Splines and the Logarithmic Function.
Abstract
Let q be fixed, q>1, and let f(x) = log x/log q in (0, pos infinity). The paper determines the unique spline function (S sub n)(x), 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 n)(qx) = (Sub n)(x) + 1 if x>0, and (S sub n)(1) = 0. It follows that (S sub n)(q nu) = f(q sup nu) for all integers nu. It is shown that (S sub n)(x) shares with f(x) its monotonicity properties of hither order. Nevertheless and against all expectations it is shown that (S sub n)(x) does not converge to f(x) as n nears infinity. Most of the paper is devoted to an analysis of the peculiar asymptotic behavior of (S sub n)(x).
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1974
- Accession Number
- ADA001664
Entities
People
- D. J. Newman
- Isaac Jacob Schoenberg
Organizations
- University of Wisconsin–Madison