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ON CERTAIN PROPERTIES OF THE COMBINATIONS OF SZASZ-DURRMEYER OPERATORS.pdf
Ana1．Theory App1．
Vo1．27，No．2(2011)，167-180
DOI10．1007／slO496．O11-0167．7
oN CEI IN PRoPERTIESoFTHE CoMBINATIoNS
，
oF SZASZ．DURRMEYER oPERAToRS
T0masZSwiderski
(PedagogicalUniversity,Poland)
ReceivedFeb．22，2010； RevisedMar．2，2011
@EditorialBoardofAnalysisinTheoryApplicationsandSpringer—VerlagBerlinHeidelberg2011
Abstract． InthepresentpaperwestudypropertiesofSz~isz-Durrmeyeroperators．These
operatorsraeintroducedin[5】andgeneralizehteintegraloperatorsproposedbyS．M．Mazhra
nadV．Totikin[12]．WealsogeneralizesomeresultsobtainedbyM．Heilmann6[]andD．一X．
Zhou[16]
．
Keywords：modifiedSzdsz-M·irakyanoperator,Baskakov-D·urremeyeroperator
AMS(2010)subjectclassification：65D99
1 Introduction
InthispaperwesutdySz~isz—Dumneyeroperatorsdefinedonfunctionsf∈Lpinthefollow—
ingform：
(加 )= M ，
k=0
wherex,nE[。， 足∈ V∈( 。o)an 一 forf∈[。， Thet—
Kn
v
， (f，)=，l∑P，k(x)p咄+v(f)
k=O
iscalledthekernelofSz~isz—Durrmeyeroperator．ThisfamilyofoperatorswasintroducedbyA．
CiupaandI．Gavrea[5lnadwasalsoindependentlyproposedbyE Wachnicki[5I
． ． Sometypical
resultscouldbefoundinpapers[5，13，14，7]．Inhtefirstsectionweplacedtheresultswhich
areusefulinhteproofofhtefurthertheorems，someofthem generalizethepropertieswhich
areknownofrhteparticularcase(ofr、，=0，n∈N)butohterscouldn’tevenbeformulatedofr
htepreviouslyconside：redfamil
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