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 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

Polynomial Interpolation on the Unit Sphere

“…Thus, for the interpolation by trigonometric polynomials, the supported sequence is a sequence that does not start from the first row, assuming that the matrix satisfies the Pólya condition. Although taking derivatives does not reduce the degree of a trigonometric polynomial, the analogue of the Atkinson-Sharma theorem ( [8] or [9, p. 23]) still holds, giving the above understanding of the support sequence. The theorem states that a Birkhoff interpolation is poised if all its odd supported sequences begin in column 0 and E satisfies the Pólya condition.…”

The Trigonometric Polynomial Like Bernstein Polynomial

“…Several other authors have addressed Hermite problems, even for arbitrary points. They were mostly interested in existence questions [ 9 ], convergence results, and formulae other than Lagrange's (see [ 10 – 13 ]). Quasi-interpolant on trigonometric splines has been discussed in [ 14 ].…”

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

“…Kress [16] has derived Lagrangian as well as remainder formulae and asymptotic convergence results for the most important case of an even number N of equidistant points. Several other authors have addressed such Hermite problems, even for arbitrary points; they were however mostly interested in existence questions [7,14], convergence results [15,20] and formulae other than Lagrange's [18].…”