Extremal Positive Splines with Applications to Interpolation and Approximation by Generalized Convex Functions

F(s, t) = wn(s)w0(t) dtiw^) dt2w2(t2) • • • </*„__!*„_!(*„_!). Jt Jtl Jtn-2 A fundamental solution for L is given by G(s, t) = F(s, t) for s ^ t, G(s, t) = 0 for s < t. A fundamental solution for L* is GJs, i) = G(t, s). To avoid cumbersome formulations results are stated for n ^ 2 (in which case G is continuous), unless indicated otherwise. By SR(T) we mean the collection of Radon measures on the locally compact Hausdorff space T; by 9£R0CO and 9W(T) , the subfamilies of measures of compact support and of positive measures. For an open interval J let 9W(7) be the set of real functions on I possessing an nth distribution derivative belonging to 301(7), « = 1 , 2 , . . . . For n ^ 2, if u e mV) then D~u e AC(I) and D~u e BV(I). One shows that a measure u e 501(7) belongs to 9K(1) iff Lu e 5R(7) in the following weak sense: There is \x e 30?(7) such that j I?<j)(t)u{dt) = J (j)(i)ix(dt) for each <j) e CJ(7); if this is so one says \x = Lu. Let 501J(R) consist of the functions in 90?"(J?) of compact support. For any interval 7 we say u e 50loO(7) if u e $0t£(jR) and supp(w) c: 7. Each u e 50l(7) has integral representations