Error Inequalities in Polynomial Interpolation and Their Applications

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...
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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< k < 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.