On some (n−1)-symmetric linear functionals

Many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle. Particular aspects are treated for quadratic or cubic decompositions. In this paper, we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality. This technique is based on the decomposition with respect to the cyclic group of order n. The main theorem is applied to the orthogonality on (n−1)-symmetric curves. Some particular cases are singled out. These results may be useful in studying Hermite–Padé approximations, vector continued fractions and dynamical systems.