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<p>The general notion of fractional derivative consists of a convolution with some kernel function. In the context of a recently introduced probabilistic operator that extends and unifies certain fractional derivatives of this type (L-fractional, normalized Caputo, scaled Hadamard, etc.), we prove the Peano–Sard theorem on the representation of continuous linear functionals. Thus, we continue the methodology developed by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024). After some additional theory on the operator is built, different versions of the theorem are included, depending on the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1271_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>-integrator, the domain of the function, and the order of differentiation.</p>

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Representation of continuous linear functionals in terms of probabilistic fractional differential operators

  • 期刊论文

  • 发表日期：

    2025年7月8日

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  • Marc Jornet,
  • Juan J. Nieto

摘要

The general notion of fractional derivative consists of a convolution with some kernel function. In the context of a recently introduced probabilistic operator that extends and unifies certain fractional derivatives of this type (L-fractional, normalized Caputo, scaled Hadamard, etc.), we prove the Peano–Sard theorem on the representation of continuous linear functionals. Thus, we continue the methodology developed by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024). After some additional theory on the operator is built, different versions of the theorem are included, depending on the \(\psi \) ψ -integrator, the domain of the function, and the order of differentiation.