support. São Paulo J. Math. Sci. 2024, 18, 1839–1860. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Negrín, E.R.; Maan, J. New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms. Axioms 2025, 14, 291. https://doi.org/10.3390/axioms14040291
Negrín ER, Maan J. New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms. Axioms. 2025; 14(4):291. https://doi.org/10.3390/axioms14040291
Chicago/Turabian StyleNegrín, Emilio R., and Jeetendrasingh Maan. 2025. "New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms" Axioms 14, no. 4: 291. https://doi.org/10.3390/axioms14040291
APA StyleNegrín, E. R., & Maan, J. (2025). New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms. Axioms, 14(4), 291. https://doi.org/10.3390/axioms14040291