Last Updated：2022/12/22
Sentence
A
central
problem
in
approximation
theory
is
to
characterize
the
best
approximation
of
a
function
by
polynomials,
or
other
classes
of
simple
functions,
in
terms
of
the
smoothness
of
the
function.
In
this
chapter,
we
study
the
characterization
of
the
best
approximation
by
polynomials
on
the
sphere.
In
the
classical
setting
of
one
variable,
the
smoothness
of
a
function
on
𝕊¹
is
described
by
the
modulus
of
smoothness,
defined
by
the
forward
difference.
Quizzes for review
A central problem in approximation theory is to characterize the best approximation of a function by polynomials, or other classes of simple functions, in terms of the smoothness of the function. In this chapter, we study the characterization of the best approximation by polynomials on the sphere. In the classical setting of one variable, the smoothness of a function on 𝕊¹ is described by the modulus of smoothness, defined by the forward difference.
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