Mechanical Models for Hermite Interpolation on the Unit Circle

Hermite polynomial interpolation problems on the real line and on the bounded interval have been widely studied by many researchers. Most of the main contributions have been obtained by using as nodal points the zeros of the classical orthogonal polynomials and some of their generalizations (see [1,2,3]). When the nodal points are zeros of general orthogonal polynomials, the papers of Freud [4,5] (see [6]) deserve to be mentioned. The first attempt to use nodal systems not linked to measures and therefore to orthogonal polynomials was carried out by Fejér who introduced the so-called normal systems. In the same direction, we must mention the contributions of Grünwald who introduced in [7] the strongly normal nodal systems. Unfortunately, the use of these systems has not been consolidated and has not been continued. More recently, some nodal systems called well-spaced ones have been used for studying Lagrange interpolation problems. Although they are not connected with measures, any reasonable choice of interpolation nodes fulfills the conditions of being well spaced (see [8]). Thus, if one wants to work on the bounded interval with nodal systems that are not connected with measures, it seems convenient that the nodes have a distribution that is not far from the Chebyshev distribution. This closeness can be established in terms of suitable separation properties. The advantages of using these types of nodal systems is that they can be obtained through a random uniform distribution (see the examples given in [9]).

During the last few decades, several problems related to Hermite polynomial interpolation on the unit circle have been studied in depth. In some cases, the study has been connected to the Hermite interpolation on the interval and to the trigonometric interpolation. The nodal systems usually employed were the equispaced ones, that is those constituted by the n roots of complex numbers with modulus one. It is well known that there are methods to compute the Laurent polynomials of Hermite interpolation in an efficient way, when considering equally spaced nodal points on the unit circumference. By using these nodal systems, the convergence of the Hermite–Fejér interpolation polynomials related to continuous functions was studied in [10], obtaining a Fejér-type theorem. The convergence of the Hermite interpolation polynomials taking nonvanishing conditions for the derivatives was studied in [11] giving several versions of the second Fejér-type theorem.

Other more general nodal systems that have been used are those formed by the zeros of para-orthogonal polynomials, associated with good measures on the unit circle. We recall that a polynomial $Q_n\left(z\right)$ of degree n is para-orthogonal if it satisfies that $Q_n\left(z\right)=z^n\overline{Q_n}(1/z)$ and it is orthogonal to ${\left\{z^k\right\}}_{k=1}^{n-1}$ with respect to a measure on the unit circle. It deserves to mention measures in the Baxter class or analytic measures and, in particular, measures in the Szegő class with the Szegő function having an analytic extension outside the unit disk (see [12,13]). In these cases, the nodal points are characterized by satisfying suitable separation properties, and it is possible to obtain the properties of the nodal polynomials without the need to compute their zeros explicitly. Most of these properties play an important role in the results for convergence and related problems that were studied.

The use of another type of nodal system adds complexity to the problem due mainly to the difficulty of computing their zeros. In this sense, some advances have been made by using nodal polynomials that are close to the equispaced ones in the unit circumference and that can be obtained through a perturbation of the uniform random distribution. The starting point is to ask the nodes for some separation properties that allow obtaining the main properties of the nodal polynomials playing a fundamental role in the interpolation processes.

If the arguments of the nodal points are ${\theta }_j,j=1,\cdots ,n$, we recall that in the Baxter class, the separation property satisfied by the nodal points is ${\theta }_{j+1}-{\theta }_j=\frac{2\pi }{n}+\mathcal{O}\left(\frac{1}{n}\right)$ (see [14]). When the nodal points are the zeros of para-orthogonal polynomials with respect to analytic weights on the unit circle or with respect to measures in the Szegő class having an analytic extension outside the unit disk, then the separation property is ${\theta }_{j+1}-{\theta }_j=\frac{2\pi }{n}+\mathcal{O}\left(\frac{1}{n^2}\right)$ (see [15]). Thus, the nodal points behave like perturbations of the roots of complex unimodular numbers.

In [9], we changed the focus of the interpolation problem, and our starting point was to use nodal systems that were not related to measures and that were only characterized to fulfil certain separation properties between the nodes. Thus, we studied the Lagrange interpolation problem on the unit circle by using nodal systems that are not connected with any measure, and they are only characterized by satisfying a separation property of the type: ${\theta }_{j+1}-{\theta }_j=\frac{2\pi }{n}+\mathcal{O}\left(\frac{1}{n^2}\right).$ In the aforementioned paper, a detailed study of their properties was done. Moreover, the Lebesgue constant of the process was obtained, as well as some results for the convergence and the rate of convergence for different smooth continuous functions.

In the present paper, we studied nodal systems like those used in [9] that meet the separation properties mentioned above, and we present mechanical models that fit these distributions. The aim of this work was to study the Hermite interpolation process on the unit circle by using these nodal systems characterized by a separation property between the nodes. The target was to develop the corresponding interpolation theory that allows us to make practical use of these nodal systems linked to certain mechanical models. Thus, in the first part of the paper (see Section 2 and Section 3), we recall the properties given in [9] satisfied by the nodal system, and we obtain some new ones that play an important role in the Hermite process. Our main result was to prove a new version of Fejér’s theorem for continuous functions on the unit circle by using these nodal systems. We also studied the complete problem, that is the Hermite interpolation problem with nonvanishing conditions for the derivatives, giving a sufficient condition on the derivatives in order to assure convergence. These results are gathered in Section 4, with the study of the convergence of the Hermite interpolation polynomial related to analytic and smooth functions (see [16]), and finally, in Section 5 we obtain the corresponding results on the bounded interval. Indeed, we transformed the problem into a new one on the bounded interval studying the Hermite interpolation problem related to continuous functions on $[-1,1]$ and by using nodal points characterized by some separation properties. In Section 6, we present some mechanical models generating the nodal systems according to our distribution, as well as some numerical examples by applying our results. Finally, Section 7 gathers the main notation concerning the different classes of polynomials used throughout the paper; Section 8 briefs the materials and methods used and Section 9 offers a discussion of the problem.

The aim of this paper was to study Hermite interpolation problems on the unit circle $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$ by using nodal systems satisfying some suitable separation properties, which are not connected with orthogonality nor para-orthogonality with respect to any measure.

We denote the nodal polynomials by $W_n\left(z\right)$ and their zeros by ${\left\{{\alpha }_{j,n}\right\}}_{j=1}^n$, that is, we assumed that $W_n\left(z\right)={\Pi }_{j=1}^n(z-{\alpha }_{j,n})$, where $|{\alpha }_{j,n}|=1$ for $j=1,\cdots ,n$, and ${\alpha }_{j,n}\ne {\alpha }_{k,n}$ for $j\ne k.$ For simplicity, we omit the subscript n and write ${\alpha }_j$ instead of ${\alpha }_{j,n}$ for $j=1,\cdots ,n.$ First, we recall some well-known definitions related to interpolation problems on the unit circle. We work in the space of Laurent polynomials and, in particular, in the subspaces ${\Lambda }_{p,q}\left[z\right]=span\{z^k:p\le k\le q\}$, with p and q integers $p\le q$.

If ${\left\{u_j\right\}}_{j=1}^n$ and ${\left\{v_j\right\}}_{j=1}^n$ are arbitrary complex numbers and $W_n\left(z\right)$ is the nodal polynomial, then the Hermite interpolation problem consists of determining the Laurent polynomial ${\mathcal{H}}_{-n,n-1}\left(z\right)$ satisfying the interpolation conditions:

This polynomial can be rewritten as ${\mathcal{H}}_{-n,n-1}\left(z\right)={\mathcal{HF}}_{-n,n-1}\left(z\right)+{\mathcal{HD}}_{-n,n-1}\left(z\right),$ where the Hermite–Fejér interpolation polynomial satisfies that: and ${\mathcal{HD}}_{-n,n-1}\left(z\right)$ satisfies that

If f is a function, $u_j=f\left({\alpha }_j\right)$ and $v_j=0$, we denote the corresponding Laurent polynomial ${\mathcal{HF}}_{-n,n-1}(f,z)$. In the same way, if $u_j=0,v_j=f^{\prime }\left({\alpha }_j\right)$, we denote the corresponding Laurent polynomial by ${\mathcal{HD}}_{-n,n-1}(f,z)$ and also denote by ${\mathcal{H}}_{-n,n-1}(f,z)={\mathcal{HF}}_{-n,n-1}(f,z)+{\mathcal{HD}}_{-n,n-1}(f,z)$. In the case, when $u_j=f\left({\alpha }_j\right)$ and $v_j$ is arbitrary, if ${\gamma }_n={\left(v_j\right)}_{j=1}^n$, we denote by ${\mathcal{H}}_{-n,n-1}(f,{\gamma }_n,z)={\mathcal{HF}}_{-n,n-1}(f,z)+{\mathcal{HD}}_{-n,n-1}\left(z\right)$.

Let us recall that the preceding polynomials can be computed by using the following expressions (see [17]). and where ${\mathfrak{h}}_{k,n}$ and ${\mathfrak{k}}_{k,n}$ are the fundamental polynomials of Hermite interpolation given by: and:

We can obtain more suitable expressions of these polynomials by using the following relations:

If $z,{\alpha }_k\in \mathbb{T}$, then ${(z-{\alpha }_k)}^2=-z{\alpha }_k{|z-{\alpha }_k|}^2$, and therefore, if $W_n\left(z\right)={\displaystyle \prod _{k=1}^n}(z-{\alpha }_k)$, then we get ${\left(W_n\left(z\right)\right)}^2={(-1)}^n{z}^n\left({\displaystyle \prod _{k=1}^n}{\alpha }_k\right){\left|W_n\left(z\right)\right|}^2.$ Moreover, since $W_n^{\prime }\left({\alpha }_k\right)={\displaystyle \prod _{j=1,j\ne k}^n}({\alpha }_k-{\alpha }_j)$, then ${\left(W_n^{\prime }\left({\alpha }_k\right)\right)}^2={(-1)}^{n-1}{\alpha }_k^{n-2}\left({\displaystyle \prod _{j=1}^n}{\alpha }_j\right){\left|W_n^{\prime }\left({\alpha }_k\right)\right|}^2$.

Hence, by substituting these relations in (5) and (6), we obtain: and: and therefore, we have the following desired expressions for the Hermite interpolation polynomials: and

In practice, it is more convenient to use the barycentric expressions for computing the interpolation polynomials. Thus, in what follows, we obtain these types of formulas.

By taking into account that ${\mathcal{H}}_{-n,n-1}\left(z\right)={\mathcal{HF}}_{-n,n-1}\left(z\right)+{\mathcal{HD}}_{-n,n-1}\left(z\right)$ and $1={\mathcal{HF}}_{-n,n-1}(1,z),$ then we get:

Thus, if we use (3) and (4), we have:

On the one hand, if we apply (5) and (6), we obtain the following barycentric expression:

On the other hand, if we use Equations (7) and (8) instead of (5) and (6), we obtain the equivalent barycentric expression:

Proceeding in a similar way, we can obtain the two following barycentric expressions for the Hermite–Fejér interpolation polynomial:

Throughout the paper, we assume that the zeros of the nodal polynomials $W_n\left(z\right)$ satisfy the following separation property: there exists a positive constant A, $A<\pi$ such that the length of the shortest arc between two consecutive nodes ${\alpha }_j$ and ${\alpha }_{j+1}$ satisfies: where ${\alpha }_{n+1}={\alpha }_1$. As we said before, we denote the length of the shortest arc between any two points of the unit circle, $z_1$ and $z_2$, by $\widehat{z_1-z_2}.$

If we use Landau’s notation for complex sequences, denoting by $a_n=\mathcal{O}\left(b_n\right)$ if $|\frac{a_n}{b_n}|$ is bounded, then the preceding property can be formulated as follows:

We use the same $\mathcal{O}$ to denote different sequences. Unless we mention it explicitly, the limits we obtained from (13) were uniform.

We also considered other nodal polynomials, ${\tilde{W}}_{n,j}\left(z\right),j=1,\cdots ,n$, well connected with $W_n\left(z\right)$. We define ${\tilde{W}}_{n,j}\left(z\right)=z^n-{\lambda }_j$, with ${\lambda }_j={\alpha }_j^n$, and we denote their zeros by ${\beta }_{k,j}=\sqrt[n]{{\lambda }_j},k=1,\cdots ,n,\mathrm{and\; it\; holds}{\beta }_{1,j}={\alpha }_j.$

In what follows, we take for simplicity $j=1$, that is we work with ${\tilde{W}}_{n,1}\left(z\right)=z^n-{\lambda }_1$, and in order to simplify the notation, we denote ${\tilde{W}}_{n,1}\left(z\right)$ by ${\tilde{W}}_n\left(z\right)$ and its zeros ${\beta }_{k,1}$ by ${\beta }_k$ for $k=1,\cdots ,n$, with ${\beta }_{1,1}={\beta }_1={\alpha }_1$.

Hence, it is clear that the separation property satisfied by the zeros $\left\{{\beta }_j\right\}$ of ${\tilde{W}}_n\left(z\right)$ is: where ${\beta }_{n+1}={\beta }_1$.

In this section, we present the properties satisfied by these nodal polynomials $W_n\left(z\right)$ that play an important role in our interpolatory scheme.

First, we recall the following well-known relation between arcs and strings that we use to obtain the nodal properties based on the convex character of the arcsin function:

If $z_1$ and $z_2$ belong to $\mathbb{T}$, then:

Secondly, we recall a separation property, given in [9], between both nodal systems $W_n\left(z\right)$ and ${\tilde{W}}_n\left(z\right)$.

If ${\left\{{\alpha }_j\right\}}_{j=1}^n$ and ${\left\{{\beta }_j\right\}}_{j=1}^n$, with ${\alpha }_1={\beta }_1$, are the nodal points satisfying the separation properties (13) and (14), respectively, and we assume they are numbered in the clockwise sense, then:

Notice that we can write the preceding relations as follows:

In the next propositions, we present the main properties of the nodal polynomials $W_n\left(z\right)$ involved in the interpolatory schemes of Lagrange and Hermite.

We finish the section with the next property, which turns out to be the key property to study the convergence of the Hermite interpolation polynomials.