Local Estimates for the Koornwinder Jacobi-Type Polynomials

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1 Available at Appl. Appl. Math. ISSN: Vol. 6 Issue (Jue 0) pp (reviously Vol. 6 Issue pp ) Appliatios ad Applied Mathematis: A Iteratioal Joural (AAM) Loal Estimates for the Koorwider Jaobi-Type olyomials Valmir Krasiqi Naim L. Braha ad Armed Sh. Shabai Departmet of Mathematis ad Computer Siees Uiversity of rishti Aveue "Mother Theresa" Nr = 5 rishtie 0000 Republi o Kosova vali.99@hotmail.om braha@yahoo.om armed_shabai@hotmail.om Reeived: July 3 00; Aepted: February 3 0 Abstrat I this paper we give some loal estimates for the Koorwider Jaobi-type polyomials by usig asymptoti properties of Jaobi orthogoal polyomials. Keywords: Koorwider Jaobi-type polyomials Jaobi orthogoal polyomials AMS (00) No.: 33C45 4C05. Itrodutio Let ( ) x[ ] 6

2 AAM: Iter. J. Vol. 6 Issue (Jue 0) [reviously Vol. 6 Issue pp ] 63 be a Jaobi weight with. Let also p x p x x ( ) ( ) 0 ( ) deote the uique Jaobi polyomials of preise degree with leadig oeffiiets fulfillig the orthogoal oditios 0 ( ) p pm m m. Felte (007) itrodued modified Jaobi weights as ( ) : x x x[ ]. () He proved the followig theorem [see Felte (007)]: Theorem.: Let ad. The p x C () ( ) 4 4 for all x [ ] with a positive ostat C C( ) beig idepedet of ad x. The above estimatio first appeared i Lubiski ad Totik (994). The for (004) exteded the previous results as follows: Felte Theorem.: Let ad. The p t C (3) ( ) 4 4 for all t U ad eah x [ ] where

3 64 Valmir Krasiqi et al. U: t[ ]: tx x x (4) for ad x [ ] with ( x ): x. Koorwider (984) itrodued the polyomials ) ( M N 0 defied as follows: Defiitio.3. Fix M N 0 ad. For 0 defie where ( M N) ( ) d ( ) ( B M A N A B ) p! dx A ( )! ( ) M ( ) ( ) ( )( ) (5) ad B ( )! ( ) N. ( ) ( ) ( )( ) (6) We all these polyomials the Koorwider s Jaobi-type polyomials. The above defied polyomials are orthogoal o the iterval [ ] with respet to the measure defied by ( ) f ( xd ) f dx ( ) ( ) + Mf Nf() where f C([ ]) ad M N 0. (7) Clearly for M N 0 oe has. (8) ( 00) ( )

4 AAM: Iter. J. Vol. 6 Issue (Jue 0) [reviously Vol. 6 Issue pp ] 65 Also ( ). (9) ( M N) ( N M) Some basi properties of ( M N ) are give as below [Varoa (989) hapter IV)]. ( M N) () ~ 3 if N 0 if N 0 (0) ad ( M N) ~ 3 if M 0 if M 0. () Theorem.4 [Varoa (989)]: Let M N 0. For every x [ ] there exists a uique ostat C suh that the followig relatio holds for eah : where M M ( M N) 4 4 h C x x ( M N) ( M N) h ( ) d. Based o Theorem.4 ad properties of Jaobi polyomials [see Lubiski ad Totik (994) ad Szego (975)] we get the followig estimatio for the Koorwider Jaobi-type polyomials: ( M N) (os ) 0( ) if ) if 0 0( () for

5 66 Valmir Krasiqi et al. ad. The aim of this paper is to prove similar results as those give i Theorem. ad Theorem. for Koorwider Jaobi-type polyomials whe respetively for.. Results The followig Theorem is the mai result of this ote. Theorem.: Let ad. The x D (3) ( M N) 4 4 for all x [ ] with a positive ostat D D( ) beig idepedet of ad x. roof: roof of the Theorem is similar to Theorem. i Felte (007). Let x [0] ad let 0 suh that os x. From () oe has the followig estimatio ( M N) (os ) C 0 if if. (4) If i the last relatio we substitute x os the we will have if 0arosx ( M N) C aros x if aros x (5)

6 AAM: Iter. J. Vol. 6 Issue (Jue 0) [reviously Vol. 6 Issue pp ] 67 where C is fixed positive ostat beig idepedet of ad. I what follows we will make use of the followig estimates x t t x si t arosx (6) ad x t t x si t arosx. (7) We differ two ases: Case.. I this ase 0. If 0 arosx the from (7) we obtai x ad from (5) we get the followig relatio ( ) M N C C C x C x. If aros x the from relatios (5) ad (7) we get ( ) M N C (aros x) C x C x. Case.. I this ase 0. If 0aros x the from relatios (5) ad (7) we obtai ( M N) C C C x. If aros x agai aordig to relatios (5) ad (7) we have

7 68 Valmir Krasiqi et al. ( ) M N C (aros x) C (aros x aros x) C x. From previous ases we have proved that ( M N) C ( ) x x for all x[0] ad. From (0) we obtai ( M N) C ( ) x x for all x [ 0) ad. The proof is ompleted. Next we will show that the loal estimates of previous theorem a be further exteded. We ( M N) ( M N) will prove that i (4) a be replaed by ( t) wheever t is i the followig Lemma [see Felte (007)]. iterval U x x. I order to do that we will make use of the Lemma.: Let ab 0 ad x. The () t 6 (8) ( ab ) ( ab) ( ab ) for all t U. Theorem.3: Let ad. The

8 AAM: Iter. J. Vol. 6 Issue (Jue 0) [reviously Vol. 6 Issue pp ] 69 t D (9) ( M N) 4 4 for all t U ad eah x [ ] where D D( ) is a positive ostat idepedet of t ad x. roof: Sie it follows that 0. Therefore by Lemma. with 4 4 a ad we obtai x for all t U. Applyig Theorem. yields iequality (4) for all t U as laimed. Corollary.4: x. The Let ad ( M N) ( ) U ( t) ( t) dt D( ). roof: Applyig Theorem.3 we obtai ( t) ( t) dt D ( t) dt. ( M N) ( ) ( ) U U x ( ) Usig the followig result from Felte (008) we obtai

9 70 Valmir Krasiqi et al. D () ( ) U x tdt ad thus the proof is ompleted. Akowledgmet The authors would like to thak aoymous referees for their suggestios whih otributed to the quality of the ote. REFERENCES Felte M. (007). Loal estimates for Jaobi polyomials J. Iequal. ure Appl. Math. Vol. 8 pp. -7. Felte M. (008). Uiform boudedess of ( C ) meas of Jaobi expasios i weighted sup orms Ata Math. Hug. Vol.8 pp Koorwider T. H. (984). Orthogoal polyomials with weight futio M N( x ) Caad. Math. Bull. Vol. 7 pp Lubiski D. S. ad Totik V. (994). Best weighted polyomial approximatio via Jaobi expasio SIAM J. Math. Aa. Vol. 5 pp Szego G. (975). Orthogoal olyomials 4 th ed. Ameria Mathematial Soiety rovidee R.I. Ameria Mathematial Soiety Colloquium ubliatios Vol. XXIII. p Varoa J. L. (989). Covergeia e L o pesos de la serie de Fourier respeto de alguos sistemas ortogoales h. D. Thesis Sem. Mat. Gari de Galdeao se. o. Zaragoza.