ON SOME PROPERTIES OF KANTOROVICH BIVARIATE OPERATORS
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1 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Matheatics ad Iforatics ICTAMI 3 Alba Iulia ON SOME PROPERTIES OF KANTOROVICH BIVARIATE OPERATORS by Lucia A. Căbulea ad Mihaela Aldea Abstract. I this paper we cotiue our earlier ivestigatios cocerig the use of probabilistic ethods for costructig liear positive operators useful i approxiatio theory of fuctios. The ai result of this paper cosists i itroducig ad ivestigatig the approxiatio properties of Katorovich bivariate operator which is a itegral liear positive operator reproducig the liear fuctios. Keywords: bivariate operator. THE OPERATORS OF KANTOROVICH Let N be fixed. The operators K : L ([] C([] defied by ( K f x ( x ( x ( f ( t dt (. are the operators of Katorovich. If we oticed χ the characteristic fuctio of the the operators (. ca be defie ( f ( t χ t dt (. ( K f x ( p ( x where p x x ( x [] are the Berstei polyoials. Lea[]. The operators of Katorovich satisfy the relatios i ( e ( x K 6
2 Lucia A. Căbulea Mihaela Aldea - O soe properties of Katorovich K ( x x ( ii ( e ( K e ( x x x. ( ( 3( iii ( iv For all f ([ ] ( K f ( x ( B F ( x L ( B f ( x x ( x f [ ] d where F( x dx x f ( t dt ad x are the operators of Berstei. Theore [8]. The operators of Katorovich have the properties li uiforly o [] ( f C[ ] i f f K. ii K li f f f L [ ] p. ( p. THE OPERATORS OF STANCU-KANTOROVICH The Stacu polyoials [7] defied by S ( f ; x w ( x f I [ ] x ( ( x ( x where w ( x ( ( x x( x...( x ( used for costructig the operators of Stacu-Katorovich[5] ca be K ( f ; x ( w ( x f ( t dt. (. It is clear that the operators defied by (. are liear ad positive too. I the th particular case the operator reduces obviously to the classical Katorovich operator defied by (.. Lea [3]. The operators defied by (. satisfy the relatios 6
3 Lucia A. Căbulea Mihaela Aldea - O soe properties of Katorovich K x i ( ; ii K ( t x; x x ( iii ( t x K ( ; x x( x ( 3( iv If C with C a positive costat it follows where ( x ( t x ; x ( x x K ( C ψ x ( x E A A ψ with E x I \ E ad A > fixed. Theore. Let ( ( K be defied by (. ad ( p ( ( β. ( / p p 3( / r If r 3 is a iteger uber ad ( p fuctio f [ ] we have with L p ( / r f f C β ( p f ω r ( p β N the for every r ( r ( f K p r β p C p r a positive costat idepedet of f ad. p 3.THE BIVARIATE OPERATORS OF KANTOROVICH We cosider the operators defied by 63
4 Lucia A. Căbulea Mihaela Aldea - O soe properties of Katorovich h h ( K f ( x y ( ( x ( x y ( y f ( u v h where f belogs to the spaces ([ ] [ ] L. The operators defied by (3. are the of Katorovich. Lea. The of Ktorovich satisfy the relatios: i ( e ( x y K K x ( ii ( e ( x y K ( x y y ( iii ( e K iv ( e ( x y x y ( ( ( K ( ( 3( v ( e ( x y x x ( K. ( ( 3( vi ( e ( x y y y Theore. The of Katorovich have the properties li uiforly o [ ] [ ] ( C( [ ] [ ] i f f K f h h dudv ii li f f ( f L ([ ] [ ] p. K p 4.THE BIVARIATE OPERATORS OF STANCU KANTOROVICH Now we cosider the operators defied by 64
5 Lucia A. Căbulea Mihaela Aldea - O soe properties of Katorovich K h h ( f ; x y ( ( w ( x w ( y f ( u v h dudv (4. ( ( x ( x ( where w ( x ( x x( x...( x ( ad ( h ( h y ( y ( w h ( x ( y h y( y...( y ( h. h The operators defied by (4. are the of Stacu- Katorovich. Lea. The operators defied by (4. satisfy the relatios K i ( ; x y ii ( u x; x y K x ( iii ( v y; x y K y ( iv (( u x( v y ; x y K x y ( ( ( ; x y x( x v ( u x K ( 3( vi ( v y K ( ; x y y( y ( 3(. REFERENCES []. Agratii O. Aproxiare pri operatori liiari Presa Uiversitară Clujeaă. []. Căbulea L. Aldea M. Geeralizatios of Katorovich type Aalele Uiversităţii Aurel Vlaicu Arad Seria Mateatica Arad 7-3. [3]. Della Vechia B. O the approxiatio offuctios by eas ofthe operators of D. 65
6 Lucia A. Căbulea Mihaela Aldea - O soe properties of Katorovich D. Stacu Studia Uiv. Babeş-Bolyai Math. XXXVII ( [4]. Della Vechia B. Mache D. H. O approxiatio properties of Stacu-Katorovich operators Rev. D'Aalyse Nu. Et de Théorie de L'Approx. 7(998 o [5]. Katorovich L. V. Surcertais dévelopets suivat Ies polyôes de la fore de S. Bestei I II C. R. Acad. URSS ( [6]. Razi Q. Approxiatio of a fuctio by Katorovich type operators Mat. Veşic. 4( [7]. Sedov B. Popov V. A. The Averaged Moduli of Soothess Pure ad Applied Matheatics Joh Wiley & Soos 988. [8]. Stacu D. D. Approxiatio of fuctios by a ew class of liear polyoial operators Rev. Rouaie Math. Pures Appl. 8( [9]. Stacu D. D. Coa Gh. Agratii O. Trîbiţaş R. Aaliză uerică şi teoria aproxiării voi. l Presa Uiversitară Clujeaă. Authors: Lucia A. Căbulea " Decebrie 98" Uiversity of Alba lulia Roâia. Departaet of Matheatics e-ail: lcabulea@uab.ro Mihaela Aldea " Decebrie 98" Uiversity of Alba lulia Roâia. Departaet of Matheatics e-ail: todea@uab.ro. 66
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