On a new kind of Birkhoff type trigonometric interpolation

Abstract

.In this paper we introduce a new kind of Birkhoff type interpolation of functions with period 27r. We find necessary and sufficient conditions for regularity of this new interpolation problem . Moreover, a closed form expression for the interpolation polynomial is given .

Birkhoff Type 2-Periodic Trigonometric Interpolation in the Family of Trigonometric Polynomial

“…Comparing with Hermite and Birkhoff trigonometric interpolation, it is a kind of more extensive interpolation in some sense . ] .The obtained result does not have the generality because the polynomial of the divided difference operator have been discussed respectively in the odd and even special polynomials; On the other hand, they [3,4] have not used "the kind" concept of the trigonometric polynomial which Professor Du proposed in [5], therefore they only discussed the issue in two special trigonometric polynomials class and did not discuss it in the general trigonometric polynomial class. But this is precisely the difference between the trigonometric interpolation and the algebraic polynomial interpolation.…”

“…Mentions specifically, when the number of interpolation condition is even, the trigonometric interpolation polynomial is not only, but in concrete trigonometric polynomial class only. In 2007, we [6] discussed Birkhoff type 1-periodic trigonometric interpolation problems in all trigonometric polynomial class by [3] profiting from certain results and the method. In the present paper, we shall discuss the Birkhoff type 2-Periodic trigonometric interpolation problem in the trigonometric polynomial class .It has cancelled the limit to the difference operator polynomial which is either even or odd and has promoted the result of [4].…”

“…In exactly the same way, we can discussion Hermite and Birkhoff trigonometric interpolation. Birkhoff type trigonometric interpolation [3] is different from Birkhoff trigonometric interpolation that we usually study. Comparing with Hermite and Birkhoff trigonometric interpolation, it is a kind of more extensive interpolation in some sense . ]…”

“…In 1995, Jiang [3]considered a new kind of Birkhoff trigonometric interpolation with equidistant interpolation points on ) 2 , 0 [ π . First, Jiang replaced the conditions of the derivative for Birkhoff trigonometric interpolation by those of the divided difference at the nodes.…”