Abstract
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.
Highlights
One of the pioneering papers concerning Hermite interpolation on the unit circle is [1]
The classical Hermite interpolation on the circle with nodal points 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
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 Fast Fourier Transform (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
Summary
One of the pioneering papers concerning Hermite interpolation on the unit circle is [1]. 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. In the present paper we study generalized Hermite interpolation problems on the unit circle considering nodal points spaced and using the values for the first two derivatives. First we obtain suitable basis for subspaces of the space of Laurent polynomials by considering appropriate interpolation conditions This enables us to express the interpolation polynomials in such a way that the coefficients can be computed by using the FFT. Like in the Lagrange interpolation (see [11]), the barycentric expressions are very useful for doing evaluations and calculus due to their stability (see [12])
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