Constrained energy problems with applications to orthogonal polynomials of a discrete variable
Since the pioneering works of Mhaskar, Saff, Gonchar and Rakhmanov in the eighties, it is known that the equilibrium distribution for the logarithmic potential in presence of an external field is an essential tool in the study of the asymptotic properties of sequences of orthogonal polynomials with respect to a varying weight. On the other hand, recently Rakhmanov has studied the solution of the classical logarithmic energy minimization problem, but considered in the subclass of the unit measures bounded from above by a fixed measure. He has also shown its connection with the limit zero distribution of (diagonal) sequences of orthogonal polynomials of a discrete variable. The authors combine both approaches to the problem of minimizing the logarithmic energy in presence of an external field in the class of unit measures subject to constraints as above. In particular, they extend to this case the classical results about existence, uniqueness, characterization and a representation of the minimal measure, as well as a saturation property. As expected, the solution describes the zero asymptotics of ray sequences of orthogonal polynomials of a discrete variable. Explicit results are obtained for Krawtchouk polynomials.
orthogonal polynomials of a discrete variable
scientific article; zbMATH DE number 1146580