Korowkin approximation

The Korowkin approximation is a mathematical convergence statement in which the approximation of functions by certain sequences of functions is examined. In an application (see below), continuous functions are approximated by polynomials. The peculiarity of the Korowkin approximation is that one arrives at convergence statements for whole approximation methods by checking the convergence of the method only on a finite number of functions. The starting point is a sentence by Pawel Petrovich Korowkin from 1953.

Korovkin's theorem

In the following theorem, let the space of continuous real-valued functions on the interval be . Furthermore, stand the function for the limitation on . For this is the constant function with the value 1, for you get the identical function , for you have the restriction of the square function to . Korovkin's theorem is as follows: ${\displaystyle C[a,b]}$ ${\displaystyle [a,b]}$ ${\displaystyle x^{k}}$ ${\displaystyle x\mapsto x^{k}}$ ${\displaystyle [a,b]}$ ${\displaystyle k=0}$ ${\displaystyle k=1}$ ${\displaystyle id_{[a,b]}}$ ${\displaystyle k=2}$ ${\displaystyle [a,b]}$

If is a sequence of positive linear operators and is evenly on for , then is evenly on for all . ${\displaystyle (P_{n})_{n\in {\mathbb {N} }}}$ ${\displaystyle C[a,b]\rightarrow C[a,b]}$ ${\displaystyle P_{n}(x^{k})\,{\stackrel {n}{\rightarrow }}\,x^{k}}$ ${\displaystyle [a,b]}$ ${\displaystyle k=0,1,2}$ ${\displaystyle P_{n}(f)\,{\stackrel {n}{\rightarrow }}\,f}$ ${\displaystyle [a,b]}$ ${\displaystyle f\in C[a,b]}$

If one understands the sequence as an approximation method, then one has to prove the convergence of the method in the sense of the above sentence only for the three functions . The convergence of the method for all functions then follows. ${\displaystyle (P_{n})_{n}}$ ${\displaystyle x^{k},\,k=0,1,2}$

application

For the sake of clarity, the most well-known application will be reproduced here, a derivation of Weierstraß's approximation theorem : For let the -th Bernstein polynomial of , i. H. ${\displaystyle f\in C[0,1]}$ ${\displaystyle B_{n}(f)}$ ${\displaystyle n}$ ${\displaystyle f}$

${\displaystyle B_{n}(f)(t)=\sum _{i=0}^{n}{n \choose i}f\left({\frac {i}{n}}\right)\,t^{i}(1-t)^{n-i},\,\,t\in [a,b]}$.

Then is a sequence of positive linear operators. The convergence for can be shown by very elementary transformations on the occurring sums. Korowkin's theorem then yields that for all continuous functions evenly . This means that every continuous function on [0,1] can be approximated uniformly by polynomials, i. That is, a comfortable derivation of Weierstraß's approximation theorem is obtained. This argument can easily be extended to the more general interval . ${\displaystyle (B_{n})_{n}}$ ${\displaystyle B_{n}(x^{k})\,{\stackrel {n}{\rightarrow }}\,x^{k}}$ ${\displaystyle k=0,1,2}$ ${\displaystyle B_{n}(f)\,{\stackrel {n}{\rightarrow }}\,f}$ ${\displaystyle f}$ ${\displaystyle [0,1]}$ ${\displaystyle [a,b]}$

Korowkin approximation

The extensions of Korowkin's theorem to more general situations form the so-called Korowkin approximation theory , which is based on functional analytical methods. The following question is asked: In which situations can one infer convergence statements of the form by having to prove the convergence for only finitely many of the functions ? ${\displaystyle P_{n}(f)\,{\stackrel {n}{\rightarrow }}\,f}$ ${\displaystyle f}$

One can see the space as a prototype of a Banach algebra and in this context come to more general convergence statements, or one tries to replace it with more general, ordered vector spaces . So z. B. The following sentence in L ^p -spaces , : ${\displaystyle C[a,b]}$ ${\displaystyle C[a,b]}$ ${\displaystyle 1\leq p<\infty }$

Is a sequence of positive linear operators and holds for all , where , then follows for all . ${\displaystyle (P_{n})_{n}}$ ${\displaystyle L^{p}[1,\infty )\rightarrow L^{p}[1,\infty )}$ ${\displaystyle \|P_{n}(f)-f\|_{p}\,{\stackrel {n}{\rightarrow }}\,0}$ ${\displaystyle f\in \{x^{-\lambda _{1}},x^{-\lambda _{2}},x^{-\lambda _{3}}\}}$ ${\displaystyle \textstyle {\frac {1}{p}}<\lambda _{1}<\lambda _{2}<\lambda _{3}}$ ${\displaystyle \|P_{n}(f)-f\|_{p}\,{\stackrel {n}{\rightarrow }}\,0}$ ${\displaystyle f\in L^{p}[1,\infty )}$

In the examples considered so far, one had convergence statements of the kind for all from a suitable space , i.e. H. point by point . Further generalizations are obtained by replacing the id operator with other operators, i.e. by examining convergence statements of the type point by point . Finally, one can generalize from the operators to operators from in other spaces, e.g. B. on functionals . The book by Altomare and Campiti given below provides a good overview. ${\displaystyle P_{n}(f)\rightarrow f}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle P_{n}\rightarrow \mathrm {id} _{X}}$ ${\displaystyle X}$ ${\displaystyle P_{n}\rightarrow S}$ ${\displaystyle X\rightarrow X}$ ${\displaystyle X}$ ${\displaystyle X\rightarrow {\mathbb {R} }}$

literature

• PP Korovkin: About the convergence of positive linear operators in the space of continuous functions . Docl. Akad. Nauk. SSSR, Vol. 90, 1953, pp. 961-964 (Russian).
• F. Altomare, M. Campiti: Korovkin-type Approximation Theory and its Applications. de Gruyter Studies in Mathematics, Volume 17, 1994, ISBN 978-3-11-014178-8 .