Given a function x(t) E c{n) [a, bj, points a = al < a2 <
...< ar = b and subsets aj of {0,1,"',n -1} with L:j=lcard(aj) =
n, the classical interpolation problem is to find a polynomial P -
(t) of degree at most (n - 1) n l such that P~~l(aj) = x{i)(aj) for
i E aj, j = 1,2," r. In the first four chapters of this monograph
we shall consider respectively the cases: the Lidstone
interpolation (a = 0, b = 1, n = 2m, r = 2, al = a2 = {a, 2", 2m -
2}), the Hermite interpolation (aj = {a, 1,' ", kj - I}), the Abel
- Gontscharoff interpolation (r = n, ai ~ ai+l, aj = {j - I}), and
the several particular cases of the Birkhoff interpolation. For
each of these problems we shall offer: (1) explicit representations
of the interpolating polynomial; (2) explicit representations of
the associated error function e(t) = x(t) - Pn-l(t); and (3)
explicit optimal/sharp constants Cn,k so that the inequalities k I
e{k)(t) I < C k(b -at- max I x{n)(t) I, 0 n - 1 n -, a$t$b - are
satisfied. In addition, for the Hermite interpolation we shall
provide explicit opti- mal/sharp constants C(n,p, v) so that the
inequality II e(t) lip:::; C(n,p, v) II x{n)(t) 1111, p, v ~ 1
holds.
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