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Lamb, W.

A distributional theory of fractional calculus. (English) Zbl 0567.46015

Proc. R. Soc. Edinb., Sect. A 99, 347-357 (1985).
A theory of fractional powers of operators is used to define arbitrary powers of some simple differential and integral operators on certain spaces of generalized functions. These powers are shown to coincide with distributional versions of the Riemann-Liouville and Weyl operators of fractional calculus. Standard results associated with fractional integration and differentiation are then deduced from corresponding results on fractional powers.

Cited in 10 Documents

MSC:

46F10 Operations with distributions and generalized functions
46F12 Integral transforms in distribution spaces

Keywords:

fractional powers of operators; differential and integral operators on certain spaces of generalized functions; distributional versions of the Riemann-Liouville and Weyl operators of fractional calculus; fractional integration and differentiation
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References:

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[2] Gel’fand, Generalised functions 1 (1964)
[3] Erdélyi, Higher transcendental functions 1 (1953)
[4] DOI: 10.1137/0503010 · Zbl 0233.45019 · doi:10.1137/0503010
[5] Zemanian, Generalized integral transformations (1968)
[6] Schwartz, Théorie des distributions (1966)
[7] Hövel, Studia Math. 42 pp 177– (1972)
[8] DOI: 10.1017/S0305004100044728 · doi:10.1017/S0305004100044728
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[10] Okikiolu, Aspects of the theory of bounded integral operators in L (1971) · Zbl 0219.44002
[11] DOI: 10.1017/S1446788700011022 · Zbl 0249.26009 · doi:10.1017/S1446788700011022
[12] Lamb, Proc. Edinburgh Math. Soc. 27 pp 165– (1984)
[13] Ross, Lecture Notes in Mathematics 457 (1975)
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