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) |
Convergence, discrete. G.M. Vainikko (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence,_discrete&oldid=13267