Gibbs–Wilbraham phenomenon on Lagrange interpolation based on analytic weights on the unit circle

Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

“…The results are consequence of the preceding Theorem 1 and they are also based on the properties satisfied by our nodal systems. Thus one can obtain these results following the same steps as in the proof of Theorems 3 and 4 in [10], where one can see the details.…”

“…Following [10] we can obtain the Lebesgue constant, (see [19]), and the convergence of this interpolatory process. Notice that this is a novelty result for our general nodal systems satisfying property (1), although the techniques that we use are the same as in [10].…”

“…For applying the interpolation it could be very useful the following results concerning the convergence and the rate of convergence for smooth continuous functions (see [10,12]).…”

“…In [10] we have studied the Lagrange interpolation process for piecewise continuous functions with suitable properties and by using as nodal points the zeros of the para-orthogonal polynomials with respect to analytic weights, which constitutes a novel approach to the Lagrange interpolation theory.…”

“…Finally we choose F(z) = χ 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 −n,n−1 (F, ). 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 −n,n−1 (F, ) in our situation. We plot the results in Figure 5.…”