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  Dirichlet splines as fractional integrals of B-splines [2 citations — 0 self]

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by Wolfgang Zu Castell
Rocky Mt. J. Math
http://www.gsf.de/ibb/preprints/2001/pp01-15.ps
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Abstract:

Abstract: Using Dirichlet averages we generalize the notion of a classical divided dierence of a function by introducing a parameter r in R k+1 +. The case r in N k+1 is related to divided dierences with multiple knots. We give an interpretation of these generalized dierences in terms of fractional operators applied to classical divided differences considered as functions of their knots. The result is then applied to show that Dirichlet splines can be seen as fractional derivatives of B-splines.

Citations

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19 Multiple hypergeometric functions and applications – Exton - 1976
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6 Multivariate Splines: A Probabilistic Perspective – Karlin, Micchelli, et al. - 1986
3 Statistical encounters with B-splines – Dahmen, Micchelli - 1986
2 A connection between elementary functions and higher transcendental functions – Carlson - 1969
2 Moments of Dirichlet splines and their applications to hypergeometric functions – Neuman, Fleet - 1994
1 1 Summability of Multiple Fourier Integrals and – Berens - 1997
1 Lauricella's hypergeometric function FD – Carlson - 1963
1 hypergeometric functions, and Dirichlet averages – Carlson, B-splines - 1991
1 Generalized Liouville dierentiations and the multiplier method in the theory of imbeddings of classes of dierentiable functions – Lizorkin - 1969
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