Abstract
Let (xk) be a bi-infinite knot sequence for which the mesh ratio is smaller than some exponential order (in particular, the local mesh ratio must be finite, but the global mesh ratio may be infinite). Let p be a non-negative real number, and (yk) a data sequence for which yk = 0(|xk|ρ) as k ∈ ± ∞. We prove the existence and uniqueness of a spline function of any previously specified odd degree, with knots (xk), which interpolates (yk) (that is, S(xk) = yk for all k) and which is dominated by exactly the same power, namely S(t) = 0 (|t|ρ) as t → ±∞. We may replace 0 by o throughout. We also obtain a series representation for S(·) in terms of (yk) and of a sequence of “fundamental splines” Lk(·) which decay exponentially near ±∞.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
de Boor, C., Odd-degree spline interpolation at a bi-infinite knot sequence, Approximation Theory, Bonn 1976, pp.30–53. Springer Lecture Notes No. 556.
de Boor, C., Splines as linear combinations of B-sp1ines:a survey, Approximation Theory 11 (Sympos.Proc.,Texas 1976, ed. G.G.Lorentz, C.K.Chui, L.L.Schumaker), pp.1–47. Academic Press, New York/London 1976.
Jakimovski, A., — Russell, D.C., On an interpolation problem and spline functions, General Inequalities 2 (Conf.Proc., Oberwolfach 1978, ed. E.F.Beckenbach), pp.205–231. Birkhäuser Verlag, Basel/Stuttgart 1980.
Jia, Rong-qing, On a conjecture of C.A.Micchelli concerning cubic spline interpolation at a bi-infinite knot sequence, J.Approx.Theory 38 (1983), 284–292.
Marsden, M.J., An identity for spline functions with applications to variation-diminishing spline approximation, J.Approx.Theory 3 (1970), 7–49.
Schoenberg, I.J., Cardinal interpolation and spline functions:II. Interpolation of data of power growth, J.Approx.Theory 6 (1972), 404–420.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Basel AG
About this chapter
Cite this chapter
Jakimovski, A., Russell, D.C., Stieglitz, M. (1984). Spline Interpolation of Power-Dominated Data. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_36
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5432-0_36
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5434-4
Online ISBN: 978-3-0348-5432-0
eBook Packages: Springer Book Archive