Concerning extended complete Tchebycheff system (Q1068305)

An extended complete Chebyshev system is a system \(u_0, u_1, \ldots, u_ n\) of class \(C^ n[a,b]\) and for each \(i\) \[ u_ i(t) = w_ 0(t) \int ^{t} _{a} w_1(s_1) \int ^{s_1} _{a} w_2(s_2) \cdots \int ^{s_{i-1}} _{a} w_ i(s_ i) ds_ i \ldots ds_1 \] where \(w_ i(t)>0\) and \(w_ i\) is of class \(C^{n-i}\) on \([a,b]\). A function \(f\) on \((a,b)\) is convex if \[ U = U \begin{pmatrix} u_0, u_1, \ldots, u_ n,f \\ t_0, t_1, \ldots, t_ n, t_{n+1} \end{pmatrix} = \left| \begin{matrix} u_0(t_0) & u_0(t_1) & \ldots & u_0(t_{n+1}) \\ \vdots & \vdots && \vdots \\ u_ n(t_0) & u_ n(t_1) & \ldots & u_ n(t_{n+1}) \\ f(t_0) & f(t_1) & \ldots & f(t_{n+1}) \end{matrix} \right| \geq 0 \] for each choice of \(t_ i\) with \(a < t_0 < t_1 < \cdots < t_{n+1} < b\). Let \(D_ i\) denote the first differential operator \((D_ if)(t) = (d/dt)(f(t)/w_ i(t))\) and \[ \begin{aligned} \| D_ n \ldots D_ 0 f\|_ p &= \left(\frac1{b-a} \int^{b}_{a} |(D_ n \ldots D_ 0 f)(t)|^ p dt \right)^{1/p} \\ \| D_ n \ldots D_ 0 f\|_{g,p} &= \left(\int^{b}_{a}g(t) |(D_ n \ldots D_ 0 f)(t)|^ p dt\right)\;sp{1/p} \end{aligned} \] where \(p>1\) and \(g(t)>0\) for all t in \([a,b]\). The author obtains several results on estimating \(U\) under appropriate conditions on \(f\). For example, if \(f:[a,b]\to R\) is such that \(D_ n \ldots D_ 0f\) is continuous on \([a,b]\) and \(a<t_1 <t_2 <\ldots <t_ n <b\), then \[ U \begin{pmatrix} u_0, u_1, \ldots, u_ n,f \\ a, t_1, \ldots, t_ n, b \end{pmatrix} \leq K_{g,n} \| D_ n \ldots D_ 0 f\|_{g,p} \] where \(K_{g,n} = \left( \int^{b}_{a} Q_ n(x)/g(x)^{q/p} dx\right) ^{1/q}\) with \(1/p+1/q=1\) and \(Q_ n(x) = U \begin{pmatrix} u_0, u_1, \ldots, u_ n, f \\ a,t_1, \ldots,t_ n,b\end{pmatrix}\). The equality holds whenever \((D_ n, \ldots, D_0f)(x) = C(Q^{n(x)}/g(x))^{1/p-1}\) with \(C\) an arbitrary real constant. Further, the author provides several applications of his results.