Difference between revisions of "Bohman-Korovkin theorem"
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The starting point is the following theorem of P.P. Korovkin (1953) [a6], [a7]: for a given sequence of positive linear operators on the space
into itself (where
is a compact interval and
denotes the Banach space of all continuous real-valued functions on
with the supremum norm
), the relations
![]() |
imply convergence:
![]() |
Here, the functions are defined by
and
is the supremum norm on the interval
. An element
is positive (denoted by
) if
for all
, and the linear operator
is positive if
implies
. One year earlier, in 1952, H. Bohman [a2] had proved this theorem for positive operators
having a representation
![]() |
with knots
in the open interval
, and
on
.
One says that the functions are a Korovkin set for (positive linear operators on)
. Korovkin proved that a set of three functions in
that replaces the set
must be a Chebyshev system; he also proved that there are no sets of only two functions such that convergence for these two functions implies convergence for all
(cf. also Korovkin theorems; Korovkin-type approximation theory).
The results have been generalized to other compact Hausdorff spaces . For example, on the circle
, the set of functions
with
,
,
is a Korovkin set for
. As usual, one identifies the functions in
with the continuous
-periodic functions on
(cf, [a1], [a4]). On the
-dimensional cube
, the set of
functions
(with
and
for
and
) is a Korovkin set for
, but not a minimal Korovkin set.
There is also the following generalization. Let be a Banach lattice, let
be the class of all positive linear operators on
, and let
be a fixed subset of
. Then the Korovkin closure (or shadow, or Korovkin hull)
of
is the set of all
with the property that for each sequence
in
the relations
,
, imply
. The problem is to find
for a given
; if
, then
is a Korovkin set (cf. [a1], [a5], [a8]).
In some cases one can prove a quantitative form of the Korovkin theorem, estimating the rate of convergence in terms of the rate of convergence for the elements of the Korovkin set. For
one has obtained estimates in terms of the first- or second-order modulus of continuity
; for example:
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with and some constant
.
If some is a polynomial operator, i.e.,
is a polynomial of degree less than or equal to
, then at least one of the functions
can not be approximated better than
.
Similar results can be obtained for . Also, it is possible to estimate
in terms of the moduli of smoothness corresponding to
![]() |
Furthermore, quantitative Korovkin theorems for positive linear operators on -spaces have been obtained (cf. [a3], [a4]).
References
[a1] | F. Altomare, M. Campiti, "Korovkin-type approximation theory and its applications" , de Gruyter (1994) |
[a2] | H. Bohman, "On approximation of continuous and of analytic functions" Arkiv. Mat. (2) , 1 (1952) pp. 43–56 |
[a3] | R.A. DeVore, "The approximation of continuous functions by positive linear operators" , Lecture Notes in Mathematics , 293 , Springer (1972) |
[a4] | R.A. DeVore, G.G. Lorentz, "Constructive approximation" , Springer (1993) |
[a5] | K. Donner, "Extension of positive operators and Korovkin theorems" , Lecture Notes in Mathematics , 904 , Springer (1982) |
[a6] | P.P. Korovkin, "On convergence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk. SSSR , 90 (1953) pp. 961–964 (In Russian) |
[a7] | P.P. Korovkin, "Linear operators and approximation theory" , Hindustan Publ. Corp. (1960) (In Russian) |
[a8] | G.G. Lorentz, M. von Golitschek, Y. Makovoz, "Constructive approximation: advanced problems" , Springer (1996) |
Bohman-Korovkin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohman-Korovkin_theorem&oldid=16494