Best complete approximation
A best approximation of a function in
variables
by algebraic or trigonometric polynomials. Let
be the space
or
of functions
,
-periodic in each variable, that are either continuous or
-summable (
) on the
-dimensional period cube with edges of length
.
The best complete approximation of a function by trigonometric polynomials is the quantity
![]() |
where the infimum is taken over all trigonometric polynomials of degree in the variable
(
). Together with the best complete approximation, one also considers best partial approximations.
A best partial approximation of a function is a best approximation by functions
that are trigonometric polynomials of degree
(
), respectively, in the fixed variables
with coefficients depending on the remaining
variables, i.e.
![]() |
It is obvious that
![]() |
S.N. Bernstein [S.N. Bernshtein] [1] proved the following inequality for continuous functions in two variables:
![]() | (1) |
![]() |
where is an absolute constant. It has been shown [3] that the term
in inequality (1) (and in the analogous relation for the space
) cannot be replaced by a factor with a slower rate of increase as
.
In the space (
) one has the inequality
![]() | (2) |
where the constant depends only on
and
.
Similar definitions yield best complete approximations and best partial approximations of functions defined on a closed bounded domain by algebraic polynomials, and in this case inequalities similar to (1) and (2) have been established.
References
[1] | S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) (In Russian) |
[2] | A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian) |
[3] | V.N. Temlyakov, "On best approximations of functions in two variables" Dokl. Akad. Nauk SSSR , 223 : 5 (1975) pp. 1079–1082 (In Russian) |
Comments
References
[a1] | G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) |
Best complete approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_complete_approximation&oldid=46041