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.


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 on the unit circle (see [10]).
In the present paper we study generalized Hermite interpolation problems on the unit circle considering nodal points equally 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.
In the second part of the paper we deduce the barycentric formulas which constitute a new contribution of the paper.Like in the Lagrange interpolation (see [11]), the barycentric expressions are very useful for doing evaluations and calculus due to their stability (see [12]).

Laurent Hermite Interpolation Polynomials
We study the generalized Hermite interpolation problem on the unit circle T := { : || = 1} considering the first two derivatives.The nodal system {  } −1 =0 is constituted by the roots of a complex number , with || = 1; that is, it consists of complex numbers equally spaced on the unit circle.The problem to solve can be posed as follows.
Finally, the polynomial L 2, () satisfying (4) can be written as By applying the first interpolation condition we obtain If  = 0 and we use the third interpolation condition we obtain 2  0 ( − 1)+ 2  0 ( + 1)(1/) = 1.By solving the system we have the following solution: By solving the corresponding system in the unknowns Next we prove that these auxiliary polynomials constitute a suitable basis of the space

Proposition 2. The system
is an orthogonal basis of the Laurent space [] with respect to the inner product defined by Now we are in conditions to obtain the expression of the polynomial H −−[/2],+[(−1)/2] () satisfying (1) that we denote for simplicity H().It is clear that it can be written as where H 0 , H 1 , and H 2 are the solutions of the following problems: [] and it satisfies H 0 (  ) =   , H  0 (  ) = 0, and H  0 (  ) = 0 for  = 0, . . .,  − 1.
(ii) Proceeding in the same way, H 1 () can be written as On one hand, if we calculate the inner product we have On the other hand, it holds that and therefore Hence, taking into account the expression of L 1, (), we obtain (ii).
(iii) We write H 2 () as follows By computing the inner product, we obtain Therefore and we obtain the expression of L 2, () given in (iii).
(iv) It is straightforward from Next we consider the particular cases in which the nodes are  roots of 1 and −1, obtaining the following results.
(b) Notice that the coefficients of the expressions given before can be computed in an easy and efficient way by using the FFT.

Barycentric Expression
In this section, our aim is to obtain a barycentric expression for the interpolation polynomial H −−[/2],+[(−1)/2] ().We distinguish two cases according to the nodal system having an even or odd number of points.

Nodal System with an Even Number of Points.
First we assume that the nodal system has an even number of points that we denote by 2 and we try to obtain the expression of H −3,3−1 () that we denote for simplicity, H().Since H() can be written in terms of the fundamental polynomials of Hermite interpolation first we obtain suitable expressions for these polynomials.Proof.Taking into account that ( 2 − ) 3 = Π 2−1 =0 ( −   ) 3 , then it is clear that E  (  ) = E   (  ) = 0, for all  = 0, . . ., 2−1, E   (  ) = 0, for all  ̸ = , and E   (  ) ̸ = 0.In the same way, it is immediate to see that F  (  ) = 0, for all  = 0, . . ., 2 − 1,

Lemma 7. The polynomials
For obtaining the exact nonvanishing values we proceed as follows.
(i) If we define   () = E  (  ), we obtain   () =    ( 2 − 1) have the following expressions Proof.It is clear that A  (), B  (), and C  () can be written in the following form: with E  (), F  (), and G  () given in Lemma 7.

ISRN Mathematical Analysis
To compute  ,2 take into account that it must be C   (  ) = 1.Then applying the preceding lemma we get that 1 =  ,2 (16    3 / 3  ), from which it follows that  ,2 =  3   /16    3 .For computing  ,1 and  ,1 we use that B   (  ) = 1 and B   (  ) = 0 and we obtain the following system: By applying Lemma 7 and solving the system we get the result.Finally, to obtain the coefficients  ,0 ,  ,0 , and  ,0 in the expression of A  (), we proceed in the same way.By applying the interpolation conditions we have the system Then, by using Lemma 7 and solving the system we conclude our result.
It is straightforward to deduce, from the preceding Proposition 8, the so-called barycentric expression for H().
(ii) Take into account that H 1 () = ∑ 2−1 =0 V  B  (), with B  () given in Proposition 8 and proceed in the same way as in (i).
(iii) Take into account that H 2 () = ∑ 2−1 =0   C  (), with C  () given in Proposition 8 and proceed in the same way as in (i).
(iv) Take into account that with A  (), B  (), and C  () given in Proposition 8 and proceed in the same way as in (i).

Nodal System with an Odd Number of
Points.Now we assume that the nodal system {  } 2 =0 is constituted by the 2 + 1 roots of , with || = 1.In this case we obtain the barycentric expression for the Laurent polynomial of Hermite interpolation H −3,3+1 () that we denote by H(), characterized by where {  } 2 =0 , {V  } 2 =0 and {  } 2 =0 are fixed complex numbers.
Proceeding like in the previous case, first we obtain the following auxiliary results.

Lemma 10. The Laurent polynomials, E
have the following expressions: (44) Proof.It is similar to the proof of Proposition 8.
It is straightforward to deduce, from the preceding Proposition, the so-called barycentric expression for H().(48) Proof.It is similar to the proof of Proposition 9.
Remark 13.Notice the novelty of the barycentric expressions that we have obtained for Hermite interpolation on the unit circle using up to the second derivative.These formulas can be implemented for use in a simple way and they are very useful for computations due to their stability.