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Interpolation of functions is one of the basic part of
Approximation Theory. There are many books on approximation theory,
including interpolation methods that - peared in the last fty
years, but a few of them are devoted only to interpolation
processes. An example is the book of J. Szabados and P. Vertesi:
Interpolation of Functions, published in 1990 by World Scienti c.
Also, two books deal with a special interpolation problem, the
so-called Birkhoff interpolation, written by G.G. Lorentz, K.
Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The
classical books on interpolation address numerous negative results,
i.e., - sultsondivergentinterpolationprocesses,
usuallyconstructedoversomeequidistant system of nodes. The present
book deals mainly with new results on convergent - terpolation
processes in uniform norm, for algebraic and trigonometric
polynomials, not yet published in other textbooks and monographs on
approximation theory and numerical mathematics. Basic tools in this
eld (orthogonal polynomials, moduli of smoothness, K-functionals,
etc.), as well as some selected applications in numerical
integration, integral equations, moment-preserving approximation
and summation of slowly convergent series are also given. The
rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic
and trigonometric polynomials introducing the most important
concepts on appro- mation of functions. Especially, in Sect. 1.4 we
give basic results on interpolation by algebraic polynomials,
including representations and computation of interpolation
polynomials, Lagrange operators, interpolation errors and uniform
convergence in some important classes of functions, as well as an
account on the Lebesgue function and some estimates for the
Lebesgue constant.
Interpolation of functions is one of the basic part of
Approximation Theory. There are many books on approximation theory,
including interpolation methods that - peared in the last fty
years, but a few of them are devoted only to interpolation
processes. An example is the book of J. Szabados and P. Vertesi:
Interpolation of Functions, published in 1990 by World Scienti c.
Also, two books deal with a special interpolation problem, the
so-called Birkhoff interpolation, written by G.G. Lorentz, K.
Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The
classical books on interpolation address numerous negative results,
i.e., - sultsondivergentinterpolationprocesses,
usuallyconstructedoversomeequidistant system of nodes. The present
book deals mainly with new results on convergent - terpolation
processes in uniform norm, for algebraic and trigonometric
polynomials, not yet published in other textbooks and monographs on
approximation theory and numerical mathematics. Basic tools in this
eld (orthogonal polynomials, moduli of smoothness, K-functionals,
etc.), as well as some selected applications in numerical
integration, integral equations, moment-preserving approximation
and summation of slowly convergent series are also given. The
rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic
and trigonometric polynomials introducing the most important
concepts on appro- mation of functions. Especially, in Sect. 1.4 we
give basic results on interpolation by algebraic polynomials,
including representations and computation of interpolation
polynomials, Lagrange operators, interpolation errors and uniform
convergence in some important classes of functions, as well as an
account on the Lebesgue function and some estimates for the
Lebesgue constant.
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