**4. Numerical Examples**

We have carried out different numerical experiments to visualize the main contributions of this article. The first examples correspond to the three cases of Theorem 2 and in all of them we work in the following way:

1. We construct the nodal systems in a quite random way. We consider four arcs or sections in the unit circumference T. The first one begins in *α*1 = 1 and its *n*4 nodes are constructed in counter clockwise sense separated by an angular length 2*πn* + , where the  are random errors determined by using the uniform distribution in [ *An*2 2*π*, 2*An*2 <sup>2</sup>*π*]. The fourth section begins in *α*1 = 1 and its *n*4 nodes are constructed in clockwise sense with arcs of angular length 2*πn* + , where the  are random errors determined by using the uniform distribution in [ *An*2 2*π*, 2*An*2 <sup>2</sup>*π*]. The second section begins after the first one and its *n*4 nodes are constructed in counter clockwise sense with arcs of angular length 2*πn* + , where the  are random errors determined by using the uniform distribution in [−<sup>2</sup>*An*2 2*π*, − *An*2 <sup>2</sup>*π*]. Finally, in the third section the arcs between the nodes are all equal.

Obviously we use different values of *n* and we must remark that we obtain always the same results, really we must say similar results because due to our random choice we never have the same nodal system.


These examples are devoted to visualize the items (i), (ii) and (iii) respectively of Theorem 2.

**Example 1.** *In this example we work with <sup>F</sup>*(*z*) = 1 + 20 *z* + *z*<sup>−</sup><sup>1</sup> 2 sin 2 *z* + *z*−<sup>1</sup> *for z* ∈ T*, which satisfies the hypotheses of Theorem 2 (i). We take n* = 1000*, A* = 2 *and we use (8) to obtain* L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)*.*

*We represent the function <sup>F</sup>*(*ei<sup>θ</sup>* ) *which takes real values and, as we have said, the real part of the interpolation polynomial. Notice that due to its variability, F is a quite difficult function to interpolate. Indeed, it is easy to check that <sup>F</sup>*(*e<sup>i</sup> θ* ) *is not differentiable at π*2*.*

 *We present in Figure 2 two graphics. On the left we have a general panoramic of the interpolation along* T *and we have added the interpolation points in green. We must point up that the interpolatory process is successful where the function has no variability. However, we have an unsuccessful situation where the function has great variability.*

*In the graphic on the right we have a detailed situation between 1.2 and 2, that is near π*2 *, which can help us to understand the problem. According to the theory presented, we must increase the number of nodes to obtain better results in this region.*

**Figure 2.** *<sup>F</sup>*(*z*) and (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*, *z*)) with F(z)= 1 + 20( *z* + *z*<sup>−</sup><sup>1</sup> 2 ) sin( 2 *z* + *z*−<sup>1</sup> ), *z* = *eiθ* , *θ* ∈ [0, <sup>2</sup>*π*], *θ* ∈ [1.2, 2] and *n* = 1000.

**Example 2.** *Now we consider the function defined on* T *by <sup>F</sup>*(*z*) = ∞ ∑ *k*=1 1 *k*6 (*zk* + *<sup>z</sup>*<sup>−</sup>*<sup>k</sup>*)*, which satisifies the hypotheses of Theorem 2. In the next Figure 3 we plot on the left <sup>F</sup>*(*ei<sup>θ</sup>* ) *and* (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,*ei<sup>θ</sup>* )) *for θ* ∈ [0, <sup>2</sup>*π*] *and n* = 60*. Notice that they are indistinguishable. On the right we plot the errors given by* (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,*ei<sup>θ</sup>* )) − *<sup>F</sup>*(*ei<sup>θ</sup>* ) *with θ* ∈ [0, <sup>2</sup>*π*]*. We point out that the errors are less or equal than* 2 × 10−8*.*

In the next example we also construct an alternative interpolation polynomial based on the equispaced nodal system on T, but using the values of the function on our nodal system. We do this because a natural criticism to our method could be that with errors as <sup>O</sup>(1/*n*<sup>2</sup>) we can be so close to the equispaced nodal system to accept this approximation. We denote by A−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,) this alternative interpolation polynomial.

**Figure 3.** *<sup>F</sup>*(*z*) and (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*, *z*)) and (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*, *z*)) − *<sup>F</sup>*(*z*) with *<sup>F</sup>*(*z*) = ∞ ∑ *k*=1 1 *k*6 (*zk* + *<sup>z</sup>*<sup>−</sup>*<sup>k</sup>*), *z* = *ei<sup>θ</sup>* , *θ* ∈ [0, <sup>2</sup>*π*] and *n* = 60.

**Example 3.** *In this example we take <sup>F</sup>*(*z*) = *ez, n* = 24*, A* = 2 *and we use (8) to obtain the interpolation polynomials* L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,) *and* A−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)*. Taking into account that F is analytic we know that F and* L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,) *must be close. In Figure 4 we plot* (*F*) *in black,* (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)) *in red and* (A−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)) *in green for z* = *ei<sup>θ</sup> with θ* ∈ [0, <sup>2</sup>*π*]*. On the left hand side we have a global vision with θ* ∈ [0, <sup>2</sup>*π*] *and we can observe that* (*F*) *and* (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)) *are indistinguishable; in fact for this example the maximum error was* 3 × 10−9*.*

*Although* (A−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)) *has a similar shape, see that it can drive us to catastrophic errors. On the right hand side we present a detail of the previous one, which give us an idea of the error. Notice that in general we cannot know the details of the nodal distribution.*

*We have done the same with the imaginary part and we obtain the same results.*

**Figure 4.** (*F*(*z*)), (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*, *z*)) and (A−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*, *z*)) with *<sup>F</sup>*(*z*) = *<sup>e</sup>z*, *z* = *ei<sup>θ</sup>* , *θ* ∈ [0, <sup>2</sup>*π*], *θ* ∈ [4.5, 4.9] and *n* = 24.

**Example 4.** *Finally we choose <sup>F</sup>*(*z*) = *<sup>χ</sup>S*(*z*) *defined on* T *as the characteristic function of the superior arc S of* T*, we take n* = 2000*, A* = 2 *and we use expression (8) to obtain* L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,)*. We know the behavior when the nodal system is related to para-orthogonal polynomials with respect to an analytic positive measure (see [10]), but we do not have a theory about the behavior of* L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,) *in our situation. We plot the results in Figure 5. Notice that the basic ideas of the Gibbs–Wilbraham phenomenon are present in this graphic, that is, the convergence of the interpolator to the function in regions which are far enough from the discontinuities and a heavy oscillation near the discontinuities. A representation of the oscillation and its amplitude, maybe, deserves a detailed study.*

**Figure 5.** *<sup>F</sup>*(*ei<sup>θ</sup>* ) and (L−*<sup>n</sup>*,*n*−<sup>1</sup>(*<sup>F</sup>*,*ei<sup>θ</sup>* )) with *<sup>F</sup>*(*z*) = *<sup>χ</sup>S*(*z*), *z* = *ei<sup>θ</sup>* , *θ* ∈ [0, <sup>2</sup>*π*], *θ* ∈ [3, 3.3] and *n* = 2000.
