By Josip E. Pečarić, Frank Proschan and Y.L. Tong (Eds.)

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**Example text**

89). 92) can be expressed in the form a, = lim - x - - - - - : . . ~ - - ' - - - - - - - " - ~ - - x->c J Wl(SI) ... Wi(Si) ds, ... dSI 11 C C thus using L'Hospital's rule yields . - - - - - - - . : : : DoRi-1(X) = 11m - - - - ' - - - - - : . : ~ - - - - I x->c w1(x) wz(sz) . . Wi(Si) ds, ... 93) and finally a, = lim Di-1R;_I(X)/Wi(x)= Di-1f(c)/wi(c). =_l-lim I w{c) Y->C JY I C Ri-1(y)/wo(Y) Y JWl(Sl) JSl Si~2 wz(sz)... J Wi-l(Si-l) dSi- 1 ... dSI ds S (1. 96) then, using L'Hospital's rule, we have a.

Definition (Dahmen and Micchelli, 1983). For {"o, ... ,xn } C ~ k with volk(["0, ... ,xn )) > 0 the multivariate B -spline M (. I"0, ... "o,... ,Xn)dx=nlf ~ . holds for all given by f ( } . ~ o + · · · + A . 56) )dA. ' "0, ... 57) with (J = ["0, ... 56) determines M(·I "0, ... ,xn ) everywhere on ~ k (see Micchelli, 1979). ,Xn are in a Following Micchelli (1979), we say that the points "0, general position if every subset of k + 1 points of {"o, , xn } forms a simplex of dimension k. For convenience we list below some well-known properties of these splines.

The case n = 2 follows from the definition of convex functions. Suppose that the result is valid for all 43 44 2. Jensen's and Jensen-Stelfensen's Inequalities k, 2:5 k :5 n - 1. t:2: pJ(Xi) :5 - holds by the result for n = 2 and the induction hypothesis. The proof for the strict inequality when f is strictly convex is easy and is omitted. 2. Remarks. (a) The condition that p is a positive n-tuple can be replaced by: "p is a nonnegative n-tuple and P; > O. " Similar remarks can be made for other results in this chapter.