Inverse Estimates for Lupas-Baskakov Operators

Transcription

1 7 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER Iverse Estmates for Lupas-Basaov Operators Zhaje Sog Departmet of Mathematcs Schoo of SceceTaj Uversty Taj 37 Cha Isttute of TV ad Image Iformato Taj Uversty Taj 37 Cha Pe Ye Schoo of Mathematcs ad LPMC Naa UverstyTaj 37 Cha e-ma: yep@aaeduc Abstract We estabsh Stech-Marchaud-type equates for some Feer operators by usg some modfed Dtza-Tot moduus of smooth -ess The we derve some verse resuts for ths famy of operators Moreover combg the verse resuts wth the drect estmates we obta a appromato equvaet theorem Ide Terms Lear operator Iverse estmate Lupas- Basaov operator equvaet theorem I INTRODUCTION AND MAIN RESULTS The appromato of fuctos by ear operators pays a mportat roe the support vector mache cassfcato ad computer aded geometrc desg Ref []-[] There have bee may terestg resuts o ths subject Ref[3]-[] [4] A typca probem from ths appromato theory s how to provde the smoothess propertes of a target fucto f from the sequece of appro -mato errors The vestgato of ths probem has a og hstory ad may verse theorems have bee deveoped It s we ow that the ey to sove ths d of probems s to estabsh varous Berste-type or Marchaud-type equa -tes Ref [3]-[] I ths paper we cotue to study ths probem We w derve Stech- Marchaud-type equates for some Feer ope -rators ad the obta some verse resuts for ths famy of operators So et's beg wth the defto of the Feer operators Let X be a sequece of radom varabes havg dstrbuto fucto F () t wth epectato EX = ad varace σ ( ) where s a rea cotuous parameter For a cotuous fucto f o the rea e R = ( ) we defe the probabstc type operator L ( f; ) Ef( X ) f( t) df ( t) where E f( X ) < To specaze () et Y Y = = () be dep -edet ad Correspodg author: Ye Pe Ths wor s supported by the Natura Scece Foudato of Cha (Grat No ) detcay dstrbuted radom varabes set Z The we defe Feer type operators as foow L ( f; ) Ef( Z ) f( t/ ) df ( t) = = = = Y Let () Y foow Berou dstrbuto PY ( = ) = PY ( = ) = [] ; () Y foows e Posso dstrbuto PY ( = ) =! = ( ) ; () Y foows geometrc dstrbuto PY ( = ) = pq = ; (v) Y foows orma dstrbuto or Gauss dstrbuto the probabty ( y ) / desty fucto of Y be f ( y) = e π < y < ; (v) Y foows Gamma dstrbuto the y/ probabty desty fucto of Y be f ( y) = e y> ; the we derve the Berste poyomas Szasz operators Basaov operators Weerstrass operators ad Gamma operators respectvey see [6] We reca some prevous resuts reated to ths probem We frst reca the fudameta resut o the Berste operator B ( f ) = f( / ) ( ) = () () To ths ed we troduce some basc otatos ψ ( ): = [ ( )] f = sup f( ) ; C : = { f C []: f () = f () = } C : = { f C : f cotuous ad bouded o ()} γ ψ γ f ( ): = f ( + h h ) f ( ) + f ( h ) Usg the weghted moduus of cotuty γ ωγ ( f t) : = sup{ ψ hψ f( ) : ± hψ ( ) [] < h t} Beres et a proved the foowg equaty ACADEMY PUBLISHER do:434/jcp67-75

2 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER 7 ε γ ε ωγ ( f ) M γ ψ ( Bf f) = (3) see [4] Wcere [8] mproved above resut ad obtaed for f C γ γ oe has γ ωγ ( f ) Mγ ψ ( B f f) = (4) Motvated by the wor of Dtza [9] o drect estmate for Berste poyomas wth moduus of cotuty ω ( f t)( ) ths resut ufed the orm estmate ad the estmate by cassca moduus of cotuty ad Dtza-Tot moduus [5] usg the hgher-order r weghted moduus of smoothess ω ( f t) Guo et a got the coverse theorems for the ear combatos of Berste operators Berste-Katorovch Szasz- Durrmeyer opera -torsref [3]-[4] A few years ater usg moduus of cotuty ω ( f t) Guo et a [5] estabshed the Stech-Marchaud-type equates coecto wth Basaov operators whch s defed as V( f ): = f( / ) v ( ) = + where v ( ) = ( ) ( + ) Now we wat to eted above resuts to a arge famy of operators To ths ed we cosder a speca type Feer operators as: S( f ) = f( / ) p ( c ) = (5) ( ) where p ( c ) = () Φ ( c ) ad! Φ e c ( c) = = c / ( + c) c > (6) Obvousy Φ ( c ) s rght cotuous c = It s cear that S ( f ) are Szasz operators ad Basaov operators for c = ad c = respectvey see [4]-[8] I other words e ( ) /! c = p ( c ) = Γ ( / c+ ) (7) ( ) ( ) / c c + c c > Γ ( / c)! ( / ) wher Γ c+ = Γ( / c) c c c c We use the property of Gamma fucto Γ ( + ) = Γ ( ) (7) Ths type of operators are ow as Lupas-Basaov operators for c > They have some mportat appromatg propertes For stace they pay a mportat roe the appromato of fuctos wth bouded varato To state our resuts we eed some ew otatos ( ) = ( + c) c ; C = { f s cotuous ad bouded o [ )}; C = { f C f() = }; For < < ad ( ) + we defe ( ) C = { f C f < }; ( ) C = { f C f C f < } We choose a modfed moduus of smooth ess ω ( f t) ( ) : = sup{ ( ) f( ) : < ht h ± h ( ) [ )} Usg ths moduus we obta the foowg Stech- Marchaud equaty for Lupas- Basaov operators Theorem For f C ad there s a costat M depedet of ad such that ω ( f ) ( ) ( ) M ( S f f ) + f = (8) Combg the verse estmate wth the drect estmate ( ) ( S f f)( ) ω f whch w be proved secto 4 we have Theorem For f C the foowg two statemets are equvaet: ( ) ( S f f)( ) = O ω ( f t) = O( t ) Throughout ths paper the sg N deotes the set of oegatve teger M deotes a costat depedet of ad but t s ot ecessary the same dfferet cases Let ( ) j j γ = + Cγ = C j = Ad for a gve rea umber y et y stad for the smaest teger ot ess tha y ad et y stad for the argest teger ot greater tha y Ⅱ UPPER ESTIMATES FOR γ S f I ths secto we w preset the upper estmates for γ S γ f terms of f f ad f f These estmates are cruca to the proof of the verse equates We beg wth some premary emmas Lemma Let be defed as before ad u / y we have ( y+ u) + c = ( y) + + ( y+ u) + c ACADEMY PUBLISHER

3 7 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER + + ( y+ u) + (9) Proof For c = (9) foows from a smpe computato Now we cosder the case c > ad u / t s cear that + + ( y+ u) + ( y) + = + ( y + u)[ + c( y+ u)] + y( + cy) + = ( y+ u)[ + c( y+ u)] + + u+ cyu+ u For c > ad / u we have + + ( y+ u) + ( y) + = ( y+ u)[ + c( y+ u)] + + u+ cyu+ u + cy ( y )[ + c( y )] + c cy y = ( y ) + + c cy ( y ) + c [( y ) + ] The proof of (9) s competed Lemma For f C γ γ there s a costat * M such that + + f( ) f( ) + f( ) p+ c ( c ) = hods for () γ * γ M f () [m( )] ; ad c + γ * γ p ( c )( ) f( ) M f = ( = ) hods for [m( ) ) c Based o Lemma ad Lemma 3 t s ot dffcut for oe to prove the foowg two Theorems We omt the detas Theorem 3 For f C γ γ oe has γ ** γ S f γ M f () ** γ / γ S f M f (3) ** * where the costat M s depedet of c ad M Theorem 4 For f C oe has c S f + f ; (4) ad for f C γ γ < oe has γ S f γ γ / + f + (+ 8 c) f (5) Ⅲ STECHKIN-MARCHAUD INEQUALITES FOR S Lemma 3 ([8] Lemma ) Suppose that for oegatve sequeces { σ }{ τ } wth σ = the equaty p σ σ + τ > (6) hods for є N The oe has σ ( p ) p p M p τ = (7) Lemma 3 ([8]Lemma ) Suppose that for oegatve sequeces { µ }{ ν}{ ψ } wth µ = ν = the equates ( < r < s ) ad r µ µ + ν + ψ (8) ν ν + ψ (9) hod for є N The oe has ν r r Mr s ψ = () Based o Lemma 3 ad Lemma 3 oe ca derve the foowg theorem buy usg the techque from [8] Theorem 33 For f C γ γ oe has γ γ γ S f M ( S f f) f + () = The we provde a upper estmate for K ( f ) terms of the sequece of the appromato error of Lupas-Basaov operators Theorem 34 For f C Ν oe has K ( f ) ( ) ( ) ( M + ) ( S f f ) + f = () s ACADEMY PUBLISHER

4 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER 73 Proof Obvousy for there ests a umber є N such that ( ) ( Sf f ) = ( ) m{ ( S f f ) } (3) Usg Theorem 33 ad otce that γ = ( ) + we have ( ) S f M ( ) ( ) ( S f f) f + = (4) Let g( ) = S f( ) we have K ( f ) ( ) ( ) = ( S f ) + S f ( ) ( S f ) + M M = ( ) ( ) ( S f f ) + f = ( ) ( S f ) + = ( ) ( ) ( S f f ) + f = ( M + ) ( ) ( ) ( S f f ) + f = (5) Thus Theorem 34 s proved Now we proceed to the proof of Stech- Marchaud equaty It s ow from Theorem 34 that Theorem ca be proved by showg ω ( f ) MK ( f ) To do ths we eed the foowg emma Lemma 35 If γ oe has s/ s/ s/ s/ u v dudv c s γ γ ( + + ) 4(+ ) ( ) (6) where <s< ad s < < s or s > Proof We prove γ = frsty (Case ) c = Step A For < s < ad s<<s we have s/ s/ dudv s/ s/ + u+ v = ( + s)( + s) + ( s)( s) + s s ( + s) ( s) = s + s s s = ( + ) ( ) s s ( + s) + ( s) 4s = s = 4 s ( ); (7) Step B For s we have s/ s/ s s dudv s ( ); s/ = s/ + u+ v s / (8) (Case ) c = For <s< ad s < < s we have s/ s/ dudv s/ s/ ( + u+ v)[ + c( + u+ v)] s/ s/ c = dudv s/ s/ + u+ v + c( + u+ v) s/ s/ dudv s/ s/ + u+ v 4 s (+ c) 4 s (+ c) s 4(+ cs ) ( ); (+ c) ( + c) (9) The we prove γ < Usg the Hoder equaty we have t/ s/ γ ( + u + v) dudv s/ s/ / / / / ( ( ) / / ) ( / / ) s s γ u v dudv s s dudv s s s s + + / γ / + (3) Now we are a posto to prove Theorem / [4( c)] γ γ s ( ) Proof of Theorem By the defto of K -fuctoa there est a m > ad g C such that ( ) ( ) K ( f ) ( f g ) + g mk ( f ) (3) The we cosder the secod symmetrc dfferece Let t = ( )/ the ew Dtza moduus of cotuty otce that ± h ( ) we have h ( ) Moreover et s = h ( ) Lemma 7 otce that we have s ( )/ = ( + c) / h ( ) h ( ) g ( u v ) dudv h ( ) h ( ) = c ( ) + ( ) ( ) ( g ) (3) Due to ( )( ) s a mootoe creas -g fucto ad ( ) + h ( ) et = ( ) + we have ( )( ) h f g ( ) ( ) ( ) + ( )( f g )[ ( + h ( )) ACADEMY PUBLISHER

5 74 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER + + h ( ) + ( ) + ( ) ( ( ))] ( ) ( ) + ( ) + ( )( f g )[ ( ) + 3 ( )] The we have (33) ( ) ( ) 7 + ( ) ( )( f g) ( ) ( f)( ) ( ) h ( ) ( ) ( ) ( f g)( ) + ( g)( ) h ( ) h ( ) ( ) ( ) (7 + 8 c) ( f g ) + g (34) Therefore ω ( f ) m(7+ 8 c) K ( f ) m(7 + 8 c)( M + ) ( ) ( ) ( S f f) f + = (35) For the case < 4( + c) ths s a obvous resut Thus Theorem s proved ( S f f)( ) M / t f( + t) f( t) + f( t) M ( ) ( ) ( S f f)( ) = O ω ( f t) = O( t ) (4) Proof of theorem Usg the equvaece reato M ω f t K f t M ω f t ( ) ( ) ( ) (4) whch was proved [3] ad the momet codto (5) we obta the foowg drect estmate of the appromato by S ( ) ( S f f)( ) ω f by the method [9] Thus combg ths drect estmate wth (4) we arrve the resut of Theorem Ⅳ SOME APPLICATIONS I ths secto we gve some appcatos of equates (8) We frst derve some verse resuts the prove Theorem Note that a mmedate cosequece of Theorem s the foowg coroary whch are Stech- Marchaud equates for S wth regard to both orm estmate ad cassca estmate Coroary 3 Let f C γ For = we have ω( f ) M ( ( S f f ) + f ); = (36) ad for = we have ( ) ω ( f ) M ( ( S f f ) + f ) = (37) Moreover appyg Coroary we ca derve the foowg verse resuts for S by usg stadard method Ref [3]-[7] For smpcty we wrte ω ( f t) for ω ( f t) ad K ( f t ) for K ( f t ) Coroary 3 Let = < < we have ( ) ( S f f)( ) = O ω ( f t) = O( t ) (38) ( S f f)( ) = O ω ( f t) = O( t ) (39) ACKNOWLEDGMENT Ths wor was supported part by grats from Natura Scece Foudato of Cha (Grat No ) Ye Pe s supported by the Program for New Cetury Eceet Taets Uversty of Cha REFERENCES [] HZ Tog DR Che LZ Peg Learg rates for reguarzed cassfers usg mutvarate poyoma eres J Compety vo [] G J Wag G Z Wag ad J M Zheg Computer Aded Geometrc Desg Bejg: Hgh Educato Press- Sprger ( chese) [3] W Feer A Itroducto to Probabty Theory ad ts Appcatos II New Yor Wey 966 [4] H Beres G G Loretz: Iverse theo -rems for Berste poyomas Idaa Uversty Math J [5] V Tot Uform appromato by Szasz-operators Acta Math Acad Sc Hugar [6] W Z Che Appromato theory of operators Xame Uv press 989 [7] R DeVore G G Loretz Costructve appromato vo 33 Sprger Grudehre Sprger Ber- Hedeberg 993 [8] E Wcere Stech-Marchaud-Type equates coecto wth Berste poyomas Costr Appro vo o pp Dec 986 [9] Z Dtza Drect estmate for Berste poyomas J Appro Theory Vo 79 o pp Oct 994 [] V Tot Strog coverse equates J Appro Theory vo 76 o 3 pp Sep 994 [] M Fete Drect ad verse estmates for Berste poyomas Costr Appro vo 4 o 3 pp Sep 998 [] Z Fta "O coverse appromato theorems" J Math Aa App vo 3 o 3 pp 59-8 Dec 5 [3] Z Dtza ad V Tot "Modu of smoothess" New Yor Sprger- Verag 987 ACADEMY PUBLISHER

6 JOURNAL OF COMPUTERS VOL 6 NO DECEMBER 75 [4] S Guo Y Ge Strog coverse equaty o smutaeous appromato by Basaov type operators Chese J Cotemp Math 8 No [5] S Guo C L X Lu ad Z Sog Potwse appromato for ear combato of Berste operators J Appro Theory vo 7 o pp 9- Nov [6] S Guo L Lu ad Z Sog Stech- Marchaud-type Iequates wth Berste-Katorovch poyomas Northeaster Mathematca Joura vo 6 o 3 pp Ju [7] S Guo H Tog ad G Zhag Stech-Marchaud-type equates for Basaov poyomas J Appro Theory vo 4 o pp Ja [8] X M Zeg J N Zhao Eact bouds for some bass fuctos of appromato operators J of Iequa ad App vo 6 o 5 pp Zhaje Sog receved the BS degree mathematcs from Hebe Uversty Baodg Cha ad the MS degree mathematcs from Hebe Norma Uversty Shjazhuag Cha ad the PhD degree probabty theory ad mathematca statstcs from the Schoo of Mathematca Scece Naa Uversty Taj Cha ad 6 respectvey He spet -9 as the Drector at Isttute of Apped Mathematca Departmet of Mathematcs Taj Uversty Taj Cha (TUTC) He spet 6-8 as a Postdoctora Feow sga ad formato processg wth Schoo of Eectroc ad Iformato Egeerg TUTC ad as a Postdoctora Feow ocea evromet motorg wth Natoa Ocea Techoogy Ceter Taj Cha He s currety a vce-drector at Isttute of TV ad Image Iformato ad a Feow the Luhu Ceter for Apped Mathematcs ad a Professor at Departmet of Mathematcs a TUTC Hs curret research terests are appromato of determstc sgas recostructo of radom sgas Pe Ye receved the BS Degree ad the MS degree mathematcs from Xame Uversty Fuja Cha 995 ad 998 respectvey He receved the PhD degree mathematcs from Bejg Norma Uversty From to 3 he wored the Isttute of mathematcs Chese academy of sceces as a postdoctora He s currety a Fu Professor at Schoo of mathematca sceces Naa Uversty He has pubshed more tha 6 joura ad coferece papers Hs curret research terests cude appromato theory umerca aayss quatum computg mache earg ad compressed sesg ACADEMY PUBLISHER