S.O.S. Mathematics CyberBoard

Dear all, I recently come to aware of this theory while reading journals. Is there anyone here knows what korovkin theory is all about and is there any particular book that I can refer to for more information? Thank You.

Your best bet may be to check, http://eom.springer.de/b/b110700.htm, as this website would have the book(s) you're looking for.

Good luck.

Along those lines, you should usually just check google for things like that. This board is for help with specific problems, not reference requests.

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It is about function approximation. For instance it gives a short proof of Weierstrass's theorem that every periodic continuous function is the uniform limit of trigonometric polynomials.

Let be the space of continuous real functions on a compact topological space.

A minimizing subspace of means a linear subspace of containing the function 1, and such that for any point in there is a function in having a strict minimum at . (Example: is any closed bounded real interval, and is the quadratic polynomials.)

A positive linear operator on is one that maps non-negative functions to non-negative functions.

Korovkin's Theorem: Let be a sequence of positive linear operators on , such that converges to uniformly for each in a minimizing subspace. Then the same happens for each in .

A proof and examples are in my 1973 book "Basic methods of linear functional analysis".

John Pryce