Limits And Bounds For Divided Differences On A Jordan Curve In The Complex Domain
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LIMITS AND BOUNDS FOR DIVIDED DIFFERENCES ON A JORDAN CURVE IN THE COMPLEX DOMAIN.
Let Sn+1 be a set of n+1 points lying on a Jordan curve C, let f be a function given on C, and let dn denote the divided difference of order n formed for f in the points Sn+1. It is proved that if the (n-1)-th derivative of f exists and satisfies a Lipschitz condition, and C satisfies a mild smoothness restriction, then /dn/ is uniformly bounded for all choices of Sn+1 in which the points are distinct. An extension to confluent points is given. In the case in which C is the unit circle, the structure of the bound is displayed. It is also shown for a general C that if the points of the sequence S2, S3 ... become everywhere dense on C in a certain way, then lim as n approaches infinity cn+1dn equals integral C f dz / 2pi i, where c is the transfinite diameter of C.