The Trigonometric Hermite-Birkhoff Interpolation Problem

Darell J. Johnson
1975 Transactions of the American Mathematical Society  
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
more » ... 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 onerowed 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. Recently there has been considerable interest in the classical Hermite-Birkhoff interpolation problem; we cite [1], [2], [4], [5], [10], [11], [13]-[16], [19]-[26], as examples. Variants of the Hermite-Birkhoff interpolation problem have also been proposed and studied, both for algebraic polynomials (e.g., [3], [8], [9]) and for more general subspaces of functions [6], [7], [17]. In particular, Dcebe [7] shows that for the classical Hermite-Birkhoff interpolation problem with a Haar subspace whose successive derivatives are Haar and decrease in magnitude with each derivative (like algebraic polynomials) replacing the algebraic polynomials, that an incidence matrix is necessarüy poised whenever it is conservative and satisfies the Pólya conditions. In this paper we replace the algebraic polynomials in the statement of the Hermite-Birkhoff interpolation problem by trigonometric polynomials and obtain an analogous result. Certain differences are present, however, due to the inherent difference between trigonometric and algebraic polynomials. One of these differences is that the degree of a trigonometric polynomial
doi:10.2307/1998632 fatcat:fi6t2kj47fhr7mmdbludp3ksja