J. Mah. Aal. Appl. 37 5 74 9 www.elsevier.com/locae/jmaa O mulivariae Basaov operaor Feilog Cao a,b,, Chumei Dig b, Zogbe Xu a a Ceer o Research or Basic Sciece, Xi a Jiaoog Uiversiy, Xi a, 749 Shaxi, PR Chia b Faculy o Sciece, Chia Isiue o Merology, 38 Hagzhou, Zhejiag, PR Chia Received 5 May 4 Available olie 9 February 5 Submied by D. Khaviso Absrac I his paper, we sudy mulivariae Basaov operaor B,d, x. We irs show ha he operaor ca reai some properies o he origial ucio, such as moooy, semi-addiiviy ad Lipschiz codiio, ec. Secodly, we discuss he moooy o he sequece o mulivariae Basaov operaor B,d, x or whe he ucio is covex. The, we propose, or esimaig he rae o approximaio, a ew modulus o smoohess ad prove he modulus o be equivale o cerai K-ucioal. Fially, wih he modulus o smoohess as meric, we esablish a srog direc heorem by usig a decomposiio echique or he operaor. 4 Elsevier Ic. All righs reserved. Keywords: Mulivariae Basaov operaor; Covexiy; Modulus o smoohess; Approximaio Suppored parly by he Naioal Naural Sciece Foudaio o Chia o. 647334, he Foudaio o Pas Docor o Chia o. 4355. * Correspodig auhor. E-mail addresses: lcao@cjlu.edu.c F. Cao, digcm66@sohu.com C. Dig, zbxu@mail.xju.edu.c Z. Xu. -47X/$ see ro maer 4 Elsevier Ic. All righs reserved. doi:.6/j.jmaa.4..6
F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 75. Iroducio Le P, x = x x, x [,, N. The Basaov operaor deied by B, = B,, x = P, x. = was iroduced by V.A. Basaov [] ad ca be used o approximae a ucio deied o [,. I is he prooype o he Basaov Kaorovich operaor c. [] ad he Basaov Durrmeyer operaor c. [5,9]. By ow, a umber o resuls abou he operaor have bee obaied c. [,3,,, 4]. Is approximaio behavior, i paricular, has bee well udersood ad characerized by he Dizia Toi s modulus o secod order c. [,3]: ωϕ, = sup hϕ hϕ, <h ϕx = x x. More precisely, or ay coiuous ad bouded ucio deied o [,, here are cosas such ha B, cos.ω ϕ,. ad, coversely, i erms o he classiicaio give by [], a srog coverse iequaliy ype B was esseially proved by Dizia ad Ivaov [], bu also a coverse iequaliy o ype A, ω ϕ, cos. B,,.3 was proved by Toi [5]. Le T R d d N, which is deied by T = T d = { x = x,x,...,x d R d : x i <, i d }. Throughou he paper, we shall use he sadard oaios: or x = x,x,...,x d R d, x = d i= x i, we shall also wrie or x R d, =,,..., d N d, ad N, x = x x x d d,!=!! d!, = ad! =!!, = = = = =. d = i, For a ucio deied o T, he mulivariae Basaov operaor is deied by B,d = B,d, x = P, x,.4 i=
76 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 where P, x = x x. Clearly, he operaor is a o-esor produc geeralizaio or he uivariae Basaov operaor give by. i a aural way. We also see ha a similar deiiio or he mulivariae Basaov operaor was give i [8]. I his paper, we address he ivesigaio or he mulivariae Basaov operaor deied by.4. I Secio, we prove ha some properies o he origial ucios, such as moooy, semi-addiiviy ad Lipschiz codiio, are reaied by he operaor. These resuls, o some exe, are similar o some ow oes o he Bersei operaor c. [7, 9,6]. Secio 3 deals wih he moooy o he Basaov operaor uder he codiio ha he approximaig ucio is covex. Namely, we prove ha B,d is moooically o-decreasig or whe is covex o T. I Secio 4, we propose a ew mulivariae K-ucioal ad a ew modulus o smoohess, which geeralize he oes or oe variable respecively, eve or Lebesgue spaces. We also prove, or esimaig he rae o approximaio, he K-ucioal o be equivale o he modulus o smoohess. The ial secio devoes o he direc heorem ollowig sadard procedure ad usig a Jacso-ype iequaliy, he iducio ad a appropriae decomposiio or he mulivariae Basaov operaor. A resul similar o. is obaied. We ried o ge a coverse resul similar o.3 ad o characerize he behavior o he order o approximaio bu wih o success. Sill, we believe ha he iverse iequaliy is valid, ad or he secod modulus o smoohess ωϕ, ha will be deied i Secio 4, here holds B,d = O α ωϕ, = O α, <α, which seems o mach he elega heorem or he oe-dimesioal case c. [,3].. Some properies reaied by mulivariae Basaov operaor Whe approximaig a eleme o a ucio space by meas o a approximaio operaor L, i is impora o ow which properies o are reaied by he approximas L. Firs, oe is ieresed i he relaio bewee global smoohess properies o ad L. Global smoohess properies o a coiuous ucio ca be expressed by he behavior o is modulus o coiuiy. The earlies reerece we were able o locae i regard o his quesio is a problem posed by Haje [3]. A more geeral resul alog hese lies was give by Lidvail [7] i 98. Usig probabilisic mehods, he showed ha or he classical Bersei operaor B deied o [, ], oe has Lip A µ, [, ] B Lip A µ, [, ].. Here Lip A µ, S deoes he se o all real-valued coiuous ucios deied o S saisyig he iequaliy x y A x i y i µ. i=
F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 77 or all x = x,x,...,x d, y = y,y,...,y d S, wih A> ad <µ idepede o x ad y. A more elemeary proo o his resul was give laer by Brow, Ellio ad Page [6], coiuig previous research by Bloom ad Ellio [5]. Recely, Li [6] made a deep ivesigaio io he problem. He proved ha he uivariae Bersei operaor ca reai he propery o geeral modulus o coiuiy. However, i is ieresig ha he mulivariae Basaov operaor deied by.4 also possesses he similar propery. Le us begi wih some deiiios ad oaios. The modulus o coiuiy o a ucio CT is deied by Ω,u = sup { x y : x i y i u i, x, y T, u = u,u,...,u d T d }. A coiuous ucio x is said o be covex i T i x y x y or all pois x ad y T, which ca also be deied equivalely by c. [4].3 m m α i x i α i x i.4 i= i= or ay x, x,...,x m i T ad or ay o-egaive umbers α,α,...,α d such ha α α α m =. We deoe by e i i =,,...,d he ui vecor i R d, i.e., is ih compoe is ad he ohers are. From [], a coiuous ad o-egaive ucio ωu deied i T is said o be he ucio o modulus o coiuiy, i i saisies he ollowig codiios: ω =, where =,,...,; ωu is a o-decreasig ucio i u, i.e., or u v, oe has ωu ωv, where u v meas ha u i v i holds or all i d, ad u = u,u,...,u d, v = v,v,...,v d T d ; 3 ωu is semi-addiive, i.e., ωu v ωu ωv. Now, we ca sae he irs resul o he secio. Theorem.. For he mulivariae Basaov operaor B,d, x give i.4 i ωu is a ucio o modulus o coiuiy, he so is B,d ω, u, ad B,d ω, u dωu. Proo. To prove Theorem., we eed a lemma, i.e., Lemma.. For ay ucio o modulus o coiuiy ωu, here is a covex ucio o modulus o coiuiy ω u, such ha ωu ω u dωu.
78 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 Proo. Le x y = d i= x i y i be ier produc o vecors x ad y i R d. From [8] we have or u = u,u,...,u d R d, { max Ω,ei u } ωu Ω,e i u dω,u. i d i= Se u = ωu, oe has Ω,u = ωu, ad ωu d i= Ω,e i u dωu.now, or each ωe i u, here is a covex ucio o modulus o coiuiy δu i, such ha c. [] ωe i u δu i ωe i u. Le ω u = d i= δu i, he i is o diicul o see ha ω u is a covex ucio o modulus o coiuiy. Thereore, ωu ωe i u i= δu i = ω u ωe i u dωu. i= The proo o Lemma. is complee. We ur o he proo o Theorem.. Le x = x,x,...,x d, y = y,y,...,y d T, ad x y, he or i = i,i,...,i d N d, B,d, y = = i= x y x y d! =!i! i! = = d = i = i = i d = x i y x i y. Exchagig he orders o he above summaios ad leig i j =, i j =,..., i d j d = d, we ge or j = j,j,...,j d N d, i j! B,d, y = x i y x j y ij i j. i!j!! O he oher had, B,d, x = = i= j= i x i y y x i i i i= i j! x i y x j y ij i. i!j!! i= j=
F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 79 So, B,d, y B,d, x = which shows ha i= j= i j! x i y x j y ij i!j!! i j, B,d ω, y B,d ω, x ory x, i ad B,d ω, y B,d ω, x = i= j= j= i j! x i y x j y ij j ω i!j!! j! y x j y x j j ω j!! = B,d ω, y x. Thus, we have proved ha B,d ω, x is semi-addiive. Moreover, i is easy o see B,d ω, = ω =. Hece, B,d ω, x is a ucio o modulus o coiuiy. By Lemma., we ow ha or ay ucio o modulus o coiuiy ωu here is a covex ucio o modulus o coiuiy ω u, such ha ωu ω u dωu. The, B,d ω, u B,d ω, u = B,d δi, u = i= B, δi, u i. So, o iish he proo o Theorem., we oly eed o show B, δ i, u i δu i, which will imply ha B,d ω, u = δu i = ω u dωu. i= I ac, le x be a cocave ucio deied o [,, he i= m i x i i= m i x i, where m i, i= m i =, ad x i [,. Recallig B,, x = x, gives x= P, x P, x = B,, x. = So, puig u i = δu i,wehaveδu i B, δ, u i. Hece, he proo o Theorem. is complee. The secod resul o he secio is as ollows. i=
8 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 Theorem.3. For he mulivariae Basaov operaor B,d, x deied by.4, i Lip A µ, T, <µ, he B,d Lip A µ, T. Proo. For he simpliciy o wrie, we will mae he coveio ha or d =, i x y x y, x y, he rom he proo o Theorem., i ollows ha B,d, x B,d, y A i= j= i= j= i j! i!j!! x i y x j y ij i j i j! x i y x j y ij i!j!! µ = AB, u uµ, y x = A µ B, u,y µ x B, u,y x. By he ial proo o Theorem 3., we see x µ µ B, u,x, which implies ha B,d, x B,d, y A y x µ y x µ. j i µ j µ Hece, B,d Lip A µ, T. The proo o he case x y, x y is similar. I x y, x y, he y,x T. From he above proo i ollows ha B,, x B,, y B,,x,x B,,y,x B,,y,y B,,y,x A x y µ. = Similarly, we ca iish he discussio o he case x y, x y. Thereore, he proo o Theorem.3 is complee. I he ollowig, we show ha he mulivariae Basaov operaor ca reai a cerai moooy, ha is, Theorem.4. Suppose x is deied o T ad x, ix/x i i =,,...,d is o-icreasig or x i o,, he B,d, x/x i i =,,...,dis also o-icreasig or x i o,. Proo. A direc compuaio gives ha or,
x i = B,d, x x i = x i F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 8 i = i = i = = x i i = i = P, x x i x i d = d = x x x i i x i i x d d i i!! i! i! d!!,..., i,, i,..., d = = i = i = i = d = x i x x i i x i i x i i x d d x x i i x i i x i i x d d x x x i i x = = i = i = i = i = i = d = x x i i x i i x d i d x i i e i,..., i,, i,..., d = i ei i d = x i i xi P, x x x i i x = i = i = d = i x x x i i x i i x d i d x i e i i,..., i,, i,..., d. Sice x ad x/x i is o-icreasig or x i o,, we have proved B,d, x, x i x i which shows B,d, x/x i is also o-icreasig or x i o,. The proo o Theorem.4 is complee.
8 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 3. Moooy or he sequece o mulivariae Basaov operaor I his secio, we discuss he moooy or he sequece o mulivariae Basaov operaor. We prove Theorem 3.. I x is a covex ucio deied o T, he he Basaov operaor B,d, x deied by.4 is sricly moooically o-decreasig i, uless is he liear ucio i which case B,d, x = B,d, x or all. Proo. To avoid heavy ad complicaed oaio, we here oly give he deails o proo o wo-dimesioal case i.e., d = because he proo or higher-dimesioal cases is similar. We ca wrie B,d, x B,d, x { = = = = x x { = x = x { x, x, x x x, = = { = = = = { = = } },, }, x x x x x x x x,, x x } x x }, x x
= F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 83 { = = =, x x x,,, x x x x, x { = = = = = = = { = = { = =, x x } x x x x x x,,, x, x x x x } x x },, x { = = = e e e {, =, } x } e x, x x x x } x x
84 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 { =,,, } x x. Se e I = e e, e I =,,, ad I 3 =,,,, we esimae, below, I,I ad I 3, respecively. For I,le α = =, α e = =, e α 3 = =, he α, α ad α 3 are o-egaive umbers, ad α α α 3 =. Meawhile, aig gives x =, x = e, x 3 = e, α x α x α 3 x 3 =. Thereore, rom he deiiio o covex ucio.3 i ollows ha I. ad For I,wele α = =, α = y =,, y =,, =
F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 85 he α α =, ad α y α y =,. So, rom he covexiy o he ucio, i ollows ha I. Similarly, we derive I 3. Thus, we have proved B,d, x B,d, x or all N. The equaliy o B,d, x ad B,d, x ca occur oly i I = or all, N ad I = I 3 = or all, N.Bui is covex, he I = implies ha is graph is plae i he riagle x, x, x 3 cosruced by he pois x, x ad x 3. We wrie he plae as P, ad le = x, x, x 3., N Sice or all, N heir correspodig riagles pairwise overlap, he ucio is liear i, which shows ha i P. O he oher had, rom I = ad I 3 =, we see ha he cross lies o he ucio ad he ordiae axis x = ad he abscissa axis x = all i he plae P. Thereore, is a liear ucio. Coversely, i is liear, he we oice ha he mulivariae Basaov operaor reproduces he liear ucio, i.e., B,d, x =, B,d i, x = x i, i =,,...,d. I is clear o see B,d, x = B,d, x. This eable us o ed he proo o Theorem 3.. 4. K-Fucioal ad modulus o smoohess Le L p T, p<, deoe he space o Lebesgue measurable ucios o T wih he orm p p = T p is iie. L T = C B T is he space o coiuous ad bouded ucios o T wih he orm = max x T x is iie. For x T,we deie weigh ucios ϕ i x = x i x, i d. Le Di r = r xi r, r N, D = D D D d d, N d deoe diereial operaor. We deie or p< he weighed Sobolev space W r,p ϕ T = { L p T : D L loc T ad ϕ r i Dr i Lp T }, where r, N d, ad T is he ierior o T. For he space C B T, we wrie C r ϕ T = { C B T : C r B T ad ϕ r i Dr i C BT, i d }.
86 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 The Peere K-ucioal o L p T, p<, ad C B T, p =, are deied by K r ϕ, r { p = i g p r ϕ i r Dr i g p }, >, i= where he iimum is ae over all g W r,p ϕ T, p<, ad g Cϕ r T, p =, respecively. For ay vecor e i R d, we wrie or he rh orward dierece o a ucio i he direcio o e { ri= r r he x = i i x ihe, x, x rhe T,, oherwise. We he deie he modulus o smoohess o L p T, p, as ωϕ r, p = sup <h We have r hϕi e i p, p. i= Theorem 4.. There exiss a posiive cosa, depede oly o p ad r, such ha or ay L p T, p cos. ωr ϕ, p Kϕ r, r p cos.ωr ϕ, p. Remar 4.. For d =, our deiiios ad saemes i Theorem 4. coicide wih he ow oe-dimesioal oes c. [, Chaper ]. We shall reduce he proo o he oe dimesio. Some ideas are rom [4]. Proo. For x = x,x,...,x d T d, we wrie x = x,...,x d ad T ={x : x = x,x T d }.Lex = x z, z<, ad Fz= Fz,x = x z, x. The ϕ x = x ϕz, D r x = x r F r z, ad r hϕ xe x = r hϕz Fz. Cosequely, or p<, r hϕ e p p = dx r hϕ xe x p dx T T = x r hϕz Fz p dzdx. From he proo o he releva iequaliies i oe variable c. [, Chaper ], we obai
r hϕ e p p cos. F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 87 x Fz p dzdx ad T = cos. T = cos. p p, r hϕ e p p cos. x, x p dx dx x h rp ϕ r zf r z p dzdx T = cos.h rp T = cos.h rp ϕ r Dr p p. ϕ r Dr x, x p dx dx Similarly, above iequaliies also are valid or i =, 3,...,d. Thus, we obai or p<, i d, r hϕ i e i { p, L p T, p cos. h r ϕi rdr i p, Wϕ p,r T. The case p = is easier, we omi i here. Addig up hese iequaliies, we have proved he irs esimae. To esimae he secod oe, we shall agai reduce i o he oe-dimesioal case. Firs we oe ha or ixed x, here exiss a ucio G Wϕ r,p T, >, such ha c. [, Chaper ] F G p p, rp ϕ r G r p p cos. r uϕ p p du. Sice he cosrucio o G i [] depeds o F coiuously, or Fz= Fz,x we have G z = G z, x as well. We ow deie g x Wϕ r,p T hrough x g x = G x, x, x T. The g p p = x Fz G z p dzdx T cos. T x r uϕz Fz p dzdudx
88 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 ad = cos. = cos. rp ϕ r Dr g p p = rp T T cos. cos. r uϕ xe x p dx dx du r uϕ e p du, x ϕ r zg r z p dzdx T x r uϕ e p p du. r uϕz Fz p dzdudx Similarly, we ca prove ha or each i, here are ucios g Wϕ r,p T, >, such ha g p p, rp ϕ i r Dr g p cos. r p uϕ i e i p p du. Addig up hese iequaliies, we iish he proo o he secod esimae. 5. Theorem o approximaio I his secio, we shall show a direc heorem o approximaio or a ucio C B T by meas o he K-ucioal ad modulus o smoohess deied i Secio 3, which will exed he esimae. o higher dimesio. Theorem 5.. I C B T, he here is a posiive cosa idepede o ad,such ha B,d cos.ωϕ,. Proo. Our proo is based o a iducio argume or he dimesio d. We will also use a mehod called decomposiio echique or he Basaov operaor. By sadard argumes, he proo o Theorem 5. will ollows rom Theorem 4. ad he esimaes {, C B T, B,d cos. d i= ϕi D i, Cϕ T. 5.
F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 89 The irs esimae or all i C B T is evide as B,d is posiive ad liear coracio o C B T. We shall show he secod oe by reducig i o he oe-dimesioal iequaliy B, cos. ϕ /, 5. which has bee proved i [,3]. Now, we give he decomposiio ormula: x B,d, x = P, x P,, 5.3 x = = where x = x,x 3,...,x d, x = x, x T, =, 3,..., d, =, N d ad = = = = d =, which ca be direcly checed ad will ae a impora role i he ollowig proo. Le g u =, u, u T d, ad x x x d z = z,z,...,z d =,,..., = x. x x x x From ormula 5.3 i ollows ha B,d, x x = P, x B,d g, z g z = B, h, x hx = J L, where x hu = hu, x = u, u, u<. x I he secod esimae o 5. is valid or d = r, r, i.e., r B,d cos. ϕ i Di, i= he we have or d = r, J cos. P, x = r ϕ i D i g. i= However, by deiiio implies ϕi ud i g u = u i u Di = ϕi D i, u., u
9 F. Cao e al. / J. Mah. Aal. Appl. 37 5 74 9 Thereore, J cos. r ϕ i Di. i= To esimae he secod erm L, we apply 5. ad coclude ha L cos. ϕ h. Deoig ϕ ij x = x i x j, i<j d, ad Dij = x i x j,wehave ϕ h = max u u D u< x i Di x i= x i Di x x i x j x i= i= j= D ij x u, u x x = max u< x ϕ D ϕi D i ϕi D i i= i= u u x i x ϕ i D i u u ϕ ij D ij i= i,j=,i j x u, u x. Recallig ha ϕ ij x is o bigger ha ϕ i x or ϕ j x ad he ac D ij x sup D i x i d proved i [8] c. [8, Lemma.], we obai ha L cos. ϕ i D i. i= So, he secod iequaliy o 5. has bee proved or ay d N ad he proo o Theorem 5. is iished. Reereces [] J.A. Adell, F.G. Badía, J. de la Cal, O he ieraes o some Bersei ype operaors, J. Mah. Aal. Appl. 9 997 59 54. [] V.A. Basaov, A example o a sequece o liear posiive operaors i he spaces o coiuous ucios, Dol. Aad. Nau SSSR 3 957 49 5. [3] M. Becer, Global approximaio heorems or Szász Miraja ad Basaov operaors i polyomial weigh spaces, Idiaa Uiv. Mah. J. 7 978 7 4.
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