EAN: 9781461271123

APPROXIMATION THEORY - GEORGE A ANASTASSIOU SOR...

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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