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FASCICULI
MATHEMATICI
Nr45
2010
IbrahimB¨
˙
uy¨
ukyazıcı
AKOROVKINTYPEAPPROXIMATIONTHEOREM
ANDITSAPPLICATION
Abstract.Inthispaper,weinvestigateaKorovkin-typeap-
proximationtheoremforsequencesofpositivelinearoperatorson
thespaceofallcontinuousrealvaluedfunctionsdefinedon[a,b].
Wealsoobtainsomeapproximationpropertiesforsequencesof
positivelinearoperatorsconstructedbymeansoftheBernstein
operator.
Keywords:positivelinearoperator,Korovkin-typetheorem,
Bernsteinoperator.
AMSMathematicsSubjectClassification:41A10,41A25,41A36.
1.Introduction
Let{Ln}beasequenceofpositivelinearoperatorsactingfromC[a,b]into
C[a,b],whichisthespaceofallcontinuousrealvaluedfunctionson[a,b].In
thiscase,Korovkin[8]firstnoticednecessaryandsufficientconditionsforthe
uniformconvergenceofLnftoafunctionfbyusingthetestfunctionseś(t)
definedbyeś(t)=tś(i=0,1,2).Latermanyresearchersinvestigatethese
conditionsforvariousoperatorsdefinedondifferentspaces.Inthepresent
paper,weobtainfollowingtheKorovkin-typetheoremandgiveresultsfor
theapproximationpropertiesofthegeneralizedBernsteinoperatorsBn,ϕ
asanapplication.Notethatthroughoutthepaper,wealwaysassumethat
ϕ:[a,b]→[a,b]beabijectionandacontinuousfunctionon[a,b].
Theorem1(Korovkin-typetheorem).Let{Ln}beasequenceofpositive
linearoperatorsactingfromC[a,b]intoC[a,b].Then,forallf,ϕ∈C[a,b]
(1)
ifandonlyif
(2)
n→∞
lim
"Ln(f)−f◦ϕ"
C[a,b]=0
n→∞
lim
"Ln(eś)−eś◦ϕ"C[a,b]=0,(i=0,1,2).
