Weighted Approximation by Videnskii and Lupas Operators

Transcription

1 Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

2 Abstract Weighted modificatios of geeralized Berstei operators i ratioal fuctios Videsii operators) are itroduced. Their covergece i weighted spaces is studied. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, 013 / 35

3 Itroductio The Berstei polyomials B f, x) = f =0 ) )x 1 x) 1) associated with a fuctio f defied o [0, 1] have bee the subject of much recet research ad have bee geeralized i may directios.see for istace [1],[],[3]). Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

4 Itroductio I 1966 J. P. Kig [4] itroduced the followig geeralizatio of the Berstei polyomials ) L f, x) = u x), ) f =0 where u x) are give by the geeratig fuctio g x, y) = i=1 h i x) y + 1 h i x))) = =0 u x) y, 3) ad h i x) = h i x) is a sequece of cotiuous fuctios defied o [0, 1], 0 h i x) 1. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

5 Itroductio Kig s operators coverge to the approximated fuctio if ad oly if 1 lim =0 Now let x i be fixed poles x i = 1 + ρ i, ρ i > 0 ad Put h i x) = φ x) = 1 h i x) = x. 4) ρ i x 1 + ρ i x. 5) =1 h x). Observe that φ x) is strictly icreasig from 0 to 1 o the iterval [0, 1]. The odes τ are well-defied by φ τ ) =, = 0, 1,..., ). Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

6 Itroductio I 1979 V. S. Videsii [5] itroduced aother geeralizatio of Berstei operators for approximatio by ratioal fuctios with fixed poles V f, x) = f τ ) u x). 6) =0 Later V. S Videsii cosidered more geeral case of the operators 6), where u are defied for arbitrary icreasig fuctios h i x). The mai differece betwee those families of operators is i odes. The advatage of Videsii s operators ca be easily see from the coditios for their covergece. Namely V. S Videsii see for istace [6] th. 3.1)) proved that sequece V f, x) uiformly coverges to arbitrary f C [0, 1] if ad oly if lim S = 7) ρ where S = i 1+ρ i. i=1 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

7 Itroductio A simple example ρ i = ρ = 1) shows that coditio 4) is essetially more restrictive tha 7). Later V. S. Videsii [7] cosidered arbitrary matrices of odes ξ istead of τ ad proved the covergece results for those operators V ξ f, x). Note that for ξ = we recover Kig s operators. Moreover recetly he observed [8] that aother well-ow geeralizatio of Berstei polyomials, amely Lupaş operators, see, for example [9]) ca be cosidered as a particular case of the operators V ξ f, x), too. Recetly may authors pay attetio to weighted approximatio by classical polyomial operators ad to costructio of their weighted modificatios. The reaso is that usual operators are ot always suitable for approximatig fuctios with sigularities i weighted spaces. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

8 Itroductio For istace, the sequece of classical Berstei operators 1) is ot bouded i the space { } C w = f C 0, 1)) : lim wf ) x) = lim wf ) x) = 0, x 0 x 1 where f Cw := wf = sup wf ) x), x [0,1] w x) = x α 1 x) β, α, β 0, α + β > 0, 0 x 1, but it s slight modificatio B f, x) = 1 x) [f 1 + f =1 ) f )] ) p x) + x [ f 1 1 ) f 1 )] is bouded. Oe ca cosult papers [10], [11] ad [1] which cotai these ad other deep results i this directio. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35 8)

9 Itroductio Followig [10] we cosider the Sobolev type space Wω defied as { } Wω := f C w : f AC 0, 1)), f ωϕ < where ϕ x) = x 1 x) ad AC I ) is the set of absolutely cotiuous fuctios i I. Observe also that modificatio 8) is ot a positive operator, so geeral results about weighted approximatio by liear positive operators o a real iterval see, for istace, [13] ad refereces therei) are ot applicable here.the mai goal of the research is to ivestigate approximatio properties of Videsii operators i the orm of C w uder some restrictios o the sequece of deomiators.i the followig C deotes a positive costat which may assume differet values i differet formulas. Moreover we write v u for two quatities v ad u depedig o some parameters, if v ±1 C with C idepedet of the parameters. u Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

10 Itroductio Note that operators 6) as well as the Berstei operators are ot bouded i fact eve ot defied) i C w. Here we cosider modificatios of the Videsii operators similar to 8): V f, x) = =1 f τ ) u x) + u 0 x) [f τ 1 ) f τ )] 9) + u x) [f τ ) f τ )]. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

11 Itroductio The mai result of the research is Theorem Suppose that ρ i satisfy ρ i > C > 0 ad a) 1 ρ i i=1 C, the b) [f V f )] Cw V f ) Cw C C f Cw ϕ f Cw if f W ω. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

12 Videsii operators ad their properties Now we ca give some basic facts about Videsii s operators 6) from [8]. Put P x) = = i=1 =0 1 + ρ i x) = α x 1 x) i=1 ρ i x ρ i ) 1 x)) the we ca write basic fuctios u x) for Videsii operators as x u x) = 1 x) α x x i i=1 It is clear that u satisfy the equality =0, α > 0 10) u x) = 1. 11) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

13 Videsii operators ad their properties Differetiate i y g x, y)) y = g x, y) i=0 h i x) h i x) y + 1 h i x) = u x) y =0 ad put y = 1 φ x) = =0 u x). 1) Note that the fuctios 1, φ x) play role of the fixed fuctios f 0, f 1 for geeralized Berstei operators i the sese of [14] for the system of ratioal fuctios of degree with deomiator P x). Rewrite 1) i the form φ τ ) φ x)) u x) = 0. 13) =0 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

14 Videsii operators ad their properties Differetiatio of l u x) 0 < x < 1) gives u x) = x 1 x) φ τ ) φ x)) u x). 14) Formula 14) shows by the way that the poit τ 1 1) is the uique poit of maximum of the fuctio u i the iterval [0, 1]. That is a reaso why the Videsii operators ca be cosidered as a atural aalogue of the Berstei operators for ratioal fuctios. Derivative of 13) with taig ito accout 14) gives φ τ ) φ x)) u x) = =0 x 1 x) φ x). 15) Next assertios are ot give i [8] but are ecessary for the followig. First we give a lemma which ca be cosidered as a excercise for a calculus textboo. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

15 Differetiate 3) ad use Lemma. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35 Videsii operators ad their properties Lemma If, m N, f i C [a, b]), i = 1,..., m the the followig equality holds d m ) dy f i y) i=1 = j 1 +j +...+j m = j 1 0,..., j m 0! j 1!j!...j m! d j 1 dy j 1 f 1 y))... d jm dy j m f m y)). 16) Corollary If h i is defied as i 5) the u x) = 1! j 1 +j +...+j = 0 j i 1! j 1!j!...j! i=1 ) 1 j i ) j i h i x) 17)

16 Videsii operators ad their properties Lemma Uder suppositios of Theorem 1, for ay x 0, 1) C φ x) x ad Proof. Firstly ad φ x) = 1 i=1 1 1 φ x) 1 x ρ i x 1 + ρ i x x 1 φ x) = 1 i=1 1 18) C. 19) i=1 ρ i ρ i x 1 + ρ i x x. 1 + ρ i Cx Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

17 Corollary Suppose that ρ i satisfy suppositios of Theorem 1 the w x) w φ x) ) ad ϕ x) ϕ φ x) ) Observe also that from defiitio of u x) it follows immediately that 0 u x) 1 = 0,..., ; = 1,... Usig 5), 17) we get u x) = x 1 x) 1 + ρ i j i ) j 1 +j +...+j = i=1 0 j i 1 i=1 1 + ρ i x) ad we ca write dow a explicit formula for the coefficiets α from 10) : α = j 1 +j +...+j = j i {0,1} i=1 ρ i + 1 j i ). 0) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35,

18 Lemma If ρ i satisfy the suppositios of Theorem 1 the α )P x) C Proof. Firstly l the i=1 = ρ i ) 1 ρ i i=1 C α )P x) = 1 ) j 1 +j +...+j = j i {0,1} 1 ) j 1 +j +...+j = j i {0,1} i=1 ρ i + 1 j i ρ i i=1 ρ i + 1 j i ρ i + 1 x i=1 ρ i + 1 ρ i C. 1) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

19 Proof of Theorem 1 We start with a) w x) f τ ) u x) = w x) =1 = w φ wf x) ) =1 =1 =1 α )x 1 x) )P f τ ) x) α.p x) f τ ) w τ ) )P x) w τ ) α.p x) w φ x) ) )P x) w φ )). The proof of p x) wx) C is cotaied i [10], p.30. Aalogously w ) =1 other terms i V f, x) are cosidered. The Corollary 5 ad Lemma 6 fiish the proof of Part a). Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

20 Proof of Theorem 1 Lemma Uder suppositios of Theorem 1 the iequalities φ x) 1 ad φ ) C hold. Proof. We start with φ x) 1 φ x) 1 =1 =1 ρ 1 + ρ ) C, 1 + ρ ρ C. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

21 Proof of Theorem 1 Proof. Put t = φ x). The φ ) t) = = φ ) 1 φ t) ) φ 1 φ φ t) )) φ t) ) φ ) t) φ x) = we prove the lemma. =1 ρ 1 + ρ ) 1 + ρ x) 3 =1 1 + ρ ρ C. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

22 Proof of Theorem 1 Lemma If f W w the f φ W w. Proof. We start with f φ t) )) = f φ = f φ Cosider firtsly 0 t 1.The f t) ϕ t) w t) = t t) ) φ t) ) ) t) ) φ t) ) ) + f φ t) ) φ t) ). ) f x) dx ϕ t) 1 w t) + f ϕ t) w t) 1 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, 013 / 35

23 Proof of Theorem 1 Proof. ad t 1 f x) dx ϕ t) w t) f ϕ w 1 t dx ϕ x) w x) ϕ t) w t) C f ϕ w [ x α] 1 t ϕ t) w t) C f ϕ w The case 1 t 1 is aalogous. Hece by Corollary 5 ad Lemma 7, the lemma is proved. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

24 Proof of Theorem 1 Lemma f α, β > 0, 0 x 1 the D x) = w x) φ x) φx) u x) φ x) ξ ϕ ξ) w ξ) dξ C Proof. Firstly let us assume that 0 x 1.The the restrictio φ x) φ x) splits i to either i.e φ x) φ x) φ x) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

25 Proof. D 1 x) = Cx α φx) Cx α p x) x φ x) u x) x by lemma 6 ad [[10], 18)]. For estimatig D x) ξ α dξ C ξ α ϕ dξ Cx ξ) w ξ) φx) φ u x) D x) w x) 3φx) = w x) = D 1) + D). u x) φ x) 3φx) 3 ξ ϕ ξ) w ξ) dξ + u x) 3 φ x) ξ ϕ ξ) w ξ) dξ Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

26 Proof of Theorem 1 Proof. Note that oe of D 1) For D 1) For D ) D ) or D) may be abset. we ca write usig Lemma 7, 15) ad 18). we have D 1) Cx α u x) 3φx) φ x) 3 = w x) u x) 3 C φ x) u x) =0 3 φ x) ξ dξ ξα+1 ) φ x) C. ξ dξ + ξ α+1 β+1 1 ξ) 3 ξ ξ α+1 1 ξ) β+1 d Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

27 Proof. C w x) u x) x α dξ 1 ξ) β ) Cx p x) 3 0 = Cx [ 3 ] 1 x) [ 3 ] [ 3 ] ) C d ξ) 1 ξ) β Cx p,[ 3 ] x) ) 3 3 ) ) x [ 3 ] 1 x) [ 3 ] x 3 x 3 1 x) 3 ) 1 3 ) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

28 Defiitio Let ψ x), P 1 f ), P f ) ad F f ; x) be defied as i [10], ψ x) = 10x 3 15x 4 + 6x 5, if 0 x 1 0, if 0 x 1, if x 1. P 1 f ) ad P f ) are the liear fuctios iterpolatig f at the poits 1, ad 1 1 ad 1 respectively 1 P 1 f, x) := x) f 1 P f, x) := [ 1 x)] f ) + x 1) f ) ) + [ 1 x) 1] f F f, x) := 1 ψ x 1)) P 1 f, x) + 1 ψ x + )) ψ x 1) f x) + ψ x + ) P f, x). Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35 ).

29 Lemma If f W ω the for F = F f φ F φ V f )) w C ) ad for all α, β 0 F ϕ w. Proof. First cosider =0 τ u x) φ t) φ τ )) F φ t)) φ t) d t) x ) = u x) + =0F F φ x)) u x) =0 ) [ ) 1 = f φ u x) f φ f φ =1 )] u,0 x) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

30 Proof. Hece [ = f φ ) 1 = F φ x)) V f, x). f φ F φ x)) V f, x) w x) w x) + φ x) φx) φ x) φx) =0 )] u, x) + F φ x)) u x) := E 1 x) + E x). φ x) ξ F ξ) d Firstly suppose that 0 x 1. For E 1 x) as i [[10], p.7], E 1 x) = 0 for 0 x 1. Now if 1 x 1 the 1 1 ad Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

31 Proof. E 1 x) φ x) φx) F ϕ w C 1 x) x c ξ w x) u x) φ x) φ x) φx) φ x) F ϕ w c O the other had by lemma 9 F ) u x) φ x) F ϕ w E x) C F ϕ w D x) C ξ) w ξ) ϕ ξ) w ξ) ϕ ξ) F ϕ w. d ξ Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

32 Proof of Theorem 1 Lemma If f Wω ) ) the for P 1 := P 1 f φ ad P := P f φ ad we have w [f P 1 φ ] [0,τ ] C w [f P φ ] [τ,,0] C f ϕ w. f ϕ w. Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

33 Proof. By Lemma 8 we have f φ W ω, the the proof of Lemma 3 i [10] gives max t [0, ] ad w φ max w φ t [1,1] ) f φ ) f φ t) ) P 1 f φ, t ) C t) ) P f φ, t ) C Now the proof of Lemma 8 gives the desired result. f φ f φ ) ϕ w [0, ] ) ϕ w [1 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

34 Proof of Theorem 1 Lemma Let F := F f φ ). If f W ω the we have F ϕ w C f ϕ w. Proof. Apply the proofs of [10], Lemma 4) ad of Lemma 8 Proof. Theorem1 b) We ow that for φ x) [, 1 ] F φ x) = F f φ, φ x) ) = f φ φ x)) = f x). The by Lemma 11 we deduce Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

35 Proof. w [f V f )] w [f F φ ] + w [F φ V f )] C F ϕ w + w [f F φ ] = C F ϕ w + max w [f F φ ] φ [0, ], w [f F φ ] φ [1 Now max [0, ] w x) f x) F f φ, φ x) ) = max t [0, ] w φ x φ ad Lemmas 11, 1 fiish the proof. t) ) f φ t Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

36 G. G. Loretz, Berstei polyomials, Uiversity of Toroto Press, Toroto, d editio: Chelsea, New Yor, R. A. DeVore, G. G. Loretz, Costructive approximatio, Spriger, Berli G. T. Tachev, The complete asymptotic expasio for Berstei operators, J. Math. Aal. Appl ) J. P. Kig, The Lototsy trasform ad Berstei polyomials, Caad. J. Math ) V.S. Videsii, Approximatio of fuctios by ratioal fractios with fixed real poles, Vesti Aad. Novu BSSR Ser. Fiz-Mat Navu 1) 1979) i Russia). V. S. Videsii, A.E. Mecher, O a certai sequece of positive liear operators ad two optimizatio. V. S. Videsii, Some ew ivestigatios o approximatio by liear positive operators, J. Soviet Math, 48, 1990, o:, Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

37 V. S. Videsii, A remar o the ratioal liear operators cosidered by A. Lupas. Some curret problems i moder mathematics ad educatio i mathematics. Ross. Gos. Ped. Uiv. St. Petersburg, 008) i Russia). S. Ostrovsa, O the Lupas q-aalogue of the Berstei poyomials, Rocy Moutai Jour. of Mathem. 36 5) 006) B. Della Vecchia, G. Mastroiai, ad J. Szabados, Weighted approximatio of fuctios with edpoit or ier sigularities by Berstei operators. Acta Math Hugar ) 004) B. Della Vecchia, G. Mastroiai, ad J. Szabados, Weighted approximatio of fuctios o the real lie by Berstei polyomials. J. Approx. Theory ) S. Guo, H. Tog ad G. Zhag, Poitwise weighted approximatio by Berstei Operators, Acta Math Hugar 101 4) 003) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35

38 F. Altomare, O some covergece criteria for etsof positive operators o cotious fuctio spaces, J. Math. Aal. Appl ) J. M. Aldaz, H. Reder, Optimality of geeralized Berstei operators, J. Approx. Theory ) Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig Sciece) by Videsii ad Lupas Operators April 5, / 35