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- Publisher
- de Gruyter
- Copyright
- © by Steffen Roch
- ISSN
- 0420-1213
- eISSN
- 2391-4661
- DOI
- 10.1515/dema-1989-0307
- Publisher site
- See Article on Publisher Site
Abstract
DEMONSTRATIO MATHEMATICAVol. XXIINo 31989Steffen Roch, Bernd SilbermannNON-STRONGLY CONVERGING APPROXIMATION METHODS1. IntroductionCommonly,alargeclassofapproximationmethodsforsolving the operator equation Ax = y can be interpretedasfollows :Choose projection operators P t and operators A^ : im P^——>im P^ and consider a approximation equation A^x^ = P^y.If there is a ToSO that, for each x A x and for each right'oside y, these equations have a unique solution x^, and if x^converges to a solution x of the equation Ax = y, then onesays that the approximation method II { c o n v e r g e soperatorcasesA(see(thinkmeasurable[4],[5]).onintegralfunctionsChandler/Graham,convergencethatthecompactdex^—>asintheHoog/Sloan,x^specialoperatorsx cannotfunctionsonpapersandinterestingspacesofofboundedAnselone/Sloan,Silbermann)be guaranteedconvergebutbutuniformlytheusualit turnstoxonouteachinterval. This observation leads to a weaker notionof convergence,projections."weakly"In somefor theInthe convergence with respect to a family ofthisconvergingpaperweapproximationstudymethodsview which includes the standard theory.-651 -thecorrespondingfroma pointofS. Roch, B. SilbermannThereby we make essentially use of Simonenko and Kozak'stechniques in the theory of operators of local type. As anapplication, we show how the results of 111 — [3], [7] can bealmost at once derived from our theory.Otherpossibleapplications would be equations in finite differencesinspaces of bounded functions and two-dimensional Wiener-Hopfequations.Moreover,it should be pointed outthatthemethods presented here apply to the matrix case withoutchanges. It is the authors'aim to return to this circle ofproblems in a further comprehensive publication.2. Convergence with
Journal
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 1989