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Full Description
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.
Contents
1 Lidstone Interpolation.- 1.1 Introduction.- 1.2 Lidstone Polynomials.- 1.3 Interpolating Polynomial Representations.- 1.4 Error Representations.- 1.5 Error Estimates.- 1.6 Lidstone Boundary Value Problems.- References.- 2 Hermite Interpolation.- 2.1 Introduction.- 2.2 Interpolating Polynomial Representations.- 2.3 Error Representations.- 2.4 Error Estimates.- 2.5 Some Applications.- References.- 3 Abel 7#x2014; Gontscharoff Interpolation.- 3.1 Introduction.- 3.2 Interpolating Polynomial Representations.- 3.3 Error Representations.- 3.4 Error Estimates.- 3.5 Some Applications.- References.- 4 Miscellaneous Interpolation.- 4.1 Introduction.- 4.2 (n, p) and (p, n) Interpolation.- 4.3 (0, 0; m, n — m) Interpolation.- 4.4 (0; m, n — m) Interpolation.- 4.5 (0, 2, 0; m, n — m) Interpolation.- 4.6 (0 : l — 1, l : l + j — 1; m, n — m) Interpolation.- 4.7 (0; Lidstone) Interpolation.- 4.8 (0, 2, 0; Lidstone) Interpolation.- 4.9 (1, 3, 0, 1; Lidstone) Interpolation.- 4.10 (0 : l — 1, l : l + j — 1; Lidstone) Interpolation.- 4.11 (0, 2, 1; Lidstone) Interpolation.- References.- 5 Piecewise — Polynomial Interpolation.- 5.1 Introduction.- 5.2 Preliminaries.- 5.3 Piecewise Hermite Interpolation.- 5.4 Piecewise Lidstone Interpolation.- 5.5 Two Variable Piecewise Hermite Interpolation.- 5.6 Two Variable Piecewise Lidstone Interpolation.- References.- 6 Spline Interpolation.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 Cubic Spline Interpolation.- 6.4 Quintic Spline Interpolation: ? = 4.- 6.5 Approximated Quintic Splines: ? = 4.- 6.6 Quintic Spline Interpolation: ? = 3.- 6.7 Approximated Quintic Splines: ? = 3.- 6.8 Cubic Lidstone — Spline Interpolation.- 6.9 Quintic Lidstone — Spline Interpolation.- 6.10 L2 — Error Bounds for Spline Interpolation.- 6.11 TwoVariable Spline Interpolation.- 6.12 Two Variable Lidstone — Spline Interpolation.- 6.13 Some Applications.- References.- Name Index.
