Maximum d'entropie et problème des moments. (Maximum entropy and the moment problem) (Q1173899)

In addition to the usual linear constraints (moment conditions) in a maximum entropy problem, i.e., maximize the entropy \(S(g)\) subject to \[ \int^ 1_ 0 \varphi(x)g(x)dx=c\quad\text{with given }\varphi\in C^ k[0,1]\quad\text{and}\quad c\in{\mathbf R}^ k,\tag{1} \] the authors consider nonlinear constraints on the probability density function \(g: [0,1]\to {\mathbf R}\). In particular, solutions of the above maximum entropy problem are obtained if in addition to (1) nonlinear constraints of the type \[ - \infty\leq a< g(x)< b\leq +\infty\quad\text{for all } 0\leq x\leq 1,\quad\text{or }\int^ 1_ 0 g(x)^ 2 dx<1 \] are considered. Several consequences of these results for the classical moment problem and various principles of optimization, e.g., least squares, are discussed.