Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators Adia Chirilă a a Departet of Matheatics ad Coputer Sciece, Trasilvaia Uiversity of Braşov, 9 Eroilor Blvd., 500036, Braşov, Roaia Abstract. I this paper we itroduce a ew -Szász-Miraka operator based o a ew -epoetial fuctio. We derive various forulae for the oets, prove the uifor covergece of the seuece of operators to the idetity operator o copact itervals ad show a Voroovskaa type result.. Itroductio Throughout this paper we use the otatios that Thoas Erst itroduced i 8. Assue that R + \{}. The -aalogues of a real uber a ad of a iteger are defied by {a} : a, a R, { {} : k k + + + + if N 0 if 0. Ja Cieśliński 6 itroduced aother -iteger, which is otivated by the for of the -epoetial E defied below { {} : + if N +. 3 0 if 0 The correspodig -factorials are defied by { {}! : k {k} + + + + + if N if 0, 4 k! : k {k} + {}! if N +k if 0. 5 We assue fro ow o that 0,. I the stadard approach to -calculus there are two epoetial fuctios e {}!, < 6 00 Matheatics Subect Classificatio. Priary 4A36; Keywords. Liear positive operators, -Szász-Miraka operators, Voroovskaa theore. Received: 8 Noveber 06; Accepted: 4 February 07 Couicated by Sezaa Živković Zlataović Eail address: adia.chirila@uitbv.ro Adia Chirilă
A. Chirilă / Filoat 3:8 07, 567 568 568 ad E {}!, R. 7 Both -epoetial fuctios ca be represeted by ifiite products e k, E A ew -epoetial fuctio E is defied by, see 6, E : e E + k ad it ca be represeted as E k!, <. + k. 8 It was proved i 6 that E has better ualitative properties tha the stadard -epoetial fuctios. Therefore, we will use E to itroduce a ew -aalogue of the Szász-Miraka operators. We recall that the Szász-Miraka operators have the followig for S f k e f k! k for, 0 ad all fuctios f : 0, + R such that the series at the right-had side is absolutely coverget, see 3. Reark.. This fuctio space icludes i particular the fuctio subspace of all f : 0, + R such that f Me α, where 0, M 0 ad α R, see 3. Recetly ore -aalogues of these operators were itroduced, see for eaple 4, ad 0. We itroduce the followig defiitio of the -Szász-Miraka operator. Defiitio.. Let f C0, be a fuctio. The S, f k k {k} f, k! E where N, 0 < < ad 0 < + +. Reark.3. The iterval fied for the variable is chose such that the -epoetial E Reark.4. If we calculate is coverget. {k} k + k +, 3 the we ote that 9 0 0 {k} < +, k N, 4 which eas that the iterval fied for the variable also suits the doai of the fuctio f.
A. Chirilă / Filoat 3:8 07, 567 568 569 We set s k ; E k k k!, 5 which has the followig properties s k ; E k k!, 6 s k ; 0, N, 0 < <, 0 <. 7 Fro 6 it follows that S,, where is the costat fuctio with costat value. Reark.5. As their classical aalogues, the -Szasz-Miraka operators S, are also liear ad positive.. Moets I this sectio we copute the oets S, t, 0,,.... We begi to state soe recurrece forulae useful i the seuel. We cosider the followig test fuctios e t t, 8 where N. For the reader s coveiece we recall forula 3 fro 6 z + z E z. E z Lea.. We cosider N, N ad 0 < <. The the followig siple recurrece forulae hold true Proof We have S, t + S, t + S,t + S, t + S, t + E E k k S,t + +E + S, t +. k! {k}+ + k k E k! E {k} + S, t, S, t E S, t + k k k! {k} k k k! k {k}
A. Chirilă / Filoat 3:8 07, 567 568 560 k k {k} E k! E k k {k} E E k! S,t + S, t The secod recurrece forula is derived by eas of the followig forula {k} + {k } + k k. 9 I fact S, t + E + k E E k k E k! k k {k}+ + k k! E k k E k + k k k! k k {k } + k! k k k! {k} k k k! k + E E The proof of the third recurrece forula is based o, see 0, {k} {k } + k k {k } k {k} k {k } + k k k k! E {k } k k {k} k! + k k. 0 Hece S, t + + k E E E E k k k k! k {k}+ + k k! + k k k k E + k k k! k k + k k! k k k! k {k} k {k } k {k } + k k k k! {k } {k } +
E A. Chirilă / Filoat 3:8 07, 567 568 56 E k k {k } + E k! k + E k + k + {k } E + k! k Lea.. Let N, 0 < < ad N. The followig forula holds true S, t + k a,, where a +, {} a, + + + a,, 0, ad a 0,0, a,0 0, > 0, a, 0, >. Proof We use the first recurrece forula to facilitate the iductio by. S, t + + k S,t + a, S, t + +
+ + + A. Chirilă / Filoat 3:8 07, 567 568 56 + + k a, {} + a +, + + k a, + a, {} + +a, + + + Lea.3. Let N ad 0 < <. The followig forulae hold true + a, + S,, 3 S, t + S, t S, t 3 S, t 4 3, 4 + k + k + k + 3 + + 3. +, 5 3 3 + 3+ +, 6 4 6 + 3 3 4 7 + 0 + 7 + Proof We use the previous lea ad the forulae result fro the schee above. Reark.4. The uatities Q,k + k, where N, k N ad 0 < < ca be rewritte as Q,k + k +. We wat to study the covergece of the seuece S, to the idetity operator. To this ed we state the followig preliiary result. Lea.5. Let N be a seuece such that 0,, N, ad a as, 0 < a. The followig forulae hold true for 0, + li S + a, t + a + a, li S, t + a + a + a, li S, t 3 + a 3 + a + a 3 3, li S, t 4 + a 4 + a + a 4 4. Proof The forulae are derived fro the oes i Lea.3 ad the fact that li + 0. 7
3. Uifor Approiatio o Copact Itervals A. Chirilă / Filoat 3:8 07, 567 568 563 I order to prove the uifor covergece of the seuece S, o copact itervals to the idetity operator, we will use the followig result fro, Theore 3.5. Cosider a etric space X, d. Below the sybol FX stads for the liear space of all real-valued fuctios defied o X. For ay X we deote by d CX the fuctio d y : d, y, y X. Theore 3.. Let X, d be a locally copact etric space ad cosider a lattice subspace E of FX cotaiig the costat fuctio ad all the fuctios d X. Let L be a seuece of positive liear operators fro E ito FX ad assue that li L uiforly o copact subsets of X; li L d 0 uiforly o copact subsets of X. The for every f E C b X Defie li L f f uiforly o copact subsets of X. C 0, { f C0, M > 0 : f M +, 0} I our case X 0,, E C 0,, d, y y. Note that d e e +. Our ai result is the followig theore. Theore 3.. Let N be a seuece such that 0,, N ad as. The the seuece S, f coverges uiforly to f, o ay copact iterval i 0,, for ay f C 0, C b 0,. Proof The calculatios hold true for 0,. Sice the epressio as, there eists a rak N 0 such that if N 0 the 0, a 0,, where a R +. Fro Lea.5 with a we have li S,, li S, t, li S, t. We prove the uifor covergece of the oet of order. + + + a 0,. + + The uifor covergece of the oet of order results i the followig way + + + + + + + 3 + 4 a + a + + a 3 + + + a4 0,.
The followig covergeces hold for 0, a A. Chirilă / Filoat 3:8 07, 567 568 564 S,,, S, t,, S, t,, where eas uifor covergece. We further have li S, t t + li S, t S, t + S, 0 The coditios i the previous theore are therefore satisfied. Fially, we cosider the case whe is costat. Reark 3.3. Let 0,, fied. The followig forulae hold for S,,, S, t +,, 0, + where eas uifor covergece. The proof for S, is obvious. Sice 0,, we have li 0. Fro the secod forula i Lea.3 it follows that Moreover, which iplies uifor covergece. li S,t +. + + + + + + + < + + 0,, + + < I the followig we will cosider oly the case whe depeds o, i.e. ad it verifies the coditio a as, 0 < a. 4. Voroovskaa Theore We will first prove the followig result. Lea 4.. Let N be a seuece such that 0,, N, ad as. The the followig forulae hold true li S, t 0, li S, t, 8 9 li S, t 4 3. 30 Moreover, the covergeces are uifor o ay copact iterval 0, a, a > 0.
Proof First we copute the followig liits + A. Chirilă / Filoat 3:8 07, 567 568 565 li li + + + li + + + 0 because ad,. 3 3 li li + + + + li + + + + + + + + 3 0 because ad,. 3 The calculatios hold true for 0,. Sice the epressio as, there eists a rak N 0 such that if N 0 the 0, a 0,, where a R +. The first forula ca be proved i the followig way S, t S, t S, + + + + + + + + a 0, as accordig to 3. The secod forula ca be proved i the followig way S, t S, t S, t + S, Q,0 Q, + + + + 3 Q,0 Q, + 3 + + + + + + + 4 Q,0 Q, Q,0 Q 4, 4 4 3 + + + + + + 4 4 + + + + 3 3 + + + + 3 + +.
A. Chirilă / Filoat 3:8 07, 567 568 566 As, by eas of the forula 3 we obtai the liit 9. S, t 4 3 Q,0 Q, + + 3 3 + + 3 a4 + a 3 + a, which teds to 0 by 3 as, which proves the uifor covergece. The third forula ca be prove aalogously. We will use the followig result. Reark 4.. If we cosider the coditios i the previous lea, the forula 9 iplies that S, t 0 as, uiforly o copact itervals. Recall that if we cosider I R to be a iterval, f : I R a bouded fuctio ad δ 0, the the odulus of cotiuity of first order is defied as ω f, δ sup{ f u f v : u, v I, u v δ}. 33 Moreover, the followig ieuality holds for f bouded: ω f, δ + δ η ω f, η, δ, η > 0. The ai result of this sectio is the followig theore. Theore 4.3. Let N be a seuece such that 0,, N, ad as. The for ay fuctio f that is cotiuous ad bouded o 0, such that f ad f are cotiuous ad bouded o 0, the followig uifor covergece holds o ay copact iterval 0, a, a > 0 li S, f f f. 34 Proof Let the fuctios f, f ad f be cotiuous ad bouded o 0,. Let 0, be fied. We use Taylor s forula with the reaider i its itegral for We use the followig otatio t f t f + f t + t u f udu. It follows that R t, t t u f udu t t u f du + t f t t + t u f u f du. t u f u f du f t f + f t + f t t + t u f u f du. We cosider the fuctio rt; t t t u f u f du. It follows that f t f + f t + f t + rt; t. 35 The we prove that li t rt; 0. We prove that rt; is bouded for all t 0, a ad that rt; is cotiuous.
The A. Chirilă / Filoat 3:8 07, 567 568 567 Accordig to 35 it results that rt; ca be rewritte i the followig way rt; f t f f t f t. t f t f f t f t li rt; li l H t t t f t f f t l li H f t f li 0 because f is cotiuous. t t t Boudedess results fro the followig forula t rt; t t u f u f du t t t u f u f du t t u f u + f t du t t f t udu f. We prove cotiuity. If t, the we cosider by covetio that r; 0. If t, the rt; t t u f u f du. t By eas of the odulus of cotiuity of first order it results that f u f ω f, u ω f, t. It follows that rt; ω f, t for t. By applyig S, to 35 we obtai The S, f f S, + f S, t + f S, t + +S, rt; t. S, f f f S, t + f S, t + + S, rt; t. By applyig the Cauchy-Schwarz ieuality we obtai S, rt; t S, r t; S, t 4. 36 By ultiplyig 36 by it results that S, rt; t S, r t; S, t 4. 37 Accordig to 30 it results that The li S, t 4 3. S, rt; S, rt; S, ω f, t
S, + A. Chirilă / Filoat 3:8 07, 567 568 568 t ω f, η η ω f, η S, + ω f, η S η, t We cosider η S, t, which iplies that S, rt; ω f, S, t. We apply Theore 3. ad the forulas 8 ad 9 to coplete the proof. The author would like to thak professor Radu Păltăea for valuable coets. The author would also like to thak the referee for useful suggestios. Refereces F. Altoare, Korovki-type Theores ad Approiatio by Positive Liear Operators, Surveys i Approiatio Theory, Volue 5 00, pages 9-64. F. Altoare, M. Capiti, Korovki-type Approiatio Theory ad its Applicatios, de Gruyter Studies i Matheatics, Berli, 994. 3 F. Altoare, M. C. Motao, V. Leoessa, O a Geeralizatio of Szasz-Miraka-Katorovich Operators, Results i Matheatics, Volue 63 03, Nuber 3-4, 837-863. 4 A. Aral, A geeralizatio of Szász-Mirakya operators based o -itegers, Matheatical ad Coputer Modellig 47 008, 05-06. 5 A. Aral, V. Gupta, The -derivative ad applicatios to -Szasz Mirakya operators, Calcolo, Volue 43 006, Nuber 3, 5-70. 6 J. L. Ciesliski, Iproved -Epoetial ad -Trigooetric Fuctios, Appl. Math. Lett. 4 0, o., 0-4. 7 A. de Sole, V. Kac, O Itegral Represetatios of -Gaa ad -Beta Fuctios, Atti Accad. Naz. Licei Cl. Sci. Fis. Mat. Natur. Red. Licei 9 Mat. Appl. 6 005, o., -9. 8 T. Erst, A Coprehesive Treatet of -Calculus, Birkhäuser, 0. 9 V. Kac, P. Cheug, Quatu Calculus, Spriger-Verlag, 00. 0 N. I. Mahudov, O -paraetric Szasz-Miraka operators, Mediterraea Joural of Matheatics, Volue 7 00, Nuber 3, 97-3. N. I. Mahudov, Approiatio by the -Szasz-Miraka Operators, Abstract ad Applied Aalysis, Volue 0 0, ID 7547, 6 pages. R. Păltăea, Approiatio Theory Usig Positive Liear Operators, Birkhäuser, Bosto, 004. 3 D. F. Sofoea, Soe ew Properties i -Calculus, Geeral Matheatics, Volue 6 008, Nuber, 47-54. 4 A. Veţeleau, About -Berstei Polyoials, Revista Electroică MateIfo.ro Septeber, 00, 6-8.