Abstract
Suppose I is an open real interval, and V a subspace of C∞(I) invariant under D, differentiation. It is shown that if V contains a polynomial of positive degree, then D has no homogeneous qth root on V, for any integer q > 1. This conclusion is not generally true if I is replaced by an arbitrary open set in the reals. Necessary and sufficient conditions for the existence of qth roots of D on finite dimensional subspaces of C∞(I) invariant under D, in terms of the differential equations of which such subspaces are the solution spaces, are obtained.
Keywords
- Characteristic Polynomial
- Open Interval
- Real Vector Space
- Complex Vector Space
- Positive Degree
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References
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Nering, E., Linear Algebra and Matrix Theory, London-Sydney; Wiley 1963.
Nomizu, K., Fundamentals of Linear Algebra, New York; Mc-Graw-Hill 1966.
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© 1975 Springer-Verlag
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Johnson, P.D. (1975). An algebraic definition of fractional differentiation. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067107
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DOI: https://doi.org/10.1007/BFb0067107
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07161-7
Online ISBN: 978-3-540-69975-0
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