A barycentric trigonometric Hermite interpolant via an iterative approach

In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and Hormann (2018) for the Floater-Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentation matrix. Finally, we conclude with various examples and a numerical study of the rate of convergence at equidistant nodes and conformally mapped nodes.


Introduction
Let consider an arbitrary 2π-periodic function f (θ). In [3], Berrut proposed the following interpolant between any set of n ordered nodes θ i 's in [0, 2π), where the function cst is cst(θ) = csc(θ), if n is odd, ctg(θ), if n is even.
In the case of equidistant nodes, this interpolant correspond to the classical trigonometric interpolant in barycentric form and therefore achieves exponential convergence to functions analytic in a strip. The latter fast convergence has been extended in [1] by Baltensperger to the interpolant (1) when the nodes are images of equidistant points under a periodic conformal map; an effective example of a periodic conformal map that clusters nodes around one (or more) location is presented and analysed in [4], this allows a faster convergence when the interpolated function present a front in one or more positions. The interpolant above, besides the fast convergence, present also a logarithmic growth of the Lebesgue constant for a wide class of nodes as proved in [5]. Another interesting barycentric rational trigonometric interpolant has been introduced in [2] and include Berrut's interpolant at equidistant nodes as a special case. Cirillo and Hormann in [6] presented an effective method to construct an Hermite interpolant starting from the basis b i (x) of the renowned Floater-Hormann family of interpolants, in fact they considered the basis to construct an interpolant that satisfies the Hermite conditions The authors investigated also the effectiveness of such construction for FHinterpolants by studying in [7] the Lebesgue constant at equidistant nodes and giving an estimate of the interpolation error in [8].
In this paper we are going to generalise this approach and use it to construct an effective barycentric rational trigonometric Hermite interpolant.
The paper is in fact construct as follows: in Section 2 is presented a generalization of the iterative approach presented by Cirillo and Hormann. In Section 3 is computed the differentation matrix that allows a fast construction and computation of the barycentric rational trigonometric Hermite interpolant. In Section 4 various numerical example are presented and analysed. Finally, we conclude in Section 5.

The general iterative approach
To generalize the lemma in [6], let us consider a set Ω ⊂ R. Assume that for a smooth function d i (x) that vanishes in x i and not in x j for j = i and such that d i (x i ) = 1.
Proof. For j = 0 the statement follows directly from the Lagrange property of b i (x). Let us now consider j > 0 and prove the statement by induction on j. Let then, it is clear that Using the Leibniz rule, we get Therefore, since c i (x p ) = 0 for p = 0, . . . , n and, furthermore, we have that b Let us observe that the sum is empty in the case when k < j, whereas when Finally, we notice that b (j) i,j (x p ) = 0 if p = i and in the case when p = i we have that the product is one since c i (x i ) = 1 by construction.
Remark 1. We remark that, given a function h i (x) vanishing only in x i and with a non-zero derivative in x i , we can always construct a function that satisfies the assumption of Lemma 1 by setting Once we have the b i,j we can construct the Hermite interpolant starting from with the correction term defined as Remark 2. Notice that we could write the interpolant as where Due to Lemma 1, the resulting interpolant r m will satisfy the Hermite conditions, which is why the following Theorem holds.

The barycentric rational trigonometric interpolant
Let us consider Berrut's trigonometric interpolant (1). To retain the periodic behaviour, we may choose which clearly satisfies the conditions we need to construct a basis for the Hermite interpolant and, since the basis element of Berrut's trigonometric interpolant are if n is even. Therefore, we can write the basis as where u k,j := 2 j (−1) k(j+1) j! and, if n is odd, and, if n is even, In this way we can compute the differentation matrices as done for the classical barycentric trigonometric interpolant in [1], which will be useful to compute the values of the derivatives of the interpolant r m at the nodes in the previous iterations.
If n is odd, considering equation (9), differentiating both side and evaluating at the points x i , we get that where we defined (D s j ) ik = b (s) k,j (θ i ). Moreover, by evaluating the derivative in a different nodes we have In (11) we can isolate the term q = 0 of the sum, that gives, which together with (10) gives us the following recursive formula for the differentation matrix Notice that since the terms b i,j satisfy (5), D j j corresponds to the identity matrix and D s j = 0 when s < j. Then, the formula becomes Similarly, when n is even, we have that Hence, Moreover, since if q odd and since the j-th derivative of tan((θ −θ i )/2) is a linear combinations of powers of tan for j > 1, we can simplify equation (15) to and equation (18) to Notice that for the construction of the Hermite interpolant in an iterative way as for (8), we need to compute first r (j) j−1 (x i ) and therefore D j j−1 which, after simplifications, is which for j = 0 corresponds to the values in [1]; for the diagonal elements, we use the relation i,j (θ) = 1, since we want to reproduce the constants for the j-th derivate, which implies that so that we obtain To compute the differentiation matrices of higher order, as done for j = 0 in [9, 1], we use

Numerical Experiments
Here we present some numerical tests. Whose Matlab demos are available at https://github.com/gelefant/TrigonometricHermite We considered the periodic test functions in [0, 2π) f 1 (θ) = e 2 sin(θ)+cos(θ) , f 2 (θ) = cos(3θ) + log(cos(θ) + 1.5) and we interpolated it by means of the interpolant (8) with the basis presented in Section 3 with N equidistant nodes, first by using the Hermite conditions up to the second derivative, secondly up to the third derivative and finally with four derivatives. An example with both functions is presented in Figure 1 together with two interpolants, one satisfying the Hermite conditions up to the second derivative and one satisfying the conditions up to the third derivative. By computing, for a fixed m, the absolute error with the interpolant with N nodes err N = max |f (x) − r m (x)|, we can notice that the interpolant, in the case of conditions up to the second derivative retains the exponential convergence of the classical interpolant (1) at equidistant nodes (see Figure 2, whereas in case of the conditions on three and four derivatives, it slows down the convergence (see Figure 3). We can numerically estimate the convergence as O(N −3 ) as we can see in Table 1

Conclusion
The present work introduced an iterative scheme to construct an Hermite interpolant starting from Berrut's trigonometric interpolant. In order to construct the values of the derivative of the precedent iterative step, we compute the differentation matrix which allows to obtain these values easily as a matrix product.
Our numerical experiments suggest that the produced interpolant retains the exponential convergence of the classical interpolant at equidistant nodes once we interpolate the first two derivatives and slows down to a rate of O(N −3 ) when we interpolate the third and the fourth derivatives.
Future work will be aimed to analyse the convergence more deeply and also for other class of nodes, including the one introduced in [4].