ABSTRACT
In this paper we are going to treat functions on the interval / = [0, 1]. In particular, we are interested in splines on / . The polynomial splines of maximal order of smoothness are determined by the knots n and by their order k . We consider a set of knots n = (f,*f - k < i < N + k) such that 0 = t0 < t\ . . . < tu - 1 < f/v = 1 with 0 and 1 being knots of multiplicity k i.e. = . . . = to = 0 and 1 = *#••• = 1 • A function / : / - > / ? is said to be a spline o f order k with knots n if it is a polynomial of degree k — 1 on each interval (f;, f/+i), 0 < i < N and of the class of smoothness Ck~2. The case of k = 1 corresponding to piecewise constant functions will not be treated here and the case of k = 2 corresponds to the continuous piecewise linear functions. Through this work we assume that k > 2 . The space of all splines over / corresponding to n and k is denoted by 5* . In the linear space S k there is a natural local basis of the B-splines {Ni , - k < i < N } (for their properties see [3],[7]). Characteristic properties of this basis are the following: the basic functions Ni are nonnegative, supp Nj = [f,-f fj+*], they form a partition of unity i.e. Nj(x) = 1 for x e I and their are linearly independent what implies that each / e S k has unique represantation
Consequently, the dimension of S* is d == N + k - 1. Over the Lebesgue space L p = L P(I) we have the norm \\f\\p = ( /7 1 f \ p) l /p , 1 < P < oo. Moreover, over L p x Lq with 1 / p 4-1 /q = 1 we have the bilinear form ( / , g) = f f f g which in case p = q = 2
becomes a scalar product over the real Hilbert space L 2. Similarly, for a = (a,*) e lp we have the norm \\a\\p = (]T, \oj\p) l/p and the corresponding bilinear form (a,b) which again becomes a scalar product for l2. If the dimension of lp is d then it will be denoted as 1%. The B-splines Nj can be almost normalized in the L p space as follows: since Vi = (Nj, 1) = (tj+k — t()/fc it follows that for the function Nj%p = Nj/v} /p we have the inequalities \ / k < \\Nj,p \\p < 1 for 1 < p < oo, thus the B-splines Nj%p are almost normalized in the L p norm. Customarily we use also the notation M, = Afci. Now, we have the fundamental result of de Boor on the stability of the B-spline basis (see [2])
THEOREM 1.1 There is a positive constant Kk depending only on k such that
holds for all n , a e /? and 1 < p < oo with d = N 4-k — 1.
