View source for Bohman-Korovkin theorem

The starting point is the following theorem of P.P. Korovkin (1953) [[#References|[a6]]], [[#References|[a7]]]: for a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107001.png" /> of positive linear operators on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107002.png" /> into itself (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107003.png" /> is a compact interval and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107004.png" /> denotes the [[Banach space|Banach space]] of all continuous real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107005.png" /> with the supremum norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107006.png" />), the relations

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imply convergence:

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Here, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107009.png" /> are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070011.png" /> is the supremum norm on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070012.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070013.png" /> is positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070015.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070017.png" />, and the [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070018.png" /> is positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070019.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070020.png" />. One year earlier, in 1952, H. Bohman [[#References|[a2]]] had proved this theorem for positive operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070021.png" /> having a representation

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070023.png" /> knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070024.png" /> in the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070027.png" />.

One says that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070028.png" /> are a Korovkin set for (positive linear operators on) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070029.png" />. Korovkin proved that a set of three functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070030.png" /> that replaces the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070031.png" /> must be a [[Chebyshev system|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070032.png" /> (cf. also [[Korovkin theorems|Korovkin theorems]]; [[Korovkin-type approximation theory|Korovkin-type approximation theory]]).

The results have been generalized to other compact Hausdorff spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070033.png" />. For example, on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070034.png" />, the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070035.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070038.png" /> is a Korovkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070039.png" />. As usual, one identifies the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070040.png" /> with the continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070041.png" />-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070042.png" /> (cf, [[#References|[a1]]], [[#References|[a4]]]). On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070043.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070044.png" />, the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070045.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070046.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070050.png" />) is a Korovkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070051.png" />, but not a minimal Korovkin set.

There is also the following generalization. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070052.png" /> be a Banach lattice, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070053.png" /> be the class of all positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070054.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070055.png" /> be a fixed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070056.png" />. Then the Korovkin closure (or shadow, or Korovkin hull) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070058.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070059.png" /> with the property that for each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070061.png" /> the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070063.png" />, imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070064.png" />. The problem is to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070065.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070066.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070067.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070068.png" /> is a Korovkin set (cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a8]]]).

In some cases one can prove a quantitative form of the Korovkin theorem, estimating the rate of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070069.png" /> in terms of the rate of convergence for the elements of the Korovkin set. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070070.png" /> one has obtained estimates in terms of the first- or second-order modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070071.png" />; for example:

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070073.png" /> and some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070074.png" />.

If some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070075.png" /> is a polynomial operator, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070076.png" /> is a polynomial of degree less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070077.png" />, then at least one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070078.png" /> can not be approximated better than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070079.png" />.

Similar results can be obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070080.png" />. Also, it is possible to estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070081.png" /> in terms of the moduli of smoothness corresponding to

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Furthermore, quantitative Korovkin theorems for positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070083.png" />-spaces have been obtained (cf. [[#References|[a3]]], [[#References|[a4]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Altomare,   M. Campiti,   "Korovkin-type approximation theory and its applications" , de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bohman,   "On approximation of continuous and of analytic functions"  ''Arkiv. Mat. (2)'' , '''1'''  (1952)  pp. 43–56</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. DeVore,   "The approximation of continuous functions by positive linear operators" , ''Lecture Notes in Mathematics'' , '''293''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.A. DeVore,   G.G. Lorentz,   "Constructive approximation" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Donner,   "Extension of positive operators and Korovkin theorems" , ''Lecture Notes in Mathematics'' , '''904''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , Hindustan Publ. Corp.  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.G. Lorentz,   M. von Golitschek,   Y. Makovoz,   "Constructive approximation: advanced problems" , Springer  (1996)</TD></TR></table>