Behaviour of Exponential Splines as Tensions Increase without Bound
✍ Scribed by Chris Grandison
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 418 KB
- Volume
- 89
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Schweikert (J. Math. Phys. 45 (1966), 312 317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Spa th (Computing 4 (1969), 225 233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pruess (J. Approx. Theory 17 (1976), 86 96) extended Spa th's result to the case where the inter-knot tensions p i may not all be the same but tend to infinity at the same asymptotic growth rate, in the sense that p i # 3( p 1 ) for all i. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a C 2 monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbslike behaviour of the spline's derivative cannot occur in the neighbourhood of the knots.
1997 Academic Press
1. NOTATION AND BASIC FACTS ABOUT NATURAL EXPONENTIAL SPLINES
It will be convenient to use the notation Int[a 1 , ..., a m ] for the smallest closed interval containing a 1 , ..., a m and Int(a 1 , ..., a m ) for its interior. Let 2=((x i , y i )) i=0, 1, ..., n be a sequence of plane points with x 0 <x 1 < } } } <x n , hereafter called the knots.
In what follows, the ith subinterval means [x i&1 , x i ] and the i th knot means (x i , y i ). All quantities related to the spline in tension in this subinterval or at this knot will bear the subscript i. Thus for example h i will denote x i &x i&1 and m i will denote ( y i &y i&1 )Â(x i &x i&1 ), the slope of the article no. AT963055 289