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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Approximation Theory - Moduli of Continuity and Global Smoothness Preservation (Paperback, Softcover reprint of the original 1st ed. 2000)
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Approximation Theory - Moduli of Continuity and Global Smoothness Preservation (Paperback, Softcover reprint of the original 1st ed. 2000)
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We study in Part I of this monograph the computational aspect of
almost all moduli of continuity over wide classes of functions
exploiting some of their convexity properties. To our knowledge it
is the first time the entire calculus of moduli of smoothness has
been included in a book. We then present numerous applications of
Approximation Theory, giving exact val ues of errors in explicit
forms. The K-functional method is systematically avoided since it
produces nonexplicit constants. All other related books so far have
allocated very little space to the computational aspect of moduli
of smoothness. In Part II, we study/examine the Global Smoothness
Preservation Prop erty (GSPP) for almost all known linear
approximation operators of ap proximation theory including:
trigonometric operators and algebraic in terpolation operators of
Lagrange, Hermite-Fejer and Shepard type, also operators of
stochastic type, convolution type, wavelet type integral opera tors
and singular integral operators, etc. We present also a sufficient
general theory for GSPP to hold true. We provide a great variety of
applications of GSPP to Approximation Theory and many other fields
of mathemat ics such as Functional analysis, and outside of
mathematics, fields such as computer-aided geometric design (CAGD).
Most of the time GSPP meth ods are optimal. Various moduli of
smoothness are intensively involved in Part II. Therefore, methods
from Part I can be used to calculate exactly the error of global
smoothness preservation. It is the first time in the literature
that a book has studied GSPP.
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