Saddle-point theorems for rational approximation (Q788932)

A sequentially compact space T being given together with the real continuous functions f, \(P=^{\Delta}(P_ 1,...,P_ n)\) and \(Q=^{\Delta}(Q_ 1,...,Q_ m)\) all defined on T, we deal with the rational approximation P(T') on compact subsets T'\(\subset T\) \[ \inf P(T')= ^{\Delta}\inf_{x,y} \{\max_{t\in T'}| f(t)-\frac{x^ TP(t)}{y^ TQ(t)}| \quad | y^ TQ(\cdot)>0\quad on\quad T'\}. \] We first focus our interest on finite subsets \(T_ 0\subset T\) (i.e. finite discretizations) having at most \(n+m\) points such that \(\inf P(T_ 0)=\inf P(T).\) The existence of such a sets (i.e. discretizations) is known only in the case where the original whole problem P(T) has a solution. Our first result is the existence of a subset \(T_ 0\subset T\) having \(n+m\) points and satisfying \(\inf P(T_ 0)=\inf P(T)\) even in the case where P(T) has no solution. Our technique is purely geometrical - it is essentially based on a Helly-type theorem - and differs from the Kolmogorov principle. Our second aim is to express all the sets \(T_ 0\) satisfying the above- mentioned property by max-inf statements, where the maximum and infimum are interchangeable and a finite number of variables are involved. One of these statements includes a Lagrangian having a differentiable property. In the case where P(T) has solutions, all of them, as well as all the above \(T_ 0\) sets, are expressed by the saddle points of our minimax statements.