Hermite Interpolation on the Unit Circle Considering up to the Second Derivative

The paper is devoted to study the Hermite interpolation problem on the unit circle. The interpolation conditions prefix the values of the polynomial and its first two derivatives at the nodal points and the nodal system is constituted by complex numbers equally spaced on the unit circle. We solve the problem in the space of Laurent polynomials by giving two different expressions for the interpolation polynomial. The first one is given in terms of the natural basis of Laurent polynomials and the remarkable fact is that the coefficients can be computed in an easy and efficient way by means of the Fast Fourier Transform (FFT). The second expression is a barycentric formula, which is very suitable for computational purposes. 1. Introduction One of the pioneering papers concerning Hermite interpolation on the unit circle is [1]. There a Fejér’s type theorem is proved (see [2, 3]), for nodal systems constituted by the roots of a complex number with modulus one. The main result asserts that the Hermite-Fejér interpolants uniformly converge for continuous functions on the unit circle. Some improvements to this result, considering nonvanishing derivatives and more smooth functions, were given in [4]. More recently, in [5], the order of convergence of Hermite-Fejér interpolants for analytic functions on a disk and on an annulus containing the unit circle was obtained. The classical Hermite interpolation on the circle with nodal points equally spaced was studied in [6]. There it was constructed an orthogonal basis for the space of polynomials in order to obtain the expression of the interpolation polynomials. The coefficients of the interpolation polynomials in this basis can be computed by using the FFT. In [7], the same problem was studied and the corresponding expressions for the Laurent polynomials of interpolation were obtained in a more simple way. Another basis was constructed and again the coefficients can be computed by using the FFT. From these formulas, suitable expressions for the fundamental polynomials were obtained and the barycentric formulas for Hermite interpolation on the unit circle were deduced for the first time. The barycentric formulas were known for Hermite interpolation on the bounded interval (see [8]), but [7] was a new contribution on the circle. A study about Hermite interpolation on two disjoint sets of nodes on the unit circle has been developed in [9] and problems considering more than one derivative were also considered. Indeed, lacunary Hermite interpolation problems have been also studied on some nonuniformly distributed nodes