Splines and the Logarithmic Function.
Technical summary rept.,
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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Let q be fixed, q1, and let fx log xlog q in 0, pos infinity. The paper determines the unique spline function S sub nx, of degree n, defined in 0, pos infinity and having as knots the points of the sequence q sup nu neg infinity nu infinity, such as to satisfy the conditions S sub nqx Sub nx 1 if x0, and S sub n1 0. It follows that S sub nq nu fq sup nu for all integers nu. It is shown that S sub nx shares with fx its monotonicity properties of hither order. Nevertheless and against all expectations it is shown that S sub nx does not converge to fx as n nears infinity. Most of the paper is devoted to an analysis of the peculiar asymptotic behavior of S sub nx.
- Theoretical Mathematics