The most classical development of Korovkin-type approximation
Theory essentially dealt with simple criteria ensuring the convergence of
arbitrary sequences of positive linear operators toward the identity
operator.
The significant and elegant results of the theory have
subsequently stimulated similar investigations on the convergence of
sequences of (positive) linear operators toward a fixed (positive) operator
on a Banach space (Banach lattice, respectively).
This generalization was first suggested by G.G.Lorentz (1972) and
has been developed by several mathematicians during these last twenty
years.
This more general approach revealed strong and fruitful
connections of the Korovkin-type approximation theory not only with
approximation theory but also with other fields such as structure theory of
Banach lattices and Banach space, potential theory and degenerate elliptic
differential equations.
On the other hand, as far as we know, the development of the
theory along these lines is still incomplete and certainly is worthy of
further investigations.
The main aim of this short series of lectures is to give a survey
on the main methods and results on the field as well as to illustrate some
significant applications.
In particular, special care will be devoted to the case when the
limit operator is a positive linear projection on the Banach space C(X) of
all continuous functions defined on a convex compact subset X.
In such a case, indeed, we have a rich theory which has strong
connections with other fields such as Feller semigroups, partial
differential equations and Markov processes.
Finally, some open problems will be briefly discussed as well.