Bull. Math. Sci. (205) 5:59 69 DOI 0.007/s3373-05-0065-y Korovki type approximatio theorems for weighted αβ-statistical covergece Vata Karakaya Ali Karaisa Received: 3 October 204 / Revised: 3 December 204 / Accepted: 7 Jauary 205 / Published olie: 2 Jauary 205 The Author(s) 205. This article is published with ope access at SprigerLik.com Abstract The cocept of αβ-statistical covergece was itroduced ad studied by Aktuğlu (Korovki type approximatio theorems proved via αβ-statistical covergece, J Comput Appl Math 259:74 8, 204). I this work, we geeralize the cocept of αβ-statistical covergece ad itroduce the cocept of weighted αβstatistical covergece of order γ, weighted αβ-summability of order γ, ad strogly weighted αβ-summable sequeces of order γ. We also establish some iclusio relatio, ad some related results for these ew summability methods. Furthermore, we prove Korovki type approximatio theorems through weighted αβ-statistical covergece ad apply the classical Berstei operator to costruct a example i support of our result. Keywords Korovki type theorems Weighted αβ-summability Rate of the weighted αβ-statistical coverget Positive liear operator Mathematics Subject Classificatio 4A0 4A25 4A36 40A30 40G5 Commuicated by S. K. Jai. V. Karakaya (B) Departmet of Mathematical Egieerig, Yıldız Techical Uiversity, Davutpasa Campus, 34750 Eseler, İstabul, Turkey e-mail: vkkaya@yahoo.com; vkkaya@yildiz.edu.tr A. Karaisa Departmet of Mathematics-Computer Scieces, Necmetti Erbaka Uiversity, Meram Campus, 42090 Meram, Koya, Turkey e-mail: alikaraisa@hotmail.com; akaraisa@koya.edu.tr 23
60 V. Karakaya, A. Karaisa Itroductio, otatios ad kow results Let K be a subset of N, the set of atural umbers ad K = {k : k K }. The atural desity of K is defied by δ(k ) = lim K provided it exists, where K deotes the cardiality of set K. A sequece x = (x k ) is called statistically coverget (st-coverget) to the umber l, deoted by st lim x = l, for each ɛ>0, the set K ɛ = {k N : x k l ɛ} has atural desity zero, that is lim {k : x k l ɛ} = 0. The cocept of statistical covergece has bee defied by Fast [5] ad studied by may other authors. It is well kow that every statistically coverget sequece is ordiary coverget, but the coverse is ot true. For example, y = (y k ) is defied as follows; {, k = m y = (y k ) = 2, 0, otherwise. It is clear that the sequece y = (y k ) is statistical coverget to zero but ot coverget. The idea αβ-statistical covergece was itroduced by Aktuğlu i [7] as follows: Let α() ad β() be two sequeces positive umber which satisfy the followig coditios (i) α ad β are both o-decreasig, (ii) β() α(), (iii) β() α() as ad let deote the set of pairs (α, β) satisfyig (i) (iii). For each pair (α, β), 0 <γ ad K N, we defie δ α,β (K,γ)i the followig way δ α,β (K,γ)= lim K P α,β (β() α() + ) γ, where P α,β i the closed iterval [α(), β()]. A sequece x = (x k ) is said to be αβ-statistically coverget of order γ to l or S γ αβ-coverget, if δ α,β ({k : x k l ɛ},γ)= lim { k P α,β : x k l ɛ} (β() α() + ) γ = 0. I this paper geeralizig above idea, we defie the weighted αβ-statistical covergece of order γ, the weighted αβ-summability of order γ ad the weighted αβsummability. 23
Korovki type approximatio theorems 6 2 The weighted αβ-summability Let s = (s k ) be a sequece of o-egative real umbers such that s 0 > 0 ad S = s k, as ad z γ (x) = S γ s k x k. Defiitio 2. (a) A sequece x = (x k ) is said to be strogly weighted αβ-summable of order γ to a umber l if lim S γ s k x k l 0 as. We deote it by x k l[n γ αβ, s]. Similarly, for γ = the sequeces x = (x k) is said to be strogly weighted αβ-summable to l. The set of all strogly weighted αβ-summable of order γ ad strogly weighted αβ-summable sequeces will be deoted [N γ αβ, s] ad [N αβ, s], respectively. (b) A sequece x = (x k ) is said to be weighted αβ-summable of order γ to l, if z γ (x) l as. Similarly, for γ = the sequece x = (x k ) is said to be weighted αβ-summable to l, ifz (x) l as.thesetofall weighted αβ-summable of order γ ad weighted αβ-summable sequeces will be deoted (N γ αβ, s) ad (N αβ, s), respectively. This defiitio icludes the followig special cases: (i) If γ =, α() = 0 ad β() =, weighted αβ-summable is reduced to weighted mea summable, ad [N αβ, s] summable sequeces are reduced to (N, p ) summable sequeces itroduced i [8,4]. (ii) Let λ be a oe-decreasig sequece of positive umbers tedig to such that λ λ +, λ =. If we take γ =, α() = λ + ad β() = the weighted αβ-summability is reduced to (N λ ; p)-summability ad [N αβ, s] summable sequeces are reduced to [N λ ; p]-summable sequeces itroduced i [2]. (iii) If we take γ =, α() = λ +, β() = ad s k = for all k the αβsummability is reduced to (V ; λ)-summability itroduced i [0] ad [N αβ, s]- summable sequeces are reduced to [V,λ]-summable sequeces itroduced i [2]. (iv) Recall that a lacuary sequece θ ={k r } is a icreasig iteger sequece such that k 0 = 0 ad h r := k r k r.ifwetakeγ =, α(r) = k r + ad β(r) = k r ; the P α,β (r) =[k r +, k r ]. But because of [k r +, k r ] N = (k r, k r ] N, we have s k x k = s k x k = s k x k. k [α(),β()] k [k r +,k r ] k (k r,k r ] 23
62 V. Karakaya, A. Karaisa This meas that [N αβ, s]-summable sequeces are reduced to weighted lacuary summable, ad [N αβ, s]-summable sequece are reduced to strog weighted lacuary summable sequeces. (v) If we take γ =, α(r) = k r +, β(r) = k r ad s k = for all k the, weighted αβ-summable is reduced to lacuary summable sequeces, ad [N αβ, s]-summable sequeces are reduced to N θ summbale sequeces itroduced i [6]. Defiitio 2.2 A sequece x = (x k ) is said to be weighted αβ-statistically coverget of order γ to l or S γ αβ-coverget, if for every ɛ>0 δ α,β ({k : s k x k l ɛ},γ)= lim S γ {k S : s k x k l ɛ} = 0 ad deote st γ αβ lim x = l or x k l[s γ αβ ], where Sγ αβ weighted αβ-statistically coverget sequeces of order γ. deotes the set of all Theorem 2. Let 0 <γ δ. The, we have [N γ αβ, s] [N δ αβ, s] ad the iclusio is strict for some γ,δ such that γ<δ. Proof Let x = (x k ) [N γ αβ, s] ad γ,δ be give such that 0 <γ δ. The, we obtai that S γ s k x k l S δ s k x k l which gives [N γ αβ, s] [N δ αβ, s]. Now, we show that this iclusio is strict. Let us cosider the sequece t = (t k ) defied by, {, β() β() α() + + k β(), t = (t k ) = 0, otherwise. If we choose s k = for all k N, it is clear that (β() α()+) γ (β() α()+) γ /2 t k 0 β() α() + (β() α() + ) γ = (β() α() + ) γ /2. Sice 0as for /2 <β, we have t = (t k ) [N γ αβ, s]. O the other had, we get β() α() + (β() α() + ) δ (β() α() + ) δ t k 0 ad β() α()+ (β() α()+) δ as for 0 <δ</2 the, we have t = (t k )/ [N δ αβ, s]. This completes the proof. 23
Korovki type approximatio theorems 63 Theorem 2.2 Let (α, β). The, we have followig statemets: (a) If a sequece x = (x k ) is strogly weighted (αβ)-summable of order γ to limit l, the it is weighted αβ-statistically coverget of order γ to l, that is [N γ αβ, s] S γ αβ ad this iclusio is strict. (b) If x = (x k ) bouded ad weighted αβ-statistically coverget of order γ to l the x k [ N γ αβ, s]. Proof (a) Let ɛ>0ad x k l[n γ αβ, s]. The, we get S γ x k l = S γ s k x k l ɛ s k x k l + S γ s k x k l <ɛ ɛ s k x k l s (ɛ). K αβ S γ K αβ s (ɛ) This implies that lim S γ = 0 which meas δ α,β (Ks αβ (ɛ), γ ) = 0, where Ks αβ (ɛ) = {k S : s k x k l ɛ}. Therefore, x = (x k ) is weighted αβ-statistically coverget of order γ to l. To prove [N γ αβ, s] Sγ αβ i (a) is strict, let the sequece x = (x k ) be defied by { k, k [(β() α() + ) x k = γ/2 ], 0, otherwise. (2.) The x is ot bouded ad for every ɛ>0. Let s k = for all k. The we have (β() α() + ) γ {k β() α() + : x k 0 ɛ} = [(β() α() + )γ/2 ] (β() α() + ) γ 0as. That is, x k 0[S γ αβ ].But (β() α() + ) γ x k 0 = [(β() α() + )γ/2 ] ( [(β() α() + ) γ/2 ]+ ) 2(β() α() + ) γ 2, i.e., x k 0[N γ αβ, s]. (b) Assume that x = (x k ) is bouded ad weighted αβ-statistically coverget of order γ to l. The for ɛ > 0, we have δ α,β (Kɛ αβ,γ) = 0. Sice x = (x k ) is bouded, there exists M > 0 such that s k x k l M for all k N. 23
64 V. Karakaya, A. Karaisa S γ s k x k l = S γ M S γ s k x k l ε s k x k l + S γ s k x k l <ε { k P α,β : s k x k l ɛ } + ε, x k l this implies that x k [ N γ αβ, s]. 3 Applicatio to Korovki type approximatio I this sectio, we get a aalogue of classical Korovki Theorem by usig the cocept of αβ-statistical covergece. Also we estimate, i rates of αβ-statistical covergece. Recetly, such types of approximatio theorems are proved, [,3,4,,3,4]. Let C[a, b] be the liear space of all real-valued cotiuous fuctios f o [a, b] ad let L be a liear operator which maps C[a, b] ito itself. We say L is positive operator, if for every o-egative f C[a, b], wehavel( f, x) 0forx [a, b]. It is well-kow that C[a, b] is a Baach space with the orm give by f C[a,b] = sup f (x). x [a,b] The classical Korovki approximatio theorem states as follows (see [7,9]) lim L ( f, x) f (x) C[a,b] = 0 lim L ( f, x) e i ) C[a,b] = 0, where e i = x i ad f C[a, b]. Theorem 3. Let (L k ) be a sequece of positive liear operator from C[a, b] i to C[a, b]. The for all f C[a, b] if ad oly if S γ αβ lim k L k( f, x) f (x)) C[a,b] = 0 (3.) S γ αβ lim k L k(e 0, x) e 0 C[a,b] = 0, (3.2) S γ αβ lim k L k(e, x) e ) C[a,b] = 0, (3.3) S γ αβ lim k L k(e 2, x) e 2 ) C[a,b] = 0. (3.4) Proof Because of e i C[a, b] for (i = 0,, 2), coditios (3.2) (3.4) follow immediately from (3.). Let the coditios (3.2) (3.4) hold ad f C[a, b]. By the cotiuity of f at x, it follows that for give ε>0 there exists δ such that for all t 23 f (x) f (t) <ε, wheever t x <δ. (3.5)
Korovki type approximatio theorems 65 Sice f is bouded, we get Hece By usig (3.5) ad (3.6), we have This implies that f (x) M, < x, t <. f (x) f (t) 2M, < x, t <. (3.6) f (x) f (t) <ε+ 2M (t x)2, t x <δ. ε 2M (t x)2 < f (x) f (t) <ε+ 2M (t x)2. By usig the positivity ad liearity of (L k ), we get ( L k (, x) ε 2M ) (t x)2 < L δ2 k (, x) ( f (x) f (t))< L k (, x) ( ε+ 2M ) (t x)2 δ2 where x is fixed ad so f (x) is costat umber. Therefore, εl k (, x) 2M L k((t x) 2, x) <L k ( f, x) f (x)l k (, x) <ɛl k (, x) + 2M L k((t x) 2, x). (3.7) O the other had L k ( f, x) f (x) = L k ( f, x) f (x)l k (, x) + f (x)l k (, x) f (x) =[L k ( f, x) f (x)l k (, x) f (x)l k ]+ f (x)[l k (, x) ]. (3.8) By iequality (3.7) ad (3.8), we obtai L k ( f, x) f (x) <εl k (, x)+ 2M L k((t x) 2, x) + f (x) + f (x)[l k (, x) ]. (3.9) Now, we compute secod momet L k ((t x) 2, x) = L k (x 2 2xt + t 2, x) = x 2 L k (, x) 2xL k (t, x) + L k (t 2, x) =[L k (t 2, x) x 2 ] 2x[L k (t, x) x]+x 2 [L k (, x) ]. 23
66 V. Karakaya, A. Karaisa By (3.9), we have L k ( f, x) f (x) <εl k (, x) + 2M {[L k(t 2, x) x 2 ] 2x[L k (t, x) x] + x 2 [L k (, x) ]} + f (x)(l k (, x) ) = ε[l (, x) ]+ε + 2M {[L k(t 2, x) x 2 ] 2x[L k (t, x) x] + x 2 [L k (, x) ]} + f (x)(l k (, x) ). Because of ε is arbitrary, we obtai ) L k ( f, x) f (x) C[a,b] (ε + M + 2Mb2 L k (e 0, x) e 0 C[a,b] + 4Mb L k (e, x) e C[a,b] + 2M L k(e 2, x) e 2 C[a,b] R ( L k (e 0, x) e 0 C[a,b] + L k (e, x) e C[a,b] + L k (e 2, x) e 2 C[a,b] ) ( where R = max ε + M + 2Mb2 ), 4Mb. For ε > 0, we ca write C := {k N : L k (e 0, x) e 0 C[a,b] + L k (e, x) e C[a,b] + L k (e 2, x) e 2 C[a,b] ε R } C := {k N : L k (e 0, x) e 0 C[a,b] ε, 3R } C 2 := {k N : L k (e, x) e C[a,b] ε, 3R } C 3 := {k N : L k (e 2, x) e 2 C[a,b] ε. 3R The, C C C 2 C 3,sowehaveδ α,β (C,γ) δ α,β (C,γ) + δ α,β (C 2,γ) + δ α,β (C 3,γ). Thus, by coditios (3.2) (3.4), we obtai S γ αβ lim k L k( f, x) f (x) C[a,b] = 0. }, which completes the proof. We remark that our Theorem 3. is stroger tha that of classical Korovki approximatio theorem. For this claim, we cosider the followig example: Example Cosiderig the sequece of Berstei operators 23 B ( f, x) = ( ) ( ) f x k ( x) k ; x [0, ]. k k k=0
Korovki type approximatio theorems 67 We defie the sequece of liear operators as T : C[0, ] C[0, ] with T ( f, x) = ( + x )B ( f, x), where x = (x ) is defied i (2.). Now let s k = for all k. Note that the sequece x = (x ) is weighted αβ-statistically coverget but ot coverget. The, B (, x) =, B (t, x) = x ad B (t 2, x) = x 2 + x x2 ad sequece (T ) satisfies the coditios (3.2) (3.4). Therefore, we get S γ αβ lim T ( f, x) f (x) C[a,b] = 0. O the other had, we have T ( f, 0) = ( + x ) f (0), sice B ( f, 0) = f (0), thus we obtai T ( f, x) f (x) T ( f, 0) f (0) x f (0). Oe ca see that (T ) does ot satisfy the classical Korovki theorem, sice x = (x ) is ot coverget. 4 Rate of weighted αβ-statistically coverget of order γ I this sectio, we estimate the rate of weighted αβ-statistically coverget of order γ of a sequece of positive liear operator which is defied from C[a, b] ito C[a, b]. Defiitio 4. Let (u ) be a positive o-icreasig sequece. We say that the sequece x = (x k ) is αβ-statistically coverget of order γ to l with the rate o(u ) if for every, ɛ>0 lim u S γ {k S : s k x k l ɛ} = 0. At this stage, we ca write x k l = S γ αβ o(u ). Before proceedig further, let us give basic defiitio ad otatio o the cocept of the modulus of cotiuity. The modulus of cotiuity of f, ω( f,δ)is defied by ω( f,δ)= sup f (x) f (y). x y δ x,y [a,b] It is well-kow that for a fuctio f C[a, b], lim ω( f,δ)= 0 0 + for ay δ>0 ( ) x y f (x) f (y) ω( f,δ) +. (4.) δ 23
68 V. Karakaya, A. Karaisa Theorem 4. Let (L k ) be sequece of positive liear operator from C[a, b] ito C[a, b]. Assume that (i) L k (, x) x C[a,b] = S γ αβ o(u ), (ii) ω(f,ψ k ) = S γ αβ o(v ) where ψ k = L k [(t x) 2, x]. The for all f C[a, b], weget where z = max{u,v }. L k ( f, x) f (x) C[a,b] = S γ αβ o(z ) Proof Let f C[a, b] ad x [a, b]. From(3.8) ad (4.), we ca write L k ( f, x) f (x) L k ( f (t) f (x) ; x) + f (x) L k (, x) ( x y L k + ; x δ L k ( (t x) 2 By choosig ψ k = δ, we get ) ω( f,δ)+ f (x) L k (, x) ) + ; x ω( f,δ)+ f (x) L k (, x) ( L k (, x)+ L ( k (t x) 2 ; x )) ω( f,δ)+ f (x) L k (, x) = L k (, x)ω( f,δ)+ L ( k (t x) 2 ; x ) ω( f,δ)+ f (x) L k (, x). L k ( f, x) f (x) C[a,b] f C[a,b] L k (, x) x C[a,b] + 2ω( f,ψ k ) + ω( f,ψ k ) L k (, x) x C[a,b] H{ L k (, x) x C[a,b] +ω( f,ψ k ) + ω( f,ψ k ) L k (, x) x C[a,b] }, where H = max{2, f C[a,b] }. By Defiitio 4. ad coditios (i) (ii), we get the desired the result. Ackowledgmets valuable commets. We thak the referees for their careful readig of the origial mauscript ad for the Ope Access This article is distributed uder the terms of the Creative Commos Attributio Licese which permits ay use, distributio, ad reproductio i ay medium, provided the origial author(s) ad the source are credited. Refereces. Aktuğlu, H.: Korovki type approximatio theorems proved via αβ-statistical covergece. J. Comput. Appl. Math. 259, 74 8 (204) 2. Bele, C., Mohiuddie, S.A.: Geeralized weighted statistical covergece ad applicatio. Appl. Math. Comput. 29(8), 982 9826 (203) 23
Korovki type approximatio theorems 69 3. Edely, O.H.H., Mohiuddie, S.A., Noma, A.K.: Korovki type approximatio theorems obtaied through geeralized statistical covergece. Appl. Math. Lett. 23(), 382 387 (200) 4. Edely, O.H.H., Mursalee, M., Kha, A.: Approximatio for periodic fuctios via weighted statistical covergece. Appl. Math. Comput. 29(5), 823 8236 (203) 5. Fast, H.: Sur la covergece statistique. Colloq. Math. Studia Math. 2, 24 244 (95) 6. Fridy, J.A., Orha, C.: Lacuary statistical covergece. Pac. J. Math. 60(), 43 5 (993) 7. Gadziev, A.D.: The covergece problems for a sequece of positive liear operators o ubouded sets, ad theorems aalogous to that of P.P. Korovki. Sov. Math. Dokl. 5, 433 436 (974) 8. Karakaya, V., Chishti, T.A.: Weighted statistical covergece. Ira. J. Sci. Techol. Tras. A Sci. 33, 29 223 (2009) 9. Korovki, P.P.: Liear Operators ad Approximatio Theory. Hidusta Publishig, New Delhi, Idia (960) 0. Leidler, L.: ber die de la Vallée Poussische summierbarkeit allgemeier orthogoalreihe. Acta Math. Acad. Sci. Hug. 6, 375 387 (965). Mohiuddie, S.A.: A applicatio of almost covergece i approximatio theorems. Appl. Math. Lett. 24(), 856 860 (20) 2. Mursalee, M.: λ-statistical covergece. Math. Slovaca 50, 5 (2000) 3. Mursalee, M., Alotaibi, A.: Statistical summability ad approximatio by de la Valle-Poussi mea. Appl. Math. Lett. 24(3), 320 324 (20) 4. Mursalee, M., Karakaya, V., Ertürk, M., Gürsoy, F.: Weighted statistical covergece ad its applicatio to Korovki type approximatio theorem. Appl. Math. Comput. 28(8), 932 937 (202) 23