WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE FOR SEQUENCES OF POSITIVE LINEAR OPERATORS

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1 Математички Билтен ISSN X Vol. 38(LXIV) No. 04 (-33) Скопје, Македонија WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE FOR SEQUENCES OF POSITIVE LIAR OPERATORS Elida Hoxha, Erem Aljimi ad Valdete Lou 3 Abstract. We itroduce the otio of weighted Norlud Euler A-Statistical Covergece of a sequece, where A represets the oegative regular matrix. We also prove the Korovi approximatio theorem by usig the otio of weighted Norlud-Euler A-statistical covergece. Further, we give a rate of weighted Norlud- Euler A-statistical covergece.. BACKGROUND, NOTATIONS AND PRELIMINARIES Suppose that E N {,, } ad E { : E}. The E lim E is called the atural desity of E provided that the limit exist, where. represets the () umber of elemets i the eclosed set. The term statistical covergece was first preseted by Fast [] which is geeralizatio of the cocept of ordiary covergece. Actually, a root of the otio of statistical covergece ca be detected by Zygmud [] (also see [3]), where he used the term almost covergece which tured out to be equivalet to the cocept of statistical covergece. The otio of Fast was further ivestigated by Schoeberg [4], Salat [5], Fridy [6], ad Coer [7]. The followig otio is due to Fast []. A sequece x ( x ) is said to be statistically coverget to L if ( K ) 0 for every 0, where K { N : x L } () equivaletly, lim { : x L } 0. (3) I symbol, we will write S lim x L. We remar that every coverget sequece is statistically coverget but ot coversely. Let X ad Y be two sequece spaces ad let A ( a, ) be a ifiite matrix. If for each x ( x ) i X the series 00 Mathematics Subject Classificatio. Primary: 47H0 Secodary: 55M0 Key words ad phrases. A-statistical covergece, oegative regular matrix

2 E. Hoxha, E. Aljimi, V. Lou A x a, x a, x (4) coverges for each N ad the sequece Ax A x belogs to Y, the we say the matrix A maps X to Y. By the symbol ( XY, ) we deote the set of all matrices which map X ito Y. A matrix A (or a matrix map A ) is called regular if A ( c, c), where the symbol deotes the spaces of all coverget sequeces ad lim Ax lim x (5) for all x c. The well-ow Silverma-Toeplitz theorem (see [8]) assert that A ( a, ) is regular if ad oly if i) lim a, 0 for each ; ii) lim a, ; iii) sup a,. Kol [9] exteded the defiitio of statistical covergece which the help of oegative regular matrix A ( a, ) callig it A -statistical covergece. The defiitio of A -statistical covergece is give by Kol as follows. For ay oegative regular matrix A, we say that a sequece is A -statistically coverget to L provided that for every 0 we have lim a, L (6) : x L I 009, the cocept of weighted statistical covergece was defied ad studied by Karaaya ad Chishti [0] ad further modified by Mursalee et al. [] i 0. I 03, Bele ad Mohiuddie [] preseted a geeralizatio of this otio through de la Vallee-Poussi mea i probabilistic ormed spaces. Let x be a give ifiite series with sequece of its 0 ( E,) trasform is defied as ad we say that this summability method is coverget if case we say the series Ad we will write S S E, E th partial sum S. If ( ) S (7) 0 E S as. I this x is ( E,) summable to a defiite umber S. (Hardy [3]). 0 as. Let ( p ) ad ( q ) be the two sequeces of o-zero real costats such that

3 Weighted Norlud-Euler A-Statistical Covergece for.. 3 P p p p P p 0, 0 0, 0 Q q q q Q q For the give sequeces ( p ) ad ( q ), covolutio p* q is defied by: *. (8) 0 R p q p q The series x or the sequece S is summable to S by geeralized Norlud 0 method ad it is deoted by S S( N, p, q) if p, q t R pvqvsv (9) v 0 teds to S as. Let us use i cosideratio the followig method of summability: If p, q, E R R v 0 0 v0 (0) t p q E p q ( ) S v p, q, E t S as, the we say that the series x or the sequece S is 0 summable to S by Norlud-Euler method ad it is deoted by S S N, p, q ( E,). Remar. If p, q, the we get Euler summability method. Now we are able to give the defiitio of the weighted statistical covergece related to the ( N, p, q)( E,) summability method. We say that E have weighted desity, deoted by ( E ), if E lim { : } R R E. () A sequece x ( x ) is said to be weighted Norlud-Euler statistical coverget (or S coverget) if for every 0 : R I these case we write lim v v () v0 lim { R : p q ( ) x L } 0 L S st x. I the other had, let us recall that C[ a, b ] is the space of all fuctios f cotiuous o [ ab, ]. We ow that f C[ a, b] is Baach spaces with orm f sup f ( x), f C[ a, b] (3) x[ a, b]

4 4 E. Hoxha, E. Aljimi, V. Lou Suppose that L is a liear operator from C[ a, b ] ito C[ a, b ]. It is clear that if f 0 implies Lf 0, the the liear operator L is positive o C[ a, b ]. We deote the value of Lf at a poit x [ a, b] by L( f ; x ). The classical Korovi approximatio theorem states the followig [4]. Theorem. Let ( T ) be a sequece of positive liear operators from C[ a, b ] ito C[ a, b ]. The, for all C[ a, b ] if oly if i where f x x i ad i 0,,. lim T ( f ; x) f ( x) 0 (4) lim T fi; x fi( x) 0 (5) May mathematicias exteded the Korovi-type approximatio theorems by usig various test fuctios i several setups, icludig Baach spaces, abstract Baach lattices, fuctio spaces, ad Baach algebras. Firstly, Gadjiev ad Orha [5] established classical Korovi theorem through statistical covergece ad display a iterestig example i support of our result. Recetly, Korovi-type theorems have bee obtaied by Mohiuddie [6] for almost covergece. Korovi-type theorems were also obtaied i [7] for -statistical covergece. The authors of [8] established these types of approximatio theorem i weighted L p spaces, where p, through A -summability which is stroger tha ordiary covergece. For these types of approximatio theorems ad related cocepts, oe ca be referred to [9 9] ad refereces therei.. KOROVKIN-TYPE THEOREMS BY WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE Kol [9] itroduced the otio of A -statistical covergece by cosiderig oegative regular matrix A istead of Cesáro matrix i the defiitio of statistical covergece due to Fast. Ispired from the wor of S. A. Mohiuddie, Abdullah Alotaibi, ad Bipa Hazaria [30] we itroduce the otio of weighted Norlud-Euler -statistical covergece of a sequece ad the we establish some Korovi-type theorems by usig this otio. Defiitio 3. Let A ( a, ) be a oegative regular matrix. A sequece x ( x ) of real or complex umbers is said to be weighted Norlud Euler A -statistical covergece, deoted by SA coverget, to L if for every 0

5 Weighted Norlud-Euler A-Statistical Covergece for.. 5 where I symbol, we will write lim a, 0 (6) E( p, ) v v (7) v0 E( p, ) { N : p q ( ) x L } A S lim x L Remar 4. Note that covergece sequece implies weighted Norlud-Euler A - statistical coverget to the same value but coverse is ot true i geeral. For example, tae p, q for all ad defie a sequece x ( x ) by, if x 0, otherwise where N. The this sequece is statistically coverget to 0 but ot coverget; i this case, weighted Norlud-Euler A -statistical covergece of a sequece coicides with statistical covergece. Theorem 5. Let A ( a, ) be a oegative regular matrix. Cosider a sequece of positive liear operators ( M ) from C[ ab, ] ito itself. The, for all f C[ a, b] bouded o whole real lie, if oly if (8) SA lim M f ; x f ( x) 0 (9) SA lim M ; x 0, SA lim M v; x x 0, SA lim M( v ; x) x 0 Proof. Equatio (0) directly follows from (9) because each of (0), xx, belogs to C[ a, b ]. Cosider a fuctio f C[ a, b]. The there is a costat C 0 such that f ( x) C for all x(, ). Therefore, f f( ),, v x C v x, () Let 0 be give. By hypothesis there is a ( ) 0 such that v Solvig () ad () ad the substitutig f f ( x), v x () v C f x, oe obtais Ω( v) ( v x) f ( ) Ω. (3) Equatio (3) ca be also writte by as

6 6 E. Hoxha, E. Aljimi, V. Lou Operatig (; ) v CΩ f f x CΩ. (4) M x to (4) sice M ( f ; x ) his liear ad mooto, oe obtais M (; )( CΩ) (; )( ( ) ( )) (; )( C x M Ω) x f v f x M x Note that x is fixed, so f( x ) is costat umber. Thus, we obtai from (5) that (5) C C i (6) ca also writte as M (; x) M (Ω; x) M ( f ; x) f ( x) M (; x) M (; x) M (Ω; x) (6) The term '' M ( f ; x) f ( x) M (; x)'' M ( f ; x) f ( x) M (; x) M ( f ; x) f ( x) f ( x)[ M (; x) ] (7) M f x f x M x i (6), we get that Now substitutig the value of ( ; ) ( ) (; ) C M ( f ; x) f ( x) M (; x) M (Ω; x) f ( x)[ M (; x) ] (8) We ca rewrite the term '' M (Ω; x )'' i (8) as follows: [ M (ν ; x) x xm ( x x x M x M (Ω; x) M (( x) ; x) M (ν ; x) xm (ν; x) x M (; x) ] ν; ) [ (; ) ] Equatio (8) with the above value of M (Ω; x ) becomes M ( ; ) ( ) (; ) C f x f x M x {[ M (ν ; ) [ (ν; ) ] x x ] x M x x x [ M(; x) ]} f ( x)[ M(; x) ] [ M (; ) ] C x {[ M (ν ; ) ] [ (ν; ) ] x x x M x x x [ M(; x) ]} f ( x)[ M(; x) ] Therefore, M ( ; ) ( ) ( Cb f x f x C) M(; x) C M (ν ; ) x x 4Cb M (ν; ) x x where b max x. Taig supremum over x [ a, b], oe obtais or M ( ; ) ( ) ( Cb f x f x C) M(; x) C M (ν ; ) x x 4Cb M (ν; ) x x M ( f ; x) f ( x) T{ M (; x) M (ν ; x) x M (ν; x) x } (9) (30) (3) (3) (33)

7 Weighted Norlud-Euler A-Statistical Covergece for.. 7 where Hece Cb C C 4 Cb T max{,, }. (34) pq ( ) ( ; ) ( ) { ( ) (; ) v M f x f x T pq v M x 0 0 pq ( ) (ν ; ) v M x x (35) 0 pq ( v ) M (ν; x) x } 0, ad we will defie the followig sets: For give 0, choose 0 It easy to see that such that E { N : pq ( v ) M ( f ; x) f ( x) } 0 E { N : p q ( v ) M (, x) } 3T 0 E { N : p q ( ) (ν; ) } v M x x 3T 0 E3 { N : p q ( ) (ν ; ) } v M x x 3T 0 (36) E EE E3 (37) Thus, for each N, we obtai from (35) that a a a a (38),,,, E E E E 3 Taig limit i (38) ad also (0) gives that lim a, 0. (39) E These yields that for all f C[ a, b]. SA lim M( f ; x) f ( x) 0 (40) We also obtai the followig Korovi-type theorem for weighted Norlud-Euler statistical covergece istead of oegative regular matrix A i Theorem 5. Theorem 6. Cosider a sequece of positive liear operators ( M ) from C[ a, b ] ito itself. The, for all f C[ a, b] bouded o whole real lie, S lim M ( f ; x) f ( x) 0 (4)

8 8 E. Hoxha, E. Aljimi, V. Lou if oly if S lim M (; x) 0 (4) S lim M ( ; x) x 0 (43) S lim M ( ; x) x 0 (44) Proof. Followig the proof of Theorem 5, oe obtais E EE E3 (45) ad so ( E) ( E ) ( E) ( E3) (46) Equatios (4)-(44) give that S lim M ( f ; x) f ( x) 0. (47) Remar 7. By the Theorem of [3], we have that if a sequece x ( x ) is weighted Norlu-Euler statistically coverget to L, the it is strogly ( N, p, q)( E,) summable to L, provided that pq ( v ) x L is bouded; that is, there exist 0 a costat C such that for all N. We write v 0 p q ( ) x L C (,, )(,) { : lim ( N p q E x x p R q ) 0 for some } v xv L L 0 0 (48) the set of all sequeces x ( x ) which are strogly ( N, p, q)( E,) summable to L. Theorem 8. Let M : C[ a, b] C[ a, b] be a sequece of positive liear operators which satisfies (43)-(44) of Theorem 6 ad the followig coditios holds: The, lim M ; x 0. (49) v 0 0 for ay f C[ a, b]. (50) lim p q ( ) M ( f ; x) f ( x) 0, R Proof. It follow from (49) that M ( f ; x) C ' for some costat C ' 0 ad for all N. Hece for f C[ a, b], oe obtais

9 Weighted Norlud-Euler A-Statistical Covergece for.. 9 pq ( ) ( ; ) ( ) ( )( (; ) ) v M f x f x pq v f M x f 0 0 (5) pq ( v ) C( C ' ). 0 Right had side of (5) is costat, so pq ( v ) M ( f ; x) f ( x) 0 is bouded. Sice (49) implies (4), by Theorem 6 we get that S lim M ( f ; x) f ( x) 0. (5) By remar 7, (5) ad (5) together give the desired result. 3. RATE OF WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE First we defie the rate of weighted Norlud-Euler A-statistical coverget sequece as follows. Defiitio 9. Let A ( a, ) be a oegative regular matrix ad let ( a ) be a positive o icreasig sequece. The, a sequece x ( x ) is weighted Norlud-Euler A- statistical coverget to L with the rate of oa ( ) if for each 0 where I symbol, we will write lim, 0 a a (53) E ( p, ) v (54) 0 E p, { N : p q ( ) x L } A We will prove the followig auxiliary result by usig the above defiitio. x L S o( a ) as (55) Lemma 0. Let A ( a, ) be a oegative regular matrix. Suppose that ( a ) ad ( b ) are two positive oicreasig sequeces. Let x ( x ) ad y ( y ) be two sequeces such that The, x L SA o( a ) ad y L SA o( b ). (i) ( x L) ( y L) SA o( c ), (ii) ( x L) ( y L) SA o( ab ),

10 30 E. Hoxha, E. Aljimi, V. Lou (iii) ( x L ) SA o( a ), for ay scala r, where c max { a, b }. Proof. Suppose that Give 0, defie A (, A x L S o a ) y L S o( b ) (56) E ' { N : pq ( ) v ( x L) ( y L) } 0 '' { : E N p q ( ) } v x L (57) 0 ''' { : E N p q ( v ) y L } 0 It easy to see that E ' E '' E ''' (58) These yields that a, a, a c, c c E ' E '' E ''' holds for all N. Sice c max { a, b }, (59) gives that (59) c a,,, a a b a E ' E '' E ''' Taig limit i (60) together with (56), we obtai Thus, (60) ( x L) y L) SA o c Similarly, we ca prove (ii) ad (iii). lim, 0 c a (6) E' ( ( ) (6) Now, we recall the otio of modulus of cotiuity of f i C[ a, b ] is defied by It is well ow that ( f, ) sup{ f ( x) f ( y) : x, y [ a, b], x y } (63) xy. (64) f ( x) f ( y) ( f, )( ) Theorem. Let A ( a, ) be a oegative regular matrix. If the sequece of positive liear operators M : C[ a, b] C[ a, b] satisfies the coditios A (i) M (; x) S o( a ), (ii) A x x ( f, ) S o( b ), with M ( ; x) ad ( y) ( y x),

11 Weighted Norlud-Euler A-Statistical Covergece for.. 3 where ( a ) ad ( b ) are two positive oicreasig sequeces, the for all f C[ a, b], where c max { a, b }. A M ( f ; x) f ( x) S o( c ) (65) Proof. Equatio (7) ca be reformed ito the followig form: M ( f ; x) f ( x) M ( f ( x) f ( y) ; x) f ( x) M (; x) yx M( ; x) ( f, ) f ( x) M(; x) ( yx) M( ; x) ( f, ) f ( x) M(; x) ( M (; x) M ( ; x)) ( f, ) f ( x) M (; x) x M (; x) ( f, ) f ( x) M (; x) ( f, ) M ( ; ) (, ) x x f Choosig M ( ; x), oe obtais x M ( f ; x) f ( x) T M (; x) ( f, ) M (; x) ( f, ) (67) T f. For give 0, we will defie the followig sets: where ' E { N : pq ( v ) M ( f ; x) f ( x) } 0 ' E { N : p q ( v ) M (, x) } 3T 0 ' E3 { N : p q ( ) (, ) } v f 6 0 ' E4 { N : p q ( ) (, ) (; ) }. v f M x 3 0 It follow from (67) that a, a, a c c c, a c, ' ' ' ' E E E3 E4 (66) (68) (69) holds for N. Sice c max{ a, b }, we obtai from (69) that c a, a a, b a, c a,. (70) ' ' ' ' E E E3 E4 Taig limit i (70) together with Lemma 0 ad our hypotheses (i) ad (ii), oe obtais These yields Refereces lim, 0 c a (7) E ' A M ( f ; x) f ( x) S o( c ) (7)

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