100 research outputs found
Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials
We survey developments, over the last thirty years, in the theory of Shape
Preserving Approximation (SPA) by algebraic polynomials on a finite interval.
In this article, "shape" refers to (finitely many changes of) monotonicity,
convexity, or q-monotonicity of a function (for definition, see Section 4). It
is rather well known that it is possible to approximate a function by algebraic
polynomials that preserve its shape (i.e., the Weierstrass approximation
theorem is valid for SPA). At the same time, the degree of SPA is much worse
than the degree of best unconstrained approximation in some cases, and it is
"about the same" in others. Numerous results quantifying this difference in
degrees of SPA and unconstrained approximation have been obtained in recent
years, and the main purpose of this article is to provide a "bird's-eye view"
on this area, and discuss various approaches used.
In particular, we present results on the validity and invalidity of uniform
and pointwise estimates in terms of various moduli of smoothness. We compare
various constrained and unconstrained approximation spaces as well as orders of
unconstrained and shape preserving approximation of particular functions, etc.
There are quite a few interesting phenomena and several open questions.Comment: 51 pages, 49 tables, survey, published in Surveys in Approximation
Theory, 6 (2011), 24-7
Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball
A new modulus of smoothness based on the Euler angles is introduced on the
unit sphere and is shown to satisfy all the usual characteristic properties of
moduli of smoothness, including direct and inverse theorem for the best
approximation by polynomials and its equivalence to a -functional, defined
via partial derivatives in Euler angles. The set of results on the moduli on
the sphere serves as a basis for defining new moduli of smoothness and their
corresponding -functionals on the unit ball, which are used to characterize
the best approximation by polynomials on the ball.Comment: 63 pages, to appear in Advances in Mat
The Lower Estimate for Bernstein Operator
MSC 2010: 41A10, 41A15, 41A25, 41A36For functions belonging to the classes C2[0; 1] and C3[0; 1], we establish the lower estimate with an explicit constant in approximation by Bernstein polynomials in terms of the second order Ditzian-Totik modulus of smoothness. Several applications to some concrete examples of functions are presented
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