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9. Theorem. 6 equality holds only if };(x) = A(j;) = const. (i = 1, ... , n) except possibly on a negligible set. 9. However, by adding a condition concerning L and A we can establish such a result even when K is not strictly convex. Thus we restrict our attention to systems of L, A such that the following condition holds: A4: If S is any negligible subset of E, and f(x) is any (real) function which vanishes on E - S, then f(x) is in the class L and AU) = O. 7(c) and (d) only the empty set is negligible.

The following two theorems are also given (1966, pp. 382-83, 395, and 387-88). 63. Theorem. f(t) 10 Karlin and Studden The function = fn(t, x) J J sf x x x = {wo(t) Wl(Sl) wz(sz) ... Wn(Sn) ds; ... 84) is contained in quo, Ul' ... , un)' For n = 0, fo(t, x) = wo(t) for x:5 t s: b and equals zero otherwise. ) More generally, we have En quo, ... , u;), n fn(t, x) ;=0 and we also have Uj E n 7 ~ C o (uo, ... , u;) for each j = 0, 1, ... , n. 4. 64. Theorem. For n e: 1 and a < c < b, every f admits a representation of the form t f(t) = E 27 C(uo, Ul , ...

A By HOlder's inequality we then have \$ = 1, ... , n. IIDn ... p quo, ... , Qn(t) > O. 0 38 1. 80. Theorem. Let f: [a, b] ~IR be a real-valued function such that D; ... Dof is continuous on [a, b]. Then for p E (1,00) Iv - ~aiuill ~ c; jlDn ... Dofll p , Ilfli = sUPastsb If(t)l, ao = f(a)/wo(a), a, = «Di- 1 ••• Dof)(a)/wi(a) for where i = 1, ... , n, and a Proof. 76 is related to the following result of Neuman (1986). 1), we have where An ~ «d - C)In)1I2 ~ n and In is defined by f d In = e (Mn(t ))2dt.

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