ABSTRACT
Assume that the finite complex function f(z) is defined on a set G that includes the pairwise different points (wk)1≤k≤n . The (unique) polynomial of degree at most n − 1, that takes the value f(zk) at the points zk, i.e., Ln−1(f ;wk) = f(wk) for 1 ≤ k ≤ n is called the Lagrange interpolant and is denoted by
Ln−1(f ; z) ≡ Ln−1(f ; z; (zk)nk=1)
Applying the notation ωn(z) := ∏n
k=1(z − zk), this Lagrange interpolant can be represented in the form
Ln−1(f ; z) = n∑
f(wk) z − wk ·
ωn(z) ω′n(wk)
. (1.1)
Suppose the set of points (zk)1≤k≤n is split into two disjoint subsets (z(1)k )1≤k≤n1 and (z
(2) k )1≤k≤n2 such that n1 + n2 = n . Then
the Lagrange polynomial Ln−1(f ; z; (zk)nk=1) is the sum of the two Lagrange polynomials, where the first polynomial Ln−1,∗(f ; z) interpolates f(z) for z = z(1)k , 1 ≤ k ≤ n1 and takes the values 0 at all the points (z(2)k )1≤k≤n2 and the second Lagrange polynomial
Ln−1,∗∗(f ; z) takes the value 0 at all the points (zk)1≤k≤n and interpolates the function f(z) at the points (z(2)k )1≤k≤n2 . It will be of interest to see the connections between properties of the Lagrange polynomial Ln−1(f ; z) and the two polynomials Ln−1,∗(z) and Ln−1,∗∗(f ; z).
