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<p>Motivated essentially by their potential for fruitful usages and applications the Lambert transform and its related <i>W</i> function, as well as the Widder–Lambert transform, in such widely- and extensively-investigated areas of applied mathematical and physical sciences as (for example) mathematical modeling involving differential, integral and integro-differential equations, and also fractional-order differential equations, this paper presents a unified framework that extends the classical Lambert and Widder–Lambert transforms to broader function spaces, including Lebesgue spaces and the space of compactly supported distributions on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}_+\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>. We establish new inversion formulas, analyze boundedness conditions, and study asymptotic behaviors within suitable weighted spaces. The developed theory generalizes several known results and provides a rigorous foundation for further investigations of Lambert-type integral transforms in the context of distributional generalized functions.</p>

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A unified presentation of the Lambert and the Widder–Lambert transforms over Lebesgue spaces and over the space of distributions of compact support on \(\mathbb {R}_+\)

  • 期刊论文

  • 发表日期：

    2026年4月17日

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  • H. M. Srivastava,
  • Jeetendrasingh Maan,
  • E. R. Negrín

摘要

Motivated essentially by their potential for fruitful usages and applications the Lambert transform and its related W function, as well as the Widder–Lambert transform, in such widely- and extensively-investigated areas of applied mathematical and physical sciences as (for example) mathematical modeling involving differential, integral and integro-differential equations, and also fractional-order differential equations, this paper presents a unified framework that extends the classical Lambert and Widder–Lambert transforms to broader function spaces, including Lebesgue spaces and the space of compactly supported distributions on \(\mathbb {R}_+\) R + . We establish new inversion formulas, analyze boundedness conditions, and study asymptotic behaviors within suitable weighted spaces. The developed theory generalizes several known results and provides a rigorous foundation for further investigations of Lambert-type integral transforms in the context of distributional generalized functions.