New estimates in Voronovskaja s theorem. Gancho Tachev. Numerical Algorithms ISSN Numer Algor DOI / s

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1 New estimates i Voroovskaja s theorem Gacho Tachev Numerical Algorithms ISSN DOI.7/ s

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3 Author's persoal copy DOI.7/s ORIGINAL PAPER New estimates i Voroovskaja s theorem Gacho Tachev Received: March 2 / Accepted: 27 May 2 Spriger Sciece+Busiess Media, LLC 2 Abstract I the preset article we establish poitwise variat of E. V. Voroovskaja s 932 result, cocerig the degree of approximatio of Berstei operator, applied to fuctios f C 3,. Keywords Berstei polyomials Voroovskaja-type theorem Moduli of cotiuity Degree of approximatio Itroductio For every fuctio f C, the Berstei polyomial operator is give by B f ; x) = f k ) ) x k x) k, x, k k= For this operator the theorem of Voroovskaja was first proved i 2 adis give i the book of DeVore ad Loretz 2 as follows: Theorem A If f is bouded o,, dif feretiable i some eighborhood of x ad has secod derivative f x) for some x,,the lim B x x) f, x) f x) = f x). 2 If f C 2,, the covergece is uiform. G. Tachev B) Uiversity of Architecture, Hr.Smireski Blvd., Sofia, 46, Bulgaria gtt_fte@uacg.bg

4 Author's persoal copy This result has attracted the attetio of may authors i the last 8 years. Ispired by the result of Voroovskaja, her scietific advisor-s.berstei geeralized Theorem A, showig i the asymptotic expasio of Berstei operator for f C q, for q eve as follows: Theorem B If q Niseve, f C q,, the uiformly i x,, q q 2 B f, x) f x) B e x) r f r) x), x),. r! r= I 5 Mamedov cosidered also the case f C q,, q eve, amely: Theorem C Let q Nbeeve, f C q,, adl : C, C, be a sequece of positive liear operators such that L e, x) =, x, ; L e x) q+2 j, x) lim = L e x) q, x) for at least oe j {, 2,...}. The L e x) q, x) q L f, x) f x) L e x) r, x) r= f r) x),. r! A complete asymptotic expasio i quatitative form was already give some 3 years ago by Sikkema ad va der Meer i 7: Theorem D Let WC q, deote the set of all fuctios o, whose q th derivative is piecewise cotiuous, q. Moreover,letL ) be a sequece of positive liear operators L : WC q, C, satisfyig L e, x) =. The for all f WC q,, q N, x,, Nadδ>oe has q L f, x) f x) L e x) r f r) x), x) r! c,qx,δ) ω f q),δ). r= Here c,q x,δ)= δ q L s q,μ e x ), x), δ μ = 2, if L e x) q, x), μ = 2, if L e x) q, x) <,

5 Author's persoal copy s q,μ u) = q! ) 2 u q + μ u q + b q+ u ) b q+ u u ) ). q + )! b q+ is the Beroulli polyomial of degree q + ad t =max{z Z : z t}. Moreover, the fuctios c,q x,δ) are best possible for each f C q,, x,, N, ad δ>. It is hardly possible to list all kow results, cocerig poitwise estimates i Voroovskaja-type theorems, but we ed this brief history metioig the result of Videskij, published i 9: For f C 2, we have B f, x) f x) x x) f x) 2 x x) ω f, 2 ). Very recetly for f C 2, a quatitative forms of Voroovskaja s theorem are established by the author i 4, 8 ad by Goska ad Raşa i 9 3. Let us cosider the upper boud for B f, x) f x) x x) f x)? ) 2 if f C 3,, x, ad i orm variat the upper boud for B f f ϕ2 f 2 C,? 2) where ϕx) = x x). I 28 S. Gal proposed that the quatity i ) for ay f C 3, is of order O 2 ),. This cojecture was motivated by the works of Gal o approximatio of aalytic fuctios by Berstei operator o sets, cotaiig the uit disc z : z ),see5 8. I the complex case this is fulfilled, but i the real case we give a egative aswer to this cojecture. We poit out that oe of the above metioed results gives aswer to the cojecture of Gal. For orm estimates i 2) we recall the result i the pioeerig work of Ditzia ad Ivaov see Lemma 8.3 i 3): B f f ϕ2 f ϕ 3 f. From this paper it is also ot possible to derive the asymptotic order O 2 ) for f C 3,, or to disprove this cojecture. Let us defie the fuctio f x) = x 2 3+α,α, ), x,.

6 Author's persoal copy Obviously f C 3,, but f / C 4,. Our mai results are the followig: Theorem The followig equivalece holds true. B f f ϕ2 f 2 C, 3+α 2, Theorem 2 For all f C 3,, the followig upper poitwise estimate holds true. B f, x) f x) x x) f x) 2 x x) f x) x x) ω f 3), ) I the ext sectio we give the proofs of Theorems ad 2. We establish several corollaries from our mai results ad ed with some cocludig remarks. 2 Proofs of the mai results Proof of Theorem First we observe that the fuctio f is covex ad it is kow see 2) that for covex fuctios it holds B f, x) f x), x,. Let A :=, 2 2 +,, 2. We proceed with the lower boud as follows: B f f ϕ2 f 2 max C, x A B f, x) f x) ϕ2 x) f x) 2. We use as a auxiliary result the formula of Averbach, give i the book of DeVore ad Loretz as see Theorem 4. i Ch. i 2): Lemma For =, 2,...,we have B f, x) B + f, x) = where k, k+ +, k+ x x) + ) k= { k, k + +, k + } f p,k x), f deotes the divided dif ferece of the secod order.

7 Author's persoal copy From Lemma it follows that B f, x) B + f, x) = x x) + ) f ε k,) p,k x), k= where ε k, k, k+ ad p,kx), k are the Berstei basis polyomials. Simple calculatios show that ε k, k for all k. Cosequetly B f, x) B + f, x) = x x) + ) ) k f x x) p,k x) + + ) k= ) k f ε k,) f p,k x) k= B f x x) x x), x) + ) + ) ω f, ) f x x) x x) x) + ) + ) ω f, ). I the last iequality we used B f, x) f x), due to the fact, that f is a covex fuctio. It is obvious that ω f, ) with Therefore cα) = B f, x) B + f, x) f =cα), 3 + α)2 + α) + α) 2 α 24. x x) + ) f 24 x). We replace by +, + 2,..., + p with some arbitrary atural umber p > ad observe that f 24 x) + k f 24 x), for k p.

8 Author's persoal copy After summig up we get B f, x) B +p f, x) = f 24 x) x x) f 24 x) x x) + ) p ) + p) p + p). By fixed ad fixed x A we take p i the last estimate to obtai B f, x) f x) f 24 x x) x). Hece with B f, x) f x) ϕ2 x) f x) 2 f x) = 3 + α)2 + α) 2 x f 48 x x) x) 2 +α, ad x A by fixed sufficietly large. Wesetx = 2.The max x A B f, x) f x) ϕ2 x) f x) 2 B f, x ) f x ) ϕ2 x ) f x ) 2 f x ) 48 x x ) 2 = 3 + α)2 + α) ) +α 48 4 ) ) 3+α 2 c, 3) where c = cα) depeds oly o α for sufficietly large. We recall that i the statemet of Theorem we take.

9 Author's persoal copy We cotiue with the upper boud. To apply the result of Sikkema ad va der Meer-Theorem D for q = 4 we recall the defiitio of the polyomials of Beroulli-see 6. It is kow, that b x) =, b x) = x, b 2 x) = x 2 x, b 3 x) = x x2 + 2 x, b 4 x) = x 2 x) 2. Hece for x we obtai 2 s 3,μ u) 6 For < x we obtai 2 s 3,μ u) 6 2 u u u2 u ) 2. 4) 2 u 3 2 u u2 u ) 2. 5) Here we recall the values of the first, secod, third ad fourth momets of the Berstei operator see 2): B e x, x) =, B e x) 2, x) = ϕ2 x), B e x) 3 x x) 2x), x) =, 2 B e x) 4 x x), x) = 2 2)x x) + 3 By e : t t, t, we deoted the moomial fuctio. Therefore ) ) e x B s 3,μ, x δ 6 2 B + 24 B. ) e x 3 x x) 2x) δ, x δ 3 e ) x 2 ) e x 2 δ δ ), x + e ) 24 B x 2, x) δ e ) x 4, x). δ x x) 2x) 2 2 δ B

10 Author's persoal copy After simple calculatios, usig the represetatio of the secod ad fourth momets we obtai ) ) e x 2x B s 3,μ, x x x) δ 2 2 δ δ δ 4 x x) δ δ ) 6 2 δ TheoremDad6) imply B x x) f f, x) f x) x) 2 It is easy to verify that We set δ = i 7). Hece B f, x) f x) Obviously f 3) x) 4! 2 x x x) 2x) f 6 2 x) + x x) ) ω f 3),δ. α ω + δ δ x x) f x) 2 x x) 4 x x) f 3), ) ) ω f 3) α, =, <α<, 7) ). 8) due to the fact that f 3) Lip α,. Cosequetly we arrive at B x x) f f, x) f x) x) 2 4 x x) + 7 ) ) 3+α. 9) 24 We take sup orm i both sides of 9) to get B f f ϕ2 f 2 3 ) 3+α. ) C, 96 The upper ad lower estimates ) ad 3) complete the proof of Theorem.

11 Author's persoal copy Proof of Theorem 2 Followig the computatios for the upper poitwise estimate 7) ad after settig δ = there, we coclude, that for ay f C 3,, the estimate i Theorem 2 holds true. Remark If we compare Theorem 2 with Theorem 3.2 i 9forq = 3 we see, that it is ot possible to apply the techique of Cauchy Schwarz iequality to estimate from above the quatity L e x 4, x) L e x 3, x), which appears i the argumet of ω f q), ) for q = 3. Thecasesq =, q =, q = 2 were completely studied i 9, followig the ideas from Theorem D. Corollary For ay f C 3 ad f a Lipschitz fuctio of order we have B f f ϕ2 f ) 2 = O,. ) C, 2 This result geeralized the result established i 3, where f C 4 was supposed to get ). It is clear that ) holds for ay f C q, for all q > 4. We poit out also, that Corollary shows that the estimate i Theorem 2 allows to get the best possible order of approximatio i Voroovskaja s formula, O 2 ). Corollary 2 From the proof of Theorem it follows B f, x ) f x ) ϕ2 x ) f x ) 2 ) 3+α 2. 2) Corollary 3 If f C 3 is such that f is Lipschitz fuctio of order ad if f is ot a polyomial of degree, the the order of uiform approximatio of f by Berstei polyomials is exactly. The proof of Corollary 3 is a immediate cosequece of the proof i the case of approximatio by complex Berstei polyomials i disks see 7 or 8), followed word for word ad simply replacig there the orm o the disk z r with the uiform orm o C,. Also the followig two cojectures are formulated by S.Gal i 28, cocerig asymptotique behaviour of B f for f C 3, : i) B f, x) f x) x x) f x) 2 C ωϕ 3 C 5 3 ω ϕ ) f, 3 ) f, 3. 3)

12 Author's persoal copy ii) Maybe we could have i fact a equivalece of the form: B f f ϕ2 f 2 C, ωϕ 3 ) f, 3 ) ω ϕ 5 f, 3. 4) 3 I 3) ad4) we used the first ad third order Ditzia-Totik moduli of smoothess, for their defiitio ad properties see 4. Theorem ad Corollary 2 give egative aswers to both cojectures i) ad ii). Remark 2 It is also clear, that the estimate i Theorem 2 does ot follow from Lemma 8.3 i 3. Ackowledgemet I am sicerely grateful to the aoymous referee for his/her thorough readig of the mauscript ad very helpful remarks. Refereces. Berstei, S.N.: Complémet à l article de E. Voroovskaya Détermiatio de la forme asymptotique de l approximatio des foctios par les polyómes de M.Berstei. C. R. Dokl.) Acad. Sci. URSS A 4, ) 2. DeVore, R.A., Loretz, G.G.: Costructive Approximatio. Spriger, New York 993) 3. Ditzia, Z., Ivaov, K.G.: Strog coverse iequalities. J. Aal. Math. 6, 6 993) 4. Ditzia, Z., Totik, V.: Moduli of Smoothess. Spriger, New York 987) 5. Gal, S.G.: Voroovskaja s theorem ad iteratios for complex Berstei polyomials i compact disks. Mediterr. J. Math. 53), ) 6. Gal, S.G.: Geeralized Voroovskaja s theorem ad approximatio by Butzer s combiatio of complex Berstei polyomials. Res. Math. 53, ) 7. Gal, S.G.: Approximatio by Complex Berstei ad Covolutio-Type Operators. World Scietific, Sigapore 29) 8. Gal, S.G.: Exact orders i simultaeous approximatio by complex Berstei polyomials. J. Cocr. Appl. Math. 7, ) 9. Goska, H.: O the degree of approximatio i Voroovskaja s theorem, Studia Uiv. Babeş Bolyai. Mathematica 523), ). Goska, H., Piţul, P., Raşa, I.: O Peao s form of the Taylor remaider, Voroovskaja s theorem ad the commutator of positive liear operators. I: Agratii, O., Blaga, P. eds.) Numerical Aalysis ad Approximatio Theory Proc. It. Cof. Cluj-Napoca), Cluj-Napoca, pp ). Goska, H., Raşa, I.: Remarks o Voroovskaja s theorem. Ge. Math. 64), ) 2. Goska, H., Raşa, I.: A Voroovskaya estimate with secod order modulus of smoothess. I: Proc. of the 5th Romaia Germa Semiar o Approx. Theory, Sibiu, Romaia, pp ) 3. Goska, H., Raşa, I.: The limitig semigroup of the Berstei iterates: degree of covergece. Acta Math. Hug., ) 4. Goska, H., Tachev, G.: A quatitative variat of Voroovskaja s theorem. Res. Math. 53, ) 5. Mamedov, R.G.: O the asymptotic value of the approximatio of repeatedly differetiable fuctios by positive liear operators Russia). Dokl. Acad. Nauk 46, ). Traslated i Soviet. Math. Dokl. 3, ) 6. Ralsto, A.: A First Course i Numerical Aalysis. McGraw-Book, New York 965)

13 Author's persoal copy 7. Sikkema, P.C., va der Meer, P.J.C.: The exact degree of local approximatio by liear positive operators ivolvig the modulus of cotiuity of the p-th derivative. Idag. Math. 4, ) 8. Tachev, G.: Voroovskaja s theorem revisited. J. Math. Aal. Appl. 343, ) 9. Videskij, V.S.: Liear Positive Operators of Fiite Rak Russia). A. L.Gerze State Pedagogical Ist., Leigrad 985) 2. Voroovskaja, E.V.: Détermiatio de la forme asymptotique de l approximatio des foctios par les polyómes de M. Berstei. C. R. Acad. Sci. URSS 4, )