The Trigonometric Hermite-Birkhoff Interpolation Problem

Abstract

The classical Hermite-Birkhoff interpolation problem, which has recently been generalized to a special class of Haar subspaces, is here considered for trigonometric polynomials. It is shown that a slight weakening of the result (conservativity and Pólya conditions) established for those special Haar subspaces also holds for trigonometric polynomials after one rephrases the statement of the problem, the underlying assumptions, and the result itself appropriately to reflect the inherent differences between algebraic polynomials (which the special class of Haar subspaces essentially are) and the periodic trigonometric polynomials. Furthermore, simple necessary and sufficient conditions for poisedness of one-rowed incidence matrices analogous to the Pólya conditions for two-rowed incidence matrices in the algebraic version are proved, and an elementary necessary condition for the poisedness of an arbitrary (trigonometric) incidence matrix stated.