On singular generalized absolutely monotone functions (Q1340308)

Let \(\{u_ i\}^ \infty_{i=0}\) be an infinite sequence of functions belonging to \(C^ \infty [a,b]\), such that for all \(n\), \(n = 0,1, \dots, \{u_ i\}^ n_{\ell = 0}\) form an extended Chebyshev system on \([a,b]\). We assume that \(u_ i(t) = \varphi_ i (t;a)\), \(i = 0,1,2, \dots\), where \[ \varphi_ 0 (t;x) = \begin{cases} 0 & a \leq t < x \\ w_ 0(t), & x \leq t \leq b \end{cases}, \quad \varphi_ i (t;x) = \begin{cases} 0 & a \leq t < x \\ \int^ t_ \alpha \omega_ i (\xi) d_ i (t;\xi) d \xi) & x \leq t \leq b \end{cases} \quad i = 1,2, \dots \] and where \(\{\omega_ i\}^ \infty_{t=0}\) is a sequence of positive \(c^ \infty [a,b]\) functions. A function \(f\) defined on \((a,b)\) is said to be convex with respect to the Chebyshev system \(\{u_ i\}^ n_{i=0}\) if for every set of \(n+2\) points, \(a < t_ 0 < t_ 1 < \cdots < t_{n+1} < b\), \[ \left | \begin{matrix} u_ 0 (t_ 0) & u_ 0 (t_ 1) & \cdots & u_ 0 (t_{n+1}) \\ u_ 1 (t_ 0) & u_ 1 (t_ 1) & \cdots & u_ 1 (t_{n+1}) \\ \vdots \\ u_ n (t_ 0) & u_ n (t_ 1) & \cdots & u_ n (t_{n+1}) \\ f(t_ 0) & f(t_ 1) & \cdots & f(t_{n+1}) \end{matrix} \right | \geq 0. \] The set of convex functions with respect to the Chebyshev system \([u_ i]^ n_{i=0}\) forms a convex cone denoted by \(c(u_ 0, u_ 1, \dots, u_ n)\) a simply \(c_ n\). Also \(c_{-1}\), denotes the cone of nonnegative functions on \((a,b) \cdot \varphi_ k (\cdot;x)\) for \(k \geq n\) is in \(c_ n\). The elements of the cone \(C_ A = \bigcap^ \infty_{n=-1} c_ n\) are called generalized absolutely monotone (GAM) functions. It is known that if \(f \in C_ A\) then the Taylor-type formulae \[ f(t) = \int^ b_ a \varphi_ n (t;x) (L_ nf) (x)dx + \sum^ n_{i=0} {(L_{i-1}f) (a^ +) \over w_ i (a)} u_ i(t), \quad a \leq t < b,\;n = 0,1, \dots \] holds. These formulae give extreme ray representations for the elements of \(\cap^ n_{i=-1} c_ i\). As shown in [\textit{D. Amir} and \textit{Z. Ziegler}, Israel J. Math. 7, 137-146 (1969; Zbl 0182.081)], a necessary and sufficient condition for all functions \(f \in C_ A\) to admit the Taylor type representation \(f(t) = \sum^ \infty_{i=0} a_ i u_ i(t)\), where \(a_ i = {(L_{i-1} f) (a^ +) \over w_ i (a)}\), \(i = 0,1, \dots\), (A) is that for every \(t\), \(a < t < b\), there exists a number \(s\), \(t < s < b\), such that \[ \lim_{i \to \infty} u_ i (t)/u_ i (s) = 0. \tag{D} \] Moreover, if we restrict ourselves to the cone \(B \cap C_ A\), where \(B\) denotes the set of bounded functions on \((a,b)\) then (D) could be replaced by \[ \lim_{i \to \infty} u_ i (t)/u_ i (b) = 0 .\tag{D'} \] Formula (A) is an extreme ray representation for \(f \in C_ A\). In this paper, the author generalize the representation (A for \(B \cap C_ A\)-functions in case (D') does not hold. This family of functions constitutes a convex cone in a generalized \(C^ \infty (a,b)\) space. The question of extreme rays of this cone as well as the extreme ray representation of its elements is discussed in this paper.