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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 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.
. The theory of difference equations, the methods used in their
solutions and their wide applications have advanced beyond their
adolescent stage to occupy a central position in Applicable
Analysis. In fact, in the last five years, the proliferation of the
subject is witnessed by hundreds of research articles and several
monographs, two International Conferences and numerous Special
Sessions, and a new Journal as well as several special issues of
existing journals, all devoted to the theme of Difference
Equations. Now even those experts who believe in the universality
of differential equations are discovering the sometimes striking
divergence between the continuous and the discrete. There is no
doubt that the theory of difference equations will continue to play
an important role in mathematics as a whole. In 1992, the first
author published a monograph on the subject entitled Difference
Equations and Inequalities. This book was an in-depth survey of the
field up to the year of publication. Since then, the subject has
grown to such an extent that it is now quite impossible for a
similar survey, even to cover just the results obtained in the last
four years, to be written. In the present monograph, we have
collected some of the results which we have obtained in the last
few years, as well as some yet unpublished ones.
In analysing nonlinear phenomena many mathematical models give rise
to problems for which only nonnegative solutions make sense. In the
last few years this discipline has grown dramatically. This
state-of-the-art volume offers the authors' recent work, reflecting
some of the major advances in the field as well as the diversity of
the subject. Audience: This volume will be of interest to graduate
students and researchers in mathematical analysis and its
applications, whose work involves ordinary differential equations,
finite differences and integral equations.
In analysing nonlinear phenomena many mathematical models give rise
to problems for which only nonnegative solutions make sense. In the
last few years this discipline has grown dramatically. This
state-of-the-art volume offers the authors' recent work, reflecting
some of the major advances in the field as well as the diversity of
the subject. Audience: This volume will be of interest to graduate
students and researchers in mathematical analysis and its
applications, whose work involves ordinary differential equations,
finite differences and integral equations.
. The theory of difference equations, the methods used in their
solutions and their wide applications have advanced beyond their
adolescent stage to occupy a central position in Applicable
Analysis. In fact, in the last five years, the proliferation of the
subject is witnessed by hundreds of research articles and several
monographs, two International Conferences and numerous Special
Sessions, and a new Journal as well as several special issues of
existing journals, all devoted to the theme of Difference
Equations. Now even those experts who believe in the universality
of differential equations are discovering the sometimes striking
divergence between the continuous and the discrete. There is no
doubt that the theory of difference equations will continue to play
an important role in mathematics as a whole. In 1992, the first
author published a monograph on the subject entitled Difference
Equations and Inequalities. This book was an in-depth survey of the
field up to the year of publication. Since then, the subject has
grown to such an extent that it is now quite impossible for a
similar survey, even to cover just the results obtained in the last
four years, to be written. In the present monograph, we have
collected some of the results which we have obtained in the last
few years, as well as some yet unpublished ones.
This volume, which presents the cumulation of the authors' research
in the field, deals with Lidstone, Hermite, Abel-Gontscharoff,
Birkhoff, piecewise Hermite and Lidstone, spline and
Lidstone-spline interpolating problems. Explicit representations of
the interpolating polynomials and associated error functions are
given, as well as explicit error inequalities in various norms.
Numerical illustrations are provided of the importance and
sharpness of the various results obtained. Also demonstrated are
the significance of these results in the theory of ordinary
differential equations such as maximum principles, boundary value
problems, oscillation theory, disconjugacy and disfocality. The
book should be useful for mathematicians, numerical analysts,
computer scientists and engineers.
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