Acta Math. Hungar. 107 (1–2) (2005), 77–87. CUBIC SPLINE INTERPOLATION WITH QUASIMINIMAL B-SPLINE COEFFICIENTS L. L ´ ASZL ´ O (Budapest) Abstract. The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline. 1. Introduction For a natural n let Ω n = {a + ih, i =0,...,n} be an equidistant (uni- form) partition of the real interval [a, b] with h =(b − a)/n. Let S 3 (Ω n ) be the linear space of cubic splines with regard to this partition. Any such spline s can be written uniquely as s = n−1  i=−3 c i B 3,i , where B 3,i are the cubic B-splines for the extended knot sequence Ω ∞ = {x i = a + ih, i ∈ Z}. For convenience, we give the derivatives of the B-spline B 3,i supported in [x i ,x i+4 ] at the relevant knots in the following table: x i x i+1 x i+2 x i+3 x i+4 B 3,i (x) 0 1/6 2/3 1/6 0 B ′ 3,i (x) 0 1/2h 0 −1/2h 0 B ′′ 3,i (x) 0 1/h 2 −2/h 2 1/h 2 0 Key words and phrases: cubic spline, B-spline, not-a-knot, interpolation, minimality, repro- ducing, convergence. 2000 Mathematics Subject Classification: 41A15, 65D05. 0236–5294/5/$ 20.00 c  2005 Akad´ emiai Kiad´ o, Budapest