Interpolation Of Functions

Approximation with interpolatory constraints

“…|}, where the infimum is taken over the space of all polynomials R of degree at most 2m, and K is a constant which is independent of f and n. Similar results regarding polynomial interpolation may be found in [16]. The proofs of these results are usually quite complicated, and depend upon the particular situation at hand.…”

“…[1,5]), or (iii) relaxing the requirement that the interpolant be of minimal degree n − 1. This last issue has been investigated by several mathematicians, including Erdős, Szabados, and Vertesi; see, for instance, [3,15,17,16]. In particular, Szabados has proved the following [15]: Suppose n is a positive integer, and let x j,n = cos θ j,n , where 0 ≤ θ 1,n < .…”

Orthogonal polynomials

“…Fejér's result that if P 2n−1 is the unique polynomial of degree at most 2n − 1 that interpolates a continuous function f at the nodes of the n-th Chebyshev polynomial and that has zero derivative at each of these nodes, then P 2n−1 uniformly converges to f on [−1, 1] as n → ∞. For the role of orthogonal polynomials in interpolation see the books Szabados-Vértesi [90] and Mastroianni-Milovanovic [55].…”

A note on the rate of convergence for Chebyshev-Lobatto and Radau systems

“…The nodal systems related to the Jacobi polynomials play an important role in the theory of Hermite interpolation on the bounded interval, (see [1][2][3]). The hungarian interpolatory school, beginning with Fejér, has used these systems for Lagrange and Hermite interpolation.…”