View source for Abstract approximation theory

Classical [[Approximation theory|approximation theory]] deals with the approximation of real-valued functions defined on real intervals by certain basic functions like ordinary or trigonometric polynomials (cf. also [[Trigonometric polynomial|Trigonometric polynomial]]), splines (see [[Spline approximation|Spline approximation]]), etc. An approximation may be obtained through coincidence on a given subset of the domain ([[Interpolation|interpolation]] and [[Trigonometric interpolation|trigonometric interpolation]]) or through more general constructive or non-constructive methods. The quality of an approximation is usually evaluated by a measure for the deviation of two functions (see [[Approximation of functions, measure of|Approximation of functions, measure of]]). These approaches may be suitably transferred to more general settings.

==Interpolation.==
A general formulation of an interpolation problem in normed spaces is due to L. Asimow [[#References|[a2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101001.png" /> be a closed linear subspace in a normed [[Linear space|linear space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101002.png" />. Given a bounded closed convex neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101003.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101004.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101005.png" /> and a bounded closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101006.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101007.png" />, their polars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a1101009.png" /> in the [[Adjoint space|adjoint space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010011.png" /> (cf. also (the editorial comments to) [[Polar|Polar]]), one defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010012.png" /> the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010014.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010015.png" /> one looks for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010016.png" /> satisfying:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010017.png" />;

2a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010018.png" /> (exact solution); respectively,

2b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010019.png" /> (approximate solution). In the classical situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010020.png" /> stands for a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010021.png" /> of continuous functions on a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010022.png" /> endowed with the supremum norm, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010023.png" /> for all functions vanishing on a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010025.png" />. Further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010026.png" /> is the class of functions used for the interpolation. Condition 1) means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010028.png" /> coincide on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010029.png" />, whereas 2a) (respectively, 2b)) impose further restrictions on the interpolating element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010030.png" />. Classical results on this problem include the Urysohn and Rudin–Carleson theorems for function spaces. The general case requires technical conditions involving the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010031.png" />.

==Approximation by linear operators.==
Approximation schemes may often be modelled by sequences (or generalized sequences) of linear operators (see [[Approximation of functions, linear methods|Approximation of functions, linear methods]]). For a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010032.png" /> of positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010033.png" />, Korovkin's theorem [[#References|[a4]]] states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010034.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010035.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010036.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010037.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010038.png" /> for the three test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010039.png" /> (cf. also [[Korovkin theorems|Korovkin theorems]]). This result was subsequently generalized to different sets of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010040.png" /> and various topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010041.png" /> replacing the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010042.png" /> (cf. also [[Korovkin-type approximation theory|Korovkin-type approximation theory]]). Classical examples include the Bernstein operators (see [[Bernstein polynomials|Bernstein polynomials]]) and the Fejér sums (cf. also [[Fejér sum|Fejér sum]]), which provide approximation schemes by polynomials and trigonometric polynomials, respectively. Further generalizations investigate the convergence of certain classes of linear operators on various domains, such as positive operators on topological vector lattices, contractive operators on normed spaces, multiplicative operators on Banach algebras, monotone operators on set-valued functions, monotone operators with certain restricting properties on spaces of stochastic processes, etc. (cf. [[#References|[a1]]] for a recent survey). Typically, for a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010043.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010044.png" />, one defines the Korovkin closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010046.png" /> as the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010048.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010049.png" /> is an equicontinuous generalized sequence (cf. [[Equicontinuity|Equicontinuity]]) in the restricted class of operators such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010050.png" /> holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010051.png" />. The main problem is to describe the Korovkin closure as explicitly as possible.

A very general setting allowing a unified approach to most of the above-mentioned cases is given by locally convex cones [[#References|[a3]]], structures that generalize locally convex ordered vector spaces (cf. [[Locally convex space|Locally convex space]]; [[Semi-ordered space|Semi-ordered space]]) in the sense that scalar multiplication is only defined for non-negative real numbers; neither the existence of negatives nor the validity of the cancellation law is required. In spite of their generality, locally convex cones allow a rich duality theory, including Hahn–Banach-type extension and separation theorems for continuous linear functionals. Various restrictions on classes of linear operator may be taken care of by proper choices for the domains and their topologies alone, and Korovkin closures may be characterized in terms of the dual cone.

A complete description of Korovkin closures is available for positive linear operators on weighted spaces of continuous functions [[#References|[a5]]], [[#References|[a6]]]: Given a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010052.png" /> and an upward-directed family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010053.png" /> of non-negative upper semi-continuous real-valued weight functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010054.png" />, one considers the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010055.png" /> of all continuous real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010056.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010058.png" /> vanishes at infinity for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010059.png" />. Endowed with the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010060.png" /> (cf. [[Semi-norm|Semi-norm]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010061.png" /> is a [[Locally convex space|locally convex space]]. (The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010062.png" />, for example, leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010063.png" /> with the topology of uniform convergence; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010064.png" /> consists of the characteristic functions of all compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010066.png" /> with the topology of compact convergence.) For a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010067.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010068.png" />, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010069.png" /> is contained in the Korovkin closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010070.png" /> if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010071.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010072.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010073.png" /> one (and therefore both) of the following two equivalent conditions hold:

a)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010074.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010075.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010076.png" /></td> </tr></table>

b) for every finite positive regular [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010077.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010078.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010081.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010083.png" />, then b) shows that the vector sublattice generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010084.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110100/a11010085.png" />. In this way the preceding criterion leads to versions of the classical [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Altomare,   M. Campiti,   "Korovkin-type approximation theory and its applications" , W. de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Asimow,   A.J. Ellis,   "Convexity theory and its applications in functional analysis" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Keimel,   W. Roth,   "Ordered cones and approximation" , ''Lecture Notes in Mathematics'' , '''1517''' , Springer  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , ''Russian Monographs and Texts on advanced Math.'' , '''III''' , Gordon&amp;Breach  (1960)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.B. Prolla,   "Approximation of vector valued functions" , North-Holland  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Roth,   "A Korovkin type theorem for weighted spaces of continuous functions"  ''Bull. Austral. Math. Soc.'' , '''55'''  (1997)  pp. 239–248</TD></TR></table>