Stud. Uiv. Babeş-Bolyai Math. 5924, No. 3, 33 35 Iverse theorem for the iterates of modified Berstei type polyomials T.A.K. Siha, P.N. Agrawal ad K.K. Sigh Abstract. Gupta ad Maheshwari [2] itroduced a ew sequece of Durrmeyer type liear positive operators P to approimate p th Lebesgue itegrable fuctios o [, ]. It is observed that these operators are saturated with O. I order to improve this slow rate of covergece, followig Agrawal et al [2], we [3] applied the techique of a iterative combiatio to the above operators P ad estimated the error i the L p approimatio i terms of the higher order itegral modulus of smoothess usig some properties of the Steklov mea. The preset paper is i cotiuatio of this work. Here we have discussed the correspodig iverse result for the above iterative combiatio T,k of the operators P. Mathematics Subject Classificatio 2: 4A25, 4A27, 4A36. Keywords: Liear positive operators, iterative combiatio, itegral modulus of smoothess, Steklov mea, iverse theorem.. Itroductio Motivated by the defiitio of Phillips operators cf. [] ad [5], Gupta ad Maheshwari [2] proposed modified Berstei type polyomials P to approimate fuctios i L p [, ] as follows: For f L p [, ], p <, where W, t = P f; = W, tft dt, [, ], p,ν p,ν t + δt, p,ν t = t ν t ν, t, ν
332 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh ad δt beig the Dirac-delta fuctio, is the kerel of the operators P. Sice the order of approimatio by the operators P is, at best, O, however smooth the fuctio may be, followig [3], the iterative combiatio T,k : L p [, ] C [, ] of these operators is defied as T,k f; = I I P k k k f; = m+ P m f;, k N, m where P I ad P m P P m for m N. I order to improve the rate of covergece, Micchelli [6] itroduced a iterative combiatio for Berstei polyomials ad obtaied some direct ad saturatio results. Goska ad Zhou [] showed that the iterative combiatios ca be regarded as iterated Boolea sums ad obtaied global direct ad iverse results i the suporm. The iterated Boolea sums have also bee studied by several other authors e.g. [4],[8],[7],[8] ad [2] wherei they have obtaied direct ad saturatio results. Dig ad Cao [7] discussed direct ad iverse theorems i the sup- orm for iterated Boolea sums of the multivariate Berstei polyomials usig the techique of K-fuctioals. Siha et al [9] proved a iverse theorem i the L p orm for the Micchelli combiatio of Berstei-Durrmeyer polyomials. Goska ad Zhou [] obtaied the results i the sup- orm usig the Ditzia Totik modulus of smoothess ad K fuctioal. Dig ad Cao [7] also obtaied the results i sup- orm usig K fuctioal. Sevy [7] ad [8] cosidered the limits of the liear combiatios of iterates of Berstei ad Durrmeyer polyomials i the sup- orm by keepig the degree of the approimats as a costat while the order of iteratio becomes ifiite ad showed that they coverge to the Lagrage iterpolatio polyomial ad the least square approimatig polyomial o [, ] respectively. The more geeral results have bee obtaied i [2]. Motivated by the work of Siha et al [9], Agrawal et al [3] cosidered the Micchelli combiatios for the operator proposed by Gupta ad Maheshwari [2] ad obtaied some direct results i L p orm. I the preset paper, we cotiue the work doe i [9] by provig a correspodig local iverse theorem i the L p orm. The iterates are defied as P m+ f; = m= W, tp m f; t dt, [, ]. At every stage it uses the etire previous operator value. The aalysis i L p case, therefore, differs from the study of operators i [] ad liear combiatios of operators i [8]. The proof of the theorem is carried out by usig the properties of Steklov meas. Due to the presece of the Dirac- delta term i the kerel of these operators, the aalysis of the proof is quite differet. It uses the multiomial theorem, Hölder s iequality ad the Fubii s theorem repeatedly. Throughout the preset paper, we assume that I = [, ], I j = [a j, b j ], j =, 2, where < a < a 2 < b 2 < b < ad by C we mea a positive costat ot ecessarily the same at each occurrece. I [3], we obtaied the followig direct theorem:
Iterates of modified Berstei type polyomials 333 Theorem.. Let f L p I, p. The, for sufficietly large values of there holds T,k f; f LpI 2 C ω 2k f,, p, I + k f LpI, where C is a costat idepedet of f ad. Remark.2. From the above theorem, it follows that if ω 2k f, τ, p, I = Oτ α, as τ the T,k f; f LpI 2 = O α/2, as, where < α < 2k. The aim of this paper is to characterize the class of fuctios for which T,k f; f LpI 2 = O α/2, as, where < α < 2k. Thus, we prove the followig theorem iverse theorem: Theorem.3. Let f L p I, p. Let < α < 2k ad T,k f; f LpI = O α/2, as. The, ω 2k f, τ, p, I 2 = Oτ α, as τ. Remark.4. We observe that without ay loss of geerality we may assume that f =. To prove it, let f t = ft f. By defiitio, k k T,k f ; = m+ P m f ;. m Further, usig liearity, m= P m f ; = P m f; fp m ; = P m f; f. This implies that T,k f ; = T,k f; f. This etails that where f =. T,k f ; f = T,k f; f, Sice f = i view of the above remark, it follows that P f =. Cosequetly, P m f =, m N. 2. Prelimiaries I this sectio, we metio some defiitios ad prove auiliary results which we eed i establishig our mai theorem. Lemma 2.. Let r > ad ν be a iteger such that ν. The for every ν there holds p,ν t ν t r dt = O, as. r 2 +
334 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh Proof. Let i be a iteger such that 2i > r. A applicatio of Hölder s iequality i itegral gives p,ν t ν t r dt It follows that t j p,ν t dt = Hece, by biomial epasio Now, p,ν t ν t 2i dt r p,ν t ν 2i t 2i dt p,ν t dt r 2i. 2. ν + ν + 2...ν + j Bν + j +, ν + = ν + + 2... + j +. 2i 2i = j ν 2i j ν + ν + 2...ν + j j + + 2... + j + j= { 2i = + 2i ν 2i ν 2i ν + + 2i 2 ν 2i 2 ν + ν + 2 +... + ν + ν + 2...ν + 2i s=2 + 2 2.2 + 2 + 3 2i+ s=2 + s. j+ + s p j = + + p 2j 2 + p 3j 3 +..., 2.3 where p j is a secod degree polyomial i j, p 2 j is a fourth degree polyomial i j ad so o. Similarly, ν + ν + 2...ν + j = ν j + q jν j + q 2 jν j 2 +... + j!, 2.4 where q j is a secod degree polyomial i j, q 2 j is a fourth degree polyomial i j ad so o. Thus from 2.2-2.4, we have
Iterates of modified Berstei type polyomials 335 p,ν t ν t 2i dt = { 2i + 2i j= + p j + p 2j = + 2i + p j = O i+ { 2i j= 2i j ν 2i j ν j + q jν j + q 2 jν j 2 +... j 2 +... 2i j j ν 2i + q jν 2i + q 2 jν 2i 2 +... + p 2j 2 +..., as. 2.5 This holds for every ν, where ν ad i view of the followig idetity: 2i j= { 2i j j m, m =,,..., 2i = j 2i!, m = 2i. Now, o combiig 2., 2.5 ad i view of p,ν t ν t r dt C i+ For m N, the m th order momet for P is defied as p,ν tdt = +, we obtai r 2i r 2i = O + r 2 +. µ,m = P t m ;. Lemma 2.2. [2]The elemetary momets are µ, =, µ, = m there holds the recurrece relatio + ad for +m+µ,m+ = { µ,m + 2m µ,m +m 2 µ,m. Cosequetly, i µ,m is a polyomial i of degree m; ii µ,m = O [m+/2], as, uiformly i I, where [β] is the iteger part of β. Corollary 2.3. There holds for r > P t r ; = O r/2, as, uiformly i I.
336 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh Proof. Let s be a eve iteger > r. A applicatio of Hölder s iequality i itegral ad Lemma 2.2 i the et step gives P t r ; = s W, t t s dt r Lemma 2.4. [3]There holds for l N l l D l p,ν = W, t t r dt W, t dt 2i+jl i,j r s C s/2 r/s = C r/2. i ν j q i,j,l p,ν, where D d d ad q i,j,l are certai polyomials i idepedet of ad ν. Lemma 2.5. [3] There holds for k, l N T,k t l ; = O k, as, uiformly i I. Lemma 2.6. Let r > ad V, t =: uiformly for all t i [, ]. Proof. Let J =: p,ν p,ν t, the V, t t r d = O r/2, as, V, t t r d ad s be a eve iteger > r. The, proceedig alog the lies of the proof of the Corollary 2.3 ad i view of we have We may write J V, t t s d = s. p,ν d = + V, t t s d s i= r s + s t s i i i r s. p,ν t p,ν i d.
Iterates of modified Berstei type polyomials 337 Sice it follows that V, t t s d = s. s i= p,ν i d = ν +...ν + i +... + i +, s t s i i ν + 2...ν + i + p,ν t i +... + i +. 2.6 ν= Now, ν + 2...ν + i + = ν i + p iν i + p 2 iν i 2 +..., where p j i is a polyomial i i of degree 2j. Moreover, ν i = q iν i + q iν i + q 2 iν i 2 +... + q i iν, 2.7 where q i = q i i =, ν j = νν ν 2...ν j +, j =,, 2..., i ad q j i is a polyomial i i of degree 2j. Utilizig 2.7 i 2.6 ad usig the properties of biomial coefficiets, we get the required order. Defiitio 2.7. Let f L p I, p. The for sufficietly small η >, the Steklov mea f η,m of m th order is defied as follows: f η,m t = η m η/2 η/2 η/2 η/2 ft + m m mi= tift dt...dt m, where t I ad m h is mth order forward differece operator of step legth h. Lemma 2.8. The fuctio f η,m satisfies the followig properties a f η,m has derivatives up to order m over I, f η,m m a.e. ad belogs to L p I ; b f η,m r LpI 2 C r η r ω r f, η, p, I, r =, 2,..., m; c f f η,m LpI 2 C m+ ω m f, η, p, I ; d f η,m LpI 2 C m+2 f LpI ; e f η,m m LpI 2 C m+3 η m f LpI, ACI ad f m η,m where C i s are certai costats that deped o i but are idepedet of f ad η. The proof follows from Theorem 8.7 [3] ad [2], Eercise 3.2, pp.65-66. Lemma 2.9. Let f L p I, p ad r, m N. The there holds Proof. Usig Remark.4 P m ftt r ; = P m ftt r ; LpI C r/2 f LpI.... eists V, t V t, t 2...V t m, t m t m r ft m dt m...dt.
338 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh A repeated use of Hölder s iequality ad i view of P m ftt r ; p... V, tdt = O makes V, t V t, t 2...V t m, t m t m rp ft m p dt m...dt. We ow cosider itegratio o both sides. O the right side by virtue of Fubii s theorem, the itegratio is doe with respect to followed by t, t 2,..., t m respectively. Thus P m ftt r ; p d... V, t t m rp d 2.8 V t, t 2...V t m, t m ft m p dt...dt m. Let s > rp be a iteger. The, usig Hölder s iequality ad i view of we have V, t t m rp d By multiomial epasio p,ν d = + rp s rp V, t t m s d s. 2.9 + t m s t m t m + t m t m 2 +... + t s s t m t m rm... t r. 2. r, r 2,.., r m r +r 2 +...+rm=s, r k, km Now, we combie 2.8-2., resort Lemma 2.6 m times ad Hölder s iequality m times to reach P m ftt r ; p d C r +r 2 +...+rm=s, r k, km C rp 2 m rp f p L pi, s r, r 2,.., r m r +r 2 +...+rm 2 rp s ft m p dt m usig boud of multiomial coefficiets. Takig p th root o both sides we complete the proof of lemma.
Iterates of modified Berstei type polyomials 339 Lemma 2.. Let m, s N ad f L p I, p have a compact support i [a, b],. The there holds d 2s d 2s P m f; C s f Lp[a,b]. Lp[a,b] Proof. A applicatio of Lemma 2.4 eables us to epress d 2s d 2s P m f; = d2s d 2s = W, vp m f; v dv p,ν 2i+j2s i,j p,ν vp m f; v dv. i ν j q i,j,s 2s 2. Whe p >, applyig Hölder s iequality twice, first for summatio ad the for itegratio, we obtai p ν j p,ν p,ν vp m f; v dv ν jp p,ν p,ν v P m f; v p dv. 2.2 The above iequality is true for p =, as well. Now, we itegrate both sides of 2.2 with respect to ad take help of Lemma 2. i et step to obtai b p ν j p,ν p,ν vp m f; v dv d a b p,ν ν jp d p,ν v P m f; v p dv Let C 2 =: a C jp/2. p,ν v P m f; v p dv C jp/2 P m f;. p L pi. 2.3 sup [a,b] sup 2i+j2s i,j q i,j,s 2s.
34 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh We ow combie 2. ad 2.3 ad coclude that d 2s d 2s P m f; Lp[a,b] C /p C 2 2i+j2s i,j i j/2 P m f;. LpI C s f LpI = C s f Lp[a,b]. Hece, the required result follows. Lemma 2.. Let m, s N ad f L p [, ], p have a compact support i [a, b],. Moreover, let f 2s AC[a, b] ad f 2s L p [a, b], the d2s d 2s P m f; C f 2s Lp[a,b]. Lp[a,b] Proof. Sice P m 2.4 we have maps polyomials ito polyomials of the same degree, usig Lemma d 2s d 2s P m f; = = 2s! d 2s... d 2s W, t W t, t 2... t m W t m, t m t m w 2s f 2s w dw dt m...dt 2s! p,ν 2i+j2s i,j... i ν j q i,j,s 2s p,ν t W t, t 2...W t m, t m t m t m w 2s f 2s w dw dt m...dt. Let us defie W, t =, t / [, ]. Now, we break the iterval of itegratio i t m i the followig way: There eists for each a iteger r such that r ma{ a, b r +.
Let C = sup [a,b] Iterates of modified Berstei type polyomials 34 sup 2i+j2s i,j d 2s d 2s P m f; C... q i,j,s. The 2s 2i+j2s i,j { i ν j p,ν 2.4 p,ν t W t, t 2...W t m 2, t m t m W t m, t m t m 2s f 2s w dw dt m dt m...dt. The epressio iside the curly bracket i 2.4, however is bouded by r { l= + l+ + l W t m, t m t m 2s + l+ f 2s w dw dt m + l l+ W t m, t m t m r { 2 l= + 2 l 4 + l 4 l + l+ 2s f l+ + l W t m, t m t m 2s+3 l+ W t m, t m t m + W t m, t m t m 2s Usig 2.5 i 2.4 d 2s d 2s P m f; C... 2s+3 2i+j2s i,j 2s w dw dt m + l+ f l+ + f 2s w dw dt m 2.5 2s w dw dt m f 2s w dw dt m. i ν j p,ν p,ν t W t, t 2...W t m 2, t m
342 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh { r 2 l= + 2 l 4 l 4 + l+ + l W t m, t m t m 2s+3 l l+ W t m, t m t m 2s+3 + l+ f l+ f 2s w dw dt m 2s w dw dt m + + W t m, t m t m 2s + f 2s w dw dt m dt m...dt, = J + J 2 + J 3, say. I order to estimate J, J 2 ad J 3, we use multiomial epasio 2s + 3 t m 2s+3 r, r 2,.., r m Thus J C 2i+j2s i,j i r +r 2 +...+rm=2s+3, r i, im r l= 2 l 4 2s + 3 r +r 2 +...+rm=2s+3, r k, km t m t m rm t m t m 2 rm... t r. + l+ r, r 2,.., r m f 2s w dw ν j p,ν... p,ν t W t, t 2...W t m, t m t m t m rm t m t m 2 rm... t r dt m dt m...dt. A repeated applicatio of Corollary 2.3 makes J C + r i 2 l+ l 4 f 2s w dw 2.6 2i+j2s i,j l= { ν j p,ν r +r 2 +...+rm=2s+3, r k, km p,ν t t r dt. 2s + 3 r, r 2,.., r m rm+...+r2/2
Iterates of modified Berstei type polyomials 343 I order to obtai a boud for J i 2.6 we require a estimate of p,ν t t r dt. This is accomplished with the help of Lemma 2. ad momets of Berstei polyomials [4], Theorem.5.. p,ν t t r dt p,ν t t ν + ν r dt = Therefore, J C r r i = 2i+j2s i,j i ν i C r l= r r i = 2 l 4 i { r ν j p,ν Cm 2s+3 s 2s+3/2 i = r l= i p,ν t t ν r i/2 ν i. + l+ 2 l 4 r i f 2s w dw r +r 2 +...+rm=2s+3, r k, km i i/2 ν + l+ r i dt 2s + 3 r, r 2,.., r m 2s+3/2 f 2s w dw. 2.7 We ow take p orm i i above. Let p, q be the cojugate epoets such that p + q = ad ψ l,. deote the characteristic fuctio of the iterval [, + l+ ]. By usig Hölder s iequality ad Fubii s theorem
344 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh b a + l+ f 2s w dw p d = = = p/q b l + a p/q b l + l + p/q + l+ a b p/q l + l + l + p f 2s p L p[,]. a f 2s w p dw d ψ l, w f 2s w p dw d ψ l, w d f 2s w p dw f 2s w p dw Hece, + l+ f 2s w dw Lp[a,b] l + f 2s LpI. This implies by 2.7, that J Lp[a,b] Cm 2s+3 f 2s LpI = Cm 2s+3 f 2s Lp[a,b]. I order to fid estimates J 2 ad J 3 we proceed i a similar maer ad obtai the required order. Combiig the estimates of J, J 2 ad J 3, we complete the proof. 3. Proof of Iverse Theorem Proof. We choose umbers i ad y i, i =, 2, 3 that satisfy < a < < 2 < 3 < a 2 < b 2 < y 3 < y 2 < y < b <. We choose a fuctio g C 2k such that supp g 2, y 2 with g = o [ 3, y 3 ] ad f = fg. Now, for all values of γ τ we have 2k γ f Lp[ 2,y 2] 2k γ f T,k f; + Lp[ 2k 2,y 2] γ T,k f; Lp[ 2,y 2] = Σ + Σ 2, say. 3. Let f η,2k deote the Steklov mea for the fuctio f. The, Lemmas 2. ad 2. etail
Iterates of modified Berstei type polyomials 345 Σ 2 = 2k γ T,k f; Lp[ { 2,y 2] T γ 2k 2k,k f f η,2k ; + T 2k,k f η,2k ; Lp[,y ] Lp[,y ] { Cγ 2k k f f 2k η,2k Lp[ 2,y 2] + f η,2k L p[ 2,y 2] Cγ 2k k + η 2k ω 2k f, η, p, [ 2, y 2 ], 3.2 for sufficietly small values of γ ad η. It follows from the hypothesis that a compoet of Σ is bouded as f T,k f; Lp[ 2,y 2] gf T,k f; + T Lp[ 2,y 2],kftgt g; Lp[ 2,y 2] C + T,kftgt g;. 3.3 α/2 Lp[ 2,y 2] We ow establish that T,k ftgt g; Lp[ 2,y 2] = O α/2. 3.4 This is a major poit i the proof of our theorem. Assumig 3.4to be true, it follows from 3.-3.4 that 2k f { γ C Lp[ + γ2k k + 2,y 2] α/2 η 2k ω 2k f, η, p, [ 2, y 2 ]. 3.5 We choose η = /2 ad take sup γτ i 3.5 to obtai { 2k ω 2k f, τ τ, p, [ 2, y 2 ] C η α + ω 2k η f, η, p, [ 2, y 2 ]. Now, makig use of the Lemma [6], p.696, we get ad therefore ω 2k f, τ, p, [ 2, y 2 ] C τ α ω 2k f, τ, p, I 2 = Oτ α, as τ. The proof of 3.4 is accomplished by iductio o α. Whe α, by mea value theorem ad Lemma 2.9 T,k ftgt g; = T Lp[ 2,y 2],kftt g ξ; Lp[ 2,y 2] C f /2 L pi, where ξ lies betwee t ad. This proves 3.4 whe α. We et assume that 3.4 is true whe α lies i [r, r ad prove that it is true for α [r, r +. Let f η,2k t deote the Steklov mea. We epress
346 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh ftgt g = {ft f η,2k t + f η,2k t f η,2k + f η,2k gt g = ft f η,2k tgt g + f η,2k t f η,2k gt g + f η,2k gt g = Σ 3 + Σ 4 + Σ 5, say. 3.6 Let ψt deote the characteristic fuctio of [ 2 δ, y 2 + δ ]. This etails that P m ft f η,2k tgt g; Lp[ 2,y 2] Therefore, P m ft f η,2k tt g ξψt; Lp[ 2,y 2] + P m ft f η,2k tt g ξ ψt; Lp[ 2,y 2] g CI P m ft f η,2k t t ψt; Lp[ 2,y 2] + g CI δ 2k P m ft f η,2k t t 2k ψt; Lp[ 2,y 2] C /2 f f η,2k ψt Lp[ 2,y 2] + C k f f η,2k LpI C /2 ω 2k f, η, f, p, [, y ] + C k f LpI i view of Lemmas 2.9 ad 2.8. 3.7 T,k Σ 3 ; = k m= k m+ P m Σ 3 ; m C /2 ω 2k f, η, f, p, [, y ] + C k f LpI. 3.8 To obtai a estimate of Σ 5, we ote that gt is a very smooth fuctio ad hece T,k gt g; = where ξ lies betwee t ad. 2k j= g j T,k t j ; j! + 2k! T,kg 2k ξt 2k ;, 3.9 Now, applyig Lemmas 2.5 ad 2.9 o the right had side of 3.9 respectively, we have T,k Σ 5 ; Lp[ 2,y 2] C k f η,2k Lp[ 2,y 2] C k f LpI. 3.
Iterates of modified Berstei type polyomials 347 Sice, by virtue of Lemma 2.8, f η,2k is 2k times differetiable, there follows 2k t i Σ 4 = f i η,2k i! + t t w 2k f 2k η,2k 2k! wdw i= 2k 2 t j g j t 2k + g 2k ξ j! 2k! = j= 2k i= 2k 2 j= + g2k ξ 2k! + 2k! { 2k 2 j= t i+j i!j! 2k i= t f i η,2k gj t 2k+i i! f i η,2k t w 2k f 2k η,2k wdw t j g j + j! t 2k g 2k ξ 2k! = Σ 6 + Σ 7 + Σ 8, say, 3. where ξ lies betwee t ad. By Lemma 2.5 ad Theorem 3. [9], p.5 Similarly, T,k Σ 7 ; Lp[ 2,y 2] 2k T,k Σ 6 ; Lp[ 2,y 2] C k i= f i η,2k L p[ 2,y 2] C k f η,2k Lp[ 2,y 2] + f 2k η,2k Lp[ 2,y 2]. 3.2 C k f η,2k Lp[ 2,y 2] + f 2k η,2k Lp[ 2,y 2] We ow eamie a typical term of T,k Σ 8 ; epressed as t P m t i t w 2k f 2k η,2k wdw; = P m + P m t t i t w 2k f 2k η,2k wψwdw; t i t t w 2k f 2k η,2k w ψwdw;. 3.3 = Σ 9 + Σ, say. 3.4
348 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh We may write Σ 9 t m... f 2k W, t W t, t 2...W t m, t m t m 2k+i 3.5 η,2k w ψwdw dt mdt m...dt 2 dt. Now, proceedig alog the lies of the proof of Lemma 2., we obtai C 2k Σ 9 Lp[ 2,y 2] f 2k+i/2 η,2k L p[ 2 δ,y 2 δ]. 3.6 The presece of ψw i Σ implies that w > δ. Therefore Σ... t m 2k+i +2k W, t W t, t 2...W t m, t m 3.7 δ 2k t m f 2k η,2k w ψwdw dt mdt m...dt 2 dt. Proceedig alog the lies of the proof of Lemma 2. agai yields C 2k Σ Lp[ 2,y 2] f 4k+i/2 η,2k L pi. 3.8 Combiig 3.4, 3.6 ad 3.8, we get T,k Σ 8 ; Lp[ { 2,y 2] 2k C f 2k+i/2 η,2k 2k L p[ 2 δ,y 2+δ ] + f 4k+i/2 η,2k L pi. 3.9 Utilizig 3.6-3.9, we are led to T,k ftgt g; Lp[ { 2,y 2] C ω 2kf, η, p, [ /2, y ] + k f L pi + k + f 2k η,2k Lp[ 2,y 2] + η,2k L p[ 2 δ,y 2 δ ] + 2k f 2k+/2 f η,2k Lp[ 2,y 2] 2k f 4k+/2 η,2k L pi This is further simplified by Lemma 2.8 by takig η = /2 for large values of as T,k ftgt g; Lp[ { 2,y 2] C ω 2kf, /2, p, [ /2, y ] + k f L pi + ω 2k f, /2, p, [ /2, y ]. 3.2 The iductio hypothesis implies that for [c, d] a, b ω 2k f, /2, p, [c, d] = O α/2,. 3.2.
Iterates of modified Berstei type polyomials 349 This iduces, by Theorem 6..2, [2], ω 2k f, /2, p, [c, d] = O α/2,. 3.22 Icorporatig 3.2 ad 3.22 i 3.2, we obtai T,k ftgt g; Lp[ 2,y 2] = O α+/2,. This proves 3.4 ad hece the proof of the theorem follows. Ackowledgemets. The last author is thakful to the Coucill of Scietific ad Idustrial research, New Delhi, Idia for providig fiacial support to carry out the above work. The authors are etremely thakful to the referee for a very careful readig of the mauscript ad makig valuable commets. Refereces [] Agrawal, P.N., Gupta, V., O the iterative combiatio of Phillips operators, Bull. Ist. Math. Acad. Siica, 899, o. 4, 36-368. [2] Agrawal, P.N., Gupta V., Gairola, A.R., O iterative combiatio of modified Bersteitype polyomials, Georgia Math. J., 528, o. 4, 59-6. [3] Agrawal, P.N., Sigh, K.K., Gairola, A.R., L p-approimatio by iterates of Berstei- Durrmeyer type polyomials, It. J. Math. Aal., Ruse, 42, o. 9-2, 469-479. [4] Agrawal, P.N., Kasaa, H.S., O the iterative combiatios of Berstei polyomials, Demostratio Math., 7984, o. 3, 777-783. [5] Becker, M., Nessel, R.J., A elemetary approach to iverse approimatio theorems, J. Appro. Theory, 23978, o. 2, 99-3. [6] Beres, H., Loretz, G.G., Iverse theorems for Berstei polyomials, Idiaa Uiv. Math. J., 297/72, 693 78. [7] Dig, C., Cao, F., K-fuctioals ad multivariate Berstei polyomials, J. Appro. Theory, 5528, o. 2, 25-35. [8] Gawroski, W., Stadtmüller, U., Liear combiatios of iterated geeralized Berstei fuctios with a applicatio to desity estimatio, Acta Sci. Math., Szeged, 47984, o. -2, 25-22. [9] Goldberg, S., Meir, A., Miimum moduli of ordiary differetial operators, Proc. Lodo Math. Soc., 397, o. 23, - 5. [] Goska, H.H., Zhou, X.L., A global iverse theorem o simultaeous approimatio by Berstei-Durrmeyer operators, J. Appro. Theory, 6799, o. 3, 284-32. [] Goska, H.H., Zhou, X.L., Approimatio theorems for the iterated Boolea sums of Berstei operators, J. Comput. Appl. Math., 53994, o., 2-3. [2] Gupta, V., Maheshwari, P., Bezier variat of a ew Durrmeyer type operators, Riv. Mat. Uiv. Parma, 7, 223, 9-2. [3] Hewitt, E., Stromberg, K., Real ad Abstract Aalysis, Narosa Publishig House, New Delhi, Idia, 978. [4] Loretz, G.G., Berstei Polyomials, Uiversity of Toroto Press, Toroto, 953. [5] May, C.P., O Phillips operator, J. Approimatio Theory, 2977, o. 4, 35-332. [6] Micchelli, C.A., The saturatio class ad iterates of the Berstei polyomials, J. Approimatio Theory, 8973, -8.
35 T.A.K. Siha, P.N. Agrawal ad K.K. Sigh [7] Sevy, J.C., Covergece of iterated Boolea sums of simultaeous approimats, Calcolo 3993, o., 4-68. [8] Sevy, J.C., Lagrage ad least-squares polyomials as limits of liear combiatios of iterates of Berstei ad Durrmeyer polyomials, J. Appro. Theory, 8995, o. 2, 267-27. [9] Siha, T.A.K. et al., Iverse theorem for a iterative combiatio of Berstei- Durrmeyer polyomials, Stud. Uiv. Babeş-Bolyai Math., 5429, o. 4, 53-65. [2] Tima, A.F., Theory of Approimatio of Fuctios of a Real Variable Eglish Traslatio, Dover Publicatios Ic., New York, 994. [2] Wez, H.J., O the limits of liear combiatios of iterates of liear operators, J. Appro. Theory, 89997, o. 2, 29 237. T.A.K. Siha S. M. D. College, Departmet of Mathematics Poopoo, Pata-8323 Bihar, Idia e-mail: thakurashok22@gmail.com P.N. Agrawal Idia Istitute of Techology Roorkee Departmet of Mathematics Roorkee-247667 Uttarakhad, Idia e-mail: pa iitr@yahoo.co.i K.K. Sigh I.C.F.A.I. Uiversity, Faculty of Sciece & Techology Dehradu-24897 Uttarakhad, Idia e-mail: kksiitr.sigh@gmail.com