Convergence, discrete

Convergence of functions and operators on lattices in corresponding spaces. Let be Banach spaces, and let and be systems of linear operators (connecting mappings) , , with the property

, for all , ,

A sequence with :

a) converges discretely (or -converges) to if ;

b) is discretely compact (or -compact) if for every infinite set there is an infinite set such that the subsequence converges discretely.

A sequence of operators :

a) converges discretely (or -converges) to an operator if for any -convergent sequence the relation

(1)

holds;

b) converges compactly to if, in addition to (1), the following condition is fulfilled: , () is -compact;

c) converges regularly (or properly) to if, in addition to (1), the following condition is fulfilled: , , is -compact is -compact;

d) converges stably to if, in addition to (1), the following condition is fulfilled: There exists an such that .

Let and be bounded linear operators. Then if and only if and if for every from a certain dense subset in .

For bounded linear operators and , the following conditions are equivalent:

1) stably, ;

2) regularly, , and the operators are Fredholm operators with index zero;

3) stably and regularly.

If one of these conditions is fulfilled, then and (for sufficiently large ) exist, and stably and regularly. If the conditions 1), 2) and 3) are fulfilled, they can be interpreted as a convergence theorem for the equations and : If 1), 2) or 3) are fulfilled, then implies that

with rate

In proving the convergence of approximate methods, 1) and 2) are used most frequently. Appropriate spaces of functions are chosen for and , while operators that transfer the functions to their values on a lattice are chosen for and .

References

[1a]	F. Stummel, "Diskrete Konvergenz linearer Operatoren I" Math. Ann. , 190 (1970) pp. 45–92
[1b]	F. Stummel, "Diskrete Konvergenz linearer Operatoren II" Math. Z. , 120 (1971) pp. 231–264
[2]	G.M. Vainikko, "Regular convergence of operators and approximate solution of equations" J. Soviet Math. , 15 (1981) pp. 675–705 Itogi Nauk. i Tekhn. Mat. Anal. , 16 (1979) pp. 5–53
[3]	G.M. Vainikko, "Funktionalanalysis der Diskretisierungsmethoden" , Teubner (1976) (Translated from Russian)