Chebyshev-Grüss- and Ostrowski-type Inequalities

Transcription

1 Chebyshev-Grüss- ad Ostrowski-type Iequalities Vo der Fakultät für Mathematik der Uiversität Duisburg-Esse zur Erlagug des akademische Grades eies Doktors der Naturwisseschafte Dr. rer. at. geehmigte Dissertatio vo Maria Daiela Rusu aus Tirgu-Mures Gutachter: Prof. Dr. rer. at. Dr. h.c. Heier Goska Prof. Dr. math. Gacho Tachev Tag der müdliche Prüfug: 4. Juli 04

2 Fakultät für Mathematik Fachgebiet Mathematische Iformatik PhD Thesis Chebyshev-Grüss- ad Ostrowski-type Iequalities Maria Daiela RUSU July 5, 04 Supervised by Prof. Dr. Dr. h. c. Heiz H. Goska.

3 There are three routes to wisdom:. reflectio, this is the oblest,. imitatio, this is the easiest, 3. experiece, this is the bitterest. CONFUCIUS

4 To my family.

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6 Cotets Itroductio 7 Notatios ad symbols Prelimiaries 3. Moduli of smoothess ad K-fuctioals Moduli of uivariate fuctios defied o compact itervals of the real axis K-fuctioals ad the coectio to the moduli Moduli of uivariate fuctios defied o compact metric spaces 5..4 K-fuctioals ad the coectio to the moduli Moduli of cotiuity of fuctios defied o the product of two metric spaces Positive liear operators The Berstei operator Kig operators The Berstei-Stacu operator The Berstei-Durrmeyer operator with Jacobi weights The Hermite-Fejér iterpolatio operator The quasi-hermite-fejér operator The almost-hermite-fejér operator Covolutio-type operators Shepard-type operators A piecewise liear iterpolatio operator S The BLaC operator: Defiitios ad Properties The Mirakja-Favard-Szász operator The Baskakov operator The Bleima-Butzer-Hah operator Lagrage iterpolatio Uivariate Chebyshev-Grüss Iequalities 35. Auxiliary ad historical results Results o compact itervals of the real axis Applicatios for positive liear operators Results o compact metric spaces Mai results O Chebyshev s Iequality Pre- Chebyshev-Grüss iequalities o compact itervals of the real axis Applicatios for positive liear operators Pre- Chebyshev-Grüss iequalities o compact metric spaces 6 5

7 Cotets..5 Applicatios for positive liear operators Chebyshev-Grüss iequalities via discrete oscillatios Applicatios for positive liear operators Chebyshev-Grüss iequalities via discrete oscillatios for more tha two fuctios Bivariate Chebyshev-Grüss Iequalities o compact metric spaces Auxiliary ad historical results Bivariate positive liear operators Bivariate Berstei operators Products of Kig operators Products of Hermite-Fejér iterpolatio operators Products of quasi-hermite-fejér iterpolatio operators Products of almost-hermite-fejér iterpolatio operators Products of covolutio operators Bivariate Shepard operators Products of piecewise liear iterpolatio operators at equidistat kots Bivariate BLaC operators Products of Mirakja-Favard-Szász operators Products of Baskakov operators Products of Lagrage operators Mai Results Pre-Chebyshev-Grüss iequalities i the bivariate case Applicatios to bivariate positive liear operators Bivariate Chebyshev-Grüss iequalities via discrete oscillatios Applicatios to bivariate positive liear operators Uivariate Ostrowski Iequalities 8 4. Auxiliary ad historical results Over-iterates of positive liear operators A result of A. Acu ad H. Goska Applicatios ivolvig iterates of positive liear operators Applicatios ivolvig differeces of positive liear operators Bivariate Ostrowski Iequalities 7 5. Auxiliary ad historical results Bivariate positive liear operators Bivariate Berstei-Stacu operators Bivariate Berstei-Durrmeyer operators with Jacobi weights Mai results A Ostrowski iequality for the bivariate Berstei-Stacu operator A Ostrowski iequality for the bivariate Berstei-Durrmeyer operators with Jacobi weights Bibliography 37 6

8 Itroductio The subject of this thesis is placed at the iterface betwee the theory of iequalities ad approximatio theory. Chebyshev-, Grüss- ad Ostrowski-type iequalities have attracted much attetio over the years, because of their applicatios i mathematical statistics, ecoometrics ad actuarial mathematics. The classical form of Grüss iequality, first published by G. Grüss i [6], gives a estimate of the differece betwee the itegral of the product ad the product of the itegrals of two fuctios i C[a, b]. I the succesive years, may variats of this iequality appeared i the literature. The aim of this thesis is to clarify the termiology that was ot exactly preseted i a trasparet way util ow, to remember well-kow Chebyshev-Grüss- ad Ostrowski-type iequalities that have already bee studied ad to itroduce ew results, i both the uivariate ad bivariate case. These results ca the be geeralized to the multivariate case, but this remais to be studied i the future. We also wat to poit out that all of the iequalities of Chebyshev-Grüss-type give here are for two or more fuctios of the same type. Whe cosiderig the classical Grüss iequality, we observe that o the left-had side of the estimate is the well kow classical Chebyshev fuctioal [5], while the right-had side is of Grüss-type, i.e., it icludes differeces of upper ad lower bouds of the two fuctios i questio. The Grüss iequality for the Chebyshev fuctioal explais the o-multiplicativity of the itegratio. I our research, we are iterested i how o-multiplicative ca a liear fuctioal i the worst case be. I order to give a aswer to this questio, we cosider the geeralized Chebyshev fuctioal T L f, g := L f g L f Lg, for a positive liear fuctioal L, ad use the termiology "Chebyshev-Grüss-type iequalities", whe we talk about Grüss iequalities for special cases of geeralized Chebyshev fuctioals. We therefore obtai a geeral form of such estimates, T L f, g EL, f, g, where the right-had side is a expressio depedig o differet properties of L ad some kid of oscillatios of the fuctios i questio. Aother reowed classical iequality was itroduced by A. M. Ostrowski i [89] ad ca be give i a variety of forms. The Ostrowski-type iequalities we recall ad itroduce all give differet upper bouds for the approximatio of the average value by a sigle value of the fuctio i questio. The approach cosidered by A. M. Ostrowski ad a lot of other ivestigatios o the topic were carried out assumig differetiability properties of the fuctios. I compariso, A. Acu ad H. Goska [] show that such coditios are ot ecessary ad give a geeralizatio of Ostrowski s iequality for a arbitrary cotiuous fuctio f C[a, b] ad certai liear operators. O the right-had side the least cocave majorat of the 7

9 Cotets modulus of cotiuity of the give fuctio is used. Thus the upper bouds of our geeral form of Ostrowski s iequalities will ivolve least cocave majorats, fuctios from certai classes, their orm or a expressio derived from the fuctios i questio. Both cases have bee itesively ivestigated by S.S. Dragomir ad other authors see Chapters 9 i the recet moography of G. Aastassiou [9], Ch. XV o "Itegral iequalities ivolvig fuctios with bouded derivatives" i the book by D. S. Mitriović et al. [84]; the reader ca also cosult the books [85], [] ad the refereces therei. I the preset thesis we cotiue to cosider the recet method i which the fuctioal L is obtaied by composig a poit evaluatio fuctioal with a positive liear operator i both the Chebyshev-Grüss ad the Ostrowski process. The first approach i this directio was made i a paper by A. Acu, H. Goska ad I. Raşa [] from 0 ad is exteded i the preset thesis. Oe essetial feature of it is the systematic use of the least cocave majorat of the first order modulus of cotiuity which first appeared i this cotext i a paper by B. Gavrea ad I. Gavrea [40] mostly i the Ostrowski case. The use of the cocave majorat has the advatage, that the deviatio i the Chebyshev-type fuctioal is also measured for all cotiuous fuctios o a compact metric space ad ot oly for those havig certai regularity properties, such as satisfyig a Lipschitz coditio with expoet. Such iequalities are obtaied via the use of a suitable K fuctioal see the paper of R. Pǎltǎea [9]. I all our estimates for the Chebyshev-Grüss-type iequalities the secod momets of the positive liear operators i questio or a closely related quatity play a essetial role. That these two quatities may lead to differet upper bouds will be show by the use of several iterpolatio operators which do ot reproduce liear fuctios. We cosider both the cases of a compact iterval [a, b] ad the half-ope semiaxis [0,. A secod approach cosidered i the preset work is that of Chebyshev-Grüss iequalities via discrete oscillatios. The latter ideed competes with that via the cocave majorat i that there are situatios i which the first is better tha the secod ad vice versa. All the results are applied to a variety of well-kow positive liear operators i both the uivariate ad the bivariate cases. I cotrast to that, we show that also for certai o-positive Lagrage operators similar results ca be obtaied which, however, are of a less elegat form. The Ostrowski-type iequalities give i the ed of the thesis complete our presetatio i the spirit of the may papers dealig with both types of iequalities. Our results give there build up a short parallel of the oes give i the Chebyshev- Grüss case. The applicatios icluded are ot oly for special positive liear operators, but also for their iterates ad for differeces of such operators. Some of these applicatios are exteded to the bivariate case. The thesis cosists of five chapters. The first chapter comprises prelimiary istrumets that will be further used for derivig our results. This thesis is based upo some mai tools: the moduli of smoothess, the K fuctioal ad its coectio to the moduli, positive ad otecessarily positive liear operators. The moduli are give i two differet settigs, i.e., for fuctios defied both o compact itervals of the real axis ad o a com- 8

10 Cotets pact metric space, i the uivariate case. K fuctioals ad the way they are coected to the moduli are give i both frames. For the bivariate case, that will be also treated later o, we are oly iterested i the moduli of cotiuity of fuctios defied o the product of two metric spaces see Subsectio..5. Sectio. recalls may positive liear operators that have itesively bee studied i the literature. All of them reproduce costat fuctios, some of them do ot reproduce liear fuctios ad this last property will be a advatage i order to improve some iequalities. The operators that we illustrate here represet a iterestig variety. Most of them are defied o compact itervals, but we also cosider operators for fuctios defied o ifiite itervals. We discuss the well-kow Berstei operators but also some iterestig geeralizatios. The BLaC operators give a exotic touch to the survey. I the ed of the chapter, the Lagrage iterpolatio operator is also studied, i order to see what happes if positivity is ot take ito cosideratio. I the secod chapter we talk about Chebyshev-Grüss-type iequalities i the uivariate case. First some auxiliary ad historical results are give, i the two settigs that were metioed before. These results are recalled i order to motivate our research ad because some of them will be slightly improved. Applicatios of the auxiliary results ivolvig some positive liear operators are reviewed. Some remarks ad results cocerig Chebyshev s iequality are also preseted. We give aother proof for a Chebyshev-Grüss-type iequality ivolvig a positive liear fuctioal L, a iequality that was prove i aother way i []. We the itroduce pre-chebyshev-grüss-type estimates i both settigs, usig secod momets, first absolute momets ad quatities ivolvig differeces of secod ad first momets. We the apply the mai results to the positive liear operators discussed i the first chapter. For these applicatios, oscillatios expressed by the least cocave majorat of the first order modulus are used i the first place. The use of such oscillatios icludes all poits i the cosidered itervals, ad this is the reaso why a ew approach ivolvig less poits arises. The discrete oscillatios defied i Subsectio..6 represet the grouds upo which this approach was costructed. The discrete liear fuctioal case is itroduced ad applied to the Lagrage operators. I case of positivity, we apply the discrete positive liear fuctioal case to some positive liear operators. Of great iterest here are the sums of squares of the fudametal fuctios of the operators, which eed to be miimized. Due to this ew approach, we ca also give Chebyshev-Grüss-type iequalities for operators defied for fuctios give o ifiite itervals see Subsectios..7.4,..7.5 ad Whe talkig about discrete oscillatios, we give Chebyshev-Grüss-type iequalities for more tha two fuctios at the ed of this chapter. This is motivated by the last sectio of article [], where the authors itroduced a iequality o a compact metric space for more tha two fuctios, usig the least cocave majorat. We compare our result to theirs. The third chapter exteds the results from the uivariate to the bivariate case. We use the method of parametric extesios ivolvig the product of two compact metric spaces. Auxiliary ad historical results are also recalled i the first part. We the choose some of the operators preseted i the begiig ad costruct their tesor products. For these operators we also defie the first, secod ad first absolute momets, which we will eed for our mai results i Sectio 3.3. The applicatios are 9

11 Cotets give for both the approach with the least cocave majorat ad the oe via discrete oscillatios. The purpose of the fourth ad fifth chapters is to complete this work, i the sese that we also cosider uivariate ad bivariate Ostrowski-type iequalities. I Sectios 4. ad 5. we agai recall some historical results, iequalities that were further studied ad modified. I Sectio 4. we give a result that modifies i some sese the iequality give by A. Acu ad H. Goska i []. Some additioal results are give i the form of corollaries. Corollary 4..4 is applied to iterates of differet positive liear operators. Moreover, Corollary 4.. is also applied i the case of differeces of positive liear operators, as ca be see i Sectio 4.4. The last chapter itroduces two examples of Ostrowski-type iequalities i the bivariate case. The two applicatios give here are for products of Berstei-Stacu ad Berstei-Durrmeyer operators with Jacobi weights. I both cases, we get Ostrowski-type iequalities with or without ivolvig the iterates of the operators. The limit of the iterates of the positive liear operators is also ivestigated. There is a coectio betwee the Ostrowski ad the Grüss iequalities, which explais the term "Ostrowski-Grüss-type iequalities" that was ofte used i the literature. For clarity, we emphasize that we exclusively use the term whe the lower boud is the error term i a rather simple quadrature formula like i Ostrowski s article [89], while the upper boud cotais differeces of bouds as used i the paper by G. Grüss [6]. I order to complete the historical remarks, the followig remiders appear to be i order. It seems that the term "Ostrowski-Grüss-type iequality" was coied by Dragomir et al. i [33]. The term also appeared i a paper by Ceroe et al. see [3]. A more substatial paper from 000 usig the term is oe by Matić et al. see [79]. For more details, the reader should cosult the papers [55], [56], [3] ad the refereces therei. Ackowledgmet Fially, I would like to express my sicere gratitude to my advisor Prof. Dr. Dr.h.c. Heier Goska for the cotiuous support of my PhD study ad research, for his patiece, motivatio, ethusiasm, ad immese kowledge. His guidace helped me throughout these very iterestig years ad I will always appreciate it. Furthermore, a very special thaks goes to Prof. Dr. Ioa Raşa for his permaet ecouragemet, patiece ad advice. I could ot have imagied havig a better metor for my PhD research. His love ad passio for mathematics ad the joy of sharig his kowledge with others have bee a real ispiratio. Besides my advisors, I would also like to thak my fellow colleagues from the Departmet of Mathematical Computer Sciece at the Uiversity of Duisburg-Esse. I particular, I am grateful to Michael Woziczka ad Elea-Doria Stăilă, for the stimulatig discussios ad for all the fu we have had i the last six years. I also wat to thak some very special metors, from whom I have leared a lot: Dr. Aa Maria Acu, Prof. Dr. Maria Mureşa, Prof. Dr. Margareta Heilma. It was a pleasure workig with them. My sicere thaks also goes to the Uiversity of Duisburg-Esse, for gratig me a scholarship for part of my PhD studies. 0

12 Notatios ad symbols I this work we shall ofte make use of the followig symbols: := is the sig idicatig equal by defiitio". a:=b" idicates that a is the quatity to be defied or explaied, ad b provides the defiitio or explaatio. b=:a" has the same meaig. N the set of atural umbers, N 0 the set of atural umbers icludig zero, R the set of real umbers, R + the set of positive real umbers, [a, b] a closed iterval, a, b a ope iterval. Let X be a iterval of the real axis. BX the set of all real-valued ad bouded fuctios defied o X. L p X the class of the p-lebesgue itegrable fuctios o X, p. f p is the orm o L p X defied by f p := X f x dx /p, p. CX the set of all real-valued ad cotiuous fuctios defied o X. C b X C[a, b] the set of all real-valued fuctios, defied by C b X := CX BX. the set of all real-valued ad cotiuous fuctios defied o the compact iterval [a, b]. For f BX or f CX X, d, X, d X Metric spaces equipped with metric d or d X. dx Diameter of the compact metric space X, d. f is the Chebyshev orm or sup-orm, amely f := sup{ f x : x X}. C r [a, b] the set of all real-valued, r-times cotiuously differetiable fuctio, r N. Lip r M the set of all C[a, b] fuctios that verify the Lipschitz coditio: f x f x M x x r, x, x [a, b], 0 < r, M > 0. [a, b], N 0 the liear space of all real polyomials with the degree at most. X R x R, X = a arbitrary set. f X, f Y Partial fuctios of bi-or multivariate fuctios. e deotes the th moomial with e : [a, b] x x R, N 0. For a fuctio f : X R, X a iterval of the real axis we have: ω... exclusively used to deote moduli of smoothess of various kids. ω d f, t Metric modulus of cotiuity, defied for fuctios f CX, X, d a compact metric space, t 0. ω d, ω Least cocave majorat of a metric modulus of cotiuity. ω f ; t uivariate -st order modulus of smoothess, defied usig

13 Cotets h f x Differece of order with icremet h ad startig poit x. ω k f ; t, t Bivariate total modulus of smoothess of order k, defied for fuctios f CX, X R compact, ad t i 0, i =,. D r or f r r th derivative of the fuctio f C r [a, b]. [x 0,..., x m ; f ] m th divided differece of f FX o the ot ecessarily distict kots x 0,..., x m X. a b are the risig factorials a b := b a + i, a R, b N 0, where :=. a b y [m,h] i=0 i=0 are the fallig factorials a b := b a i, a R, b N 0, where :=. i=0 the factorial power of step h R defied by: y [m,h] := m y ih, m N 0. As above :=. i=0 X, X, Fuctio space X equipped with the orm X X, X the semiorm X. BX, B R X Space of all real-valued ad bouded fuctios om the set X =. CX, C R X Space of all real-valued ad cotiuous fuctios o the topological space X. Lip r Space of Lipschitz cotiuous fuctios with expoet r. C r, C r I ad Space of all real-valued fuctios o I = [a, b] havig cotiuous derivatives C r [a, b] up to order r. If ot otherwise idicated, deotes the Chebyshev max, sup orm., X Sometimes used to deote the Chebyshev orm, ad the Chebyshev orm over the set X, respectively. g Lipr Lipschitz semiorm of a fuctio g CX, d; smallest Lipschitz costat. I, I X, I Y Idetity operator caoical embeddig o a fuctio space. X L, Y L Parametric extesio of the uivariate operator L. L [X,Y], L Caoical orm of a operator L, usually mappig a ormed space X ito a ormed space Y; the secod otatio is used whe it is clear what X ad Y are. O, o Ladau otatios. [x] the itegral part of a real umber x i.e., the greatest itegral umber, that does t exceed x. Supp f Support of a fuctio f. Suppµ Support of a measure µ. i=0 i=0

14 Prelimiaries. Moduli of smoothess ad K-fuctioals The moduli of smoothess cotiuity ad K-fuctioals, used i coectio to the moduli, will be of iterest i the whole thesis. We recall defiitios ad properties of these moduli for real-valued ad cotiuous fuctios defied both o a compact metric space X, d ad o a compact iterval [a, b], a < b, of the real axis. We will exted the results obtaied i the compact metric space to the bivariate case... Moduli of uivariate fuctios defied o compact itervals of the real axis Whe we wat to establish the degree of covergece of positive liear operators towards the idetity operator, we use first kid moduli of smoothess. Defiitio... For a fuctio f C[a, b] ad t 0, we have ω f ; t := sup{ f x + h f x : x, x + h [a, b], 0 h t}. Above we gave oe defiitio of the first moduli of smoothess. This was preseted i the Ph. D. thesis of D. Jackso [65]. The ame of the modulus comes from the followig propositio. Propositio... Let f C[a, b] ad t > 0. The the followig properties hold: a If lim t 0 + ω f ; t = 0 the f is cotiuous o [a, b]. b The followig equivalece is give: f Lip r M if ad oly if ω f ; t M t r, for 0 < r ad M > 0. A very importat tool that we use is the least cocave majorat of the modulus of cotiuity ω f ;. This is give by ω f ; t = ω f ; t t xω sup f ;y+y tω f ;x y x, for 0 t b a, := 0 x t y b a,x =y ω f ; b a = ω f ; b a, if t > b a, from which we get a relatioship betwee the differet moduli: ω f ; ω f ; ω f ;... For more properties of the moduli, icludig ω f ;, see [45]. N.P. Koreičuk gave a proof i [70] for the relatioship betwee the fuctio ω f ; ad its least cocave majorat ω f ; ω f ; ξ ε + ξ ω f ; ε, 3

15 Prelimiaries for ay ε 0 ad ξ > 0. It was also showed that this iequality caot be improved for each ε > 0 ad ξ =,, K-fuctioals ad the coectio to the moduli Whe we are iterested i measurig the smoothess of fuctios, we ca also use the so-called Peetre s K-fuctioal. It was itroduced, as the ame suggests, by J. Peetre i 968 [9] ad ca be defied i a very geeral settig. However, we eed the followig form i this thesis. Defiitio..3. For ay f C[a, b], t 0 ad s =, we deote K s= f ; t [a,b] := K f ; t; C[a, b], C [a, b] to be Peetre s K-fuctioal of order. := if { f g + t g g C [a,b] } We first recall some properties of the above K-fuctioal, prove by P.L. Butzer ad H. Beres for s see [9]. For other refereces, see [3] ad [0]. Lemma..4. see Propositio 3..3 i [9] i The mappig K f ; t : R + R + is cotiuous especially at t = 0, i.e., lim K f ; t = 0 = K f ; 0. t 0 + ii For each fixed f C[a, b], the applicatio K f ; : R + R + is mootoically icreasig ad a cocave fuctio. iii For arbitrary t, t 0 ad f C[a, b], oe has the iequality K f ; t t max{, t } K f ; t. iv For arbitrary f, f C[a, b] ad t 0, we have K f + f ; t K f ; t + K f ; t. v For each t 0 fixed, K ; t is a semiorm o C[a, b], such that holds, for all f C[a, b]. K f ; t f vi For a fixed f C[a, b] ad t 0, the idetity K f ; t = K f ; t is true. The followig equivalece relatio gives a first importat lik betwee the K- fuctioal ad the moduli see [66]. Theorem..5. There exist costats c ad c depedig oly o the iteger s = ad [a, b], such that c ω f ; t K f ; t c ω f ; t, for all f C[a, b] ad t > 0. 4

16 . Moduli of smoothess ad K-fuctioals Remark..6. I geeral, for s, o sharp costats c ad c are kow, that satisfy the above iequality. I particular, for the cases s =,, so also for our case, such sharp costats exist, as we will see i the followig result kow as Brudyǐ s represetatio theorem. This theorem is of crucial importace for the rest of this thesis, as a ice coectio betwee K f ; t [a,b] ad the least cocave majorat defied i... Lemma..7. Every fuctio f C[a, b] satisfies the equality K f ; t; C[a, b], C [a, b] = ω f ; t, t 0... For details ad proofs cocerig this lemma, see R. Păltăea s article [9], the book [03], the book of R.T. Rockafellar [98] or the book [3]. The above equality.. ca be writte i the followig way K f ; t ; C[a, b], C [a, b] = ω f ; t, t Moduli of uivariate fuctios defied o compact metric spaces I this subsectio we cosider real-valued, cotiuous fuctios of oe variable ad recall defiitios ad properties i compact metric spaces. Let f CX = C R X, d, where C R X, d is the space of all real-valued ad cotiuous fuctios defied o the compact metric space X, d, with diameter dx > 0. We have the followig defiitio for the metric modulus of cotiuity see [45] ad its least cocave majorat. This is a geeralizatio of Defiitio.. ad equality... Defiitio..8. Let f CX. If, for t [0,, the quatity ω d f ; t := sup { f x f y ; x, y X, dx, y t} is the metric modulus of cotiuity, the its least cocave majorat is give by t xω sup d f ;y+y tω d f ;x y x for 0 t dx, ω d f ; t = 0 x t y dx,x =y ω d f ; dx if t > dx...4 K-fuctioals ad the coectio to the moduli For 0 < r, let Lip r be the set of all fuctios g CX with the property that g Lipr := sup gx gy /d r x, y <. dx,y>0 Lip r is a dese subspace of CX equipped with the supremum orm, ad Lipr is a semiorm o Lip r. We also eed to defie the K-fuctioal with respect to Lip r, Lipr, which is give by Kt; f ; CX, Lip r := if { f g + t g Lipr }, g Lip r 5

17 Prelimiaries for f CX ad t 0. The lemma of Brudyǐ [83] that gives the relatioship betwee the K-fuctioal ad the least cocave majorat of the metric modulus of cotiuity will also be used i the proofs that follow. Lemma..9. Every cotiuous fuctio f o X satisfies t K ; f ; CX, Lip = ω d f ; t, 0 t dx. For more details about the metric moduli of smoothess, see [45]...5 Moduli of cotiuity of fuctios defied o the product of two metric spaces I this sectio we cosider products of two compact metric spaces ad describe various moduli of smoothess cotiuity of fuctios defied o such products. For more details about parametric extesios ad tesor products, see [45] ad [30]. We take X, d X ad Y, d Y two compact metric spaces. The cartesia product X Y, equipped with the product topology, ad d X Y a metric o X Y that geerates this topology, is also a compact metric space. The metric satisfies the followig properties: d X Y x, r, ˆx, r = d X x, ˆx for all r Y ad for all x, ˆx X, d X Y s, y, s, ŷ = d Y y, ŷ for all s X ad for all y, ŷ Y. Also for the metric d X Y it holds d X Y x, y, ˆx, ŷ = d X x, ˆx + d Y y, ŷ, for x, y, ˆx, ŷ X Y. These properties isure that we have iterestig relatioships betwee ω dx Y ad moduli defied usig d X ad d Y, as we will see i the sequel. We ow defie the total modulus of cotiuity of a fuctio f CX Y. Defiitio..0. For ay fuctio f CX Y ad t, t R +, the total modulus of cotiuity of f with respect to d X ad d Y is give by ω total,dx,d Y f ; t, t := sup{ f x, y f ˆx, ŷ : d X x, ˆx t, d Y y, ŷ t }. Regardig the relatioship betwee ω X Y ad the total modulus of cotiuity, we have the followig result. Propositio... see Lemma. i [45] Let X, d X, Y, d Y ad d X Y give as above. The for ay t, t R + we have ω total,dx,d Y f ; t, t ω dx Y f ; t + t. I our applicatios we will mostly use the Euclidea metric. For x = x i i= m ad y = y i i= m, this is give by d x, y = m i= x i y i. 6

18 . Positive liear operators The Euclidea metric ca be geeralized, i the sese that we get the more geeral metrics d p, p, give by d p x, y = m x i y i p p, p < ad i= d x, y = max{ x i y i : i m}. I the same way, oe ca obtai differet metrics, all geeratig the same topology ad correspodig moduli, deoted by ω dp. Remark... For details cocerig the relatioship betwee moduli of smoothess ad K-fuctioals of differet orders i the bivariate multivariate case, we give as refereces the book of L.L. Schumaker [0] ad the refereces therei.. Positive liear operators We give some defiitios ad properties regardig positive liear operators see [0]. Some examples of such operators will also be cosidered, operators that will be used i order to illustrate our results. Defiitio... Let X, Y be two liear spaces of real fuctios. The the mappig L : X Y is a liear operator if Lα f + βg = αl f + βlg, for all f, g X ad α, β R. If for all f X, f 0, it follows L f 0, the L is a positive operator. X ad Y ca be differet kids of spaces, as we will show i the sequel. Propositio.. Properties of positive liear operators. Let L : X Y be a positive liear operator. The we have the followig iequalities: i If f, g X with f g, the L f Lg. ii For all f X, we have L f L f. Defiitio..3. Let L : X Y, where X Y are two liear ormed spaces of real fuctios. To each operator L we assig a o-egative umber L, give by L := sup L f = f X, f = L f sup f X,0< f f. We make a covetio, that if X is the zero liear space, the ay operator L : X Y must be the zero operator ad has the zero orm assiged to it. is called the operator orm. If we take X = Y = C[a, b], we ca state the followig: Corollary..4. For L : C[a, b] C[a, b] beig positive ad liear, it follows that L is also cotiuous ad it holds: L = Le 0. 7

19 Prelimiaries.. The Berstei operator The Berstei operator is maybe the most well-kow example of a positive liear operator. It was itroduced by S. N. Berstei i 9 see [5] ad it was used to prove the fudametal theorem of Weierstrass see [9]. For properties ad further details about the Berstei polyomials, see the book of R. A. devore ad G. G. Loretz [3]. Cosiderig the degree N, x [0, ] ad a fuctio f R [0,], we have the followig defiitio: Defiitio..5. The th degree Berstei polyomial B f : [0, ] R of the fuctio f is defied by k B f := f b,k, k=0 where the Berstei fudametal fuctios are { b,k x := k xk x k, for 0 k, 0, otherwise. Propositio..6. Properties of the Berstei operator see [3] a R [0,] is edowed with the caoical operatios of additio ad scalar multiplicatio for fuctios, so the Berstei operator B is a liear operator from R [0,] oto the subspace Π [0, ] of polyomials of highest degree o the iterval [0, ]. b The Berstei operator is discretely defied, sice B f oly depeds o the + fuctio values f k, 0 k. c For the case = 0, the Berstei operator is ot defied. Sometimes, it is set to be B 0 f := f 0. d If f is o-egative, the this also holds for B f. e We have edpoit iterpolatio as follows: B f ; 0 = f 0, B f ; = f. f The Berstei operator reproduces liear polyomials, i.e., for every liear polyomial L Π [0, ], we have B L = L. Remark..7. Because of the above propositio, we say that the Berstei operator is a positive, liear operator. The secod momet of the Berstei operator is give i the sequel. Propositio..8. It is well kow that the secod momet of the Berstei polyomial is equal to B e x x x ; x =, where e i x = x i, for i 0. 8

20 . Positive liear operators Defiitio..9 Forward differeces of r th order with icremet. Let a k a fiite or ifiite sequece of real umbers. For suitable idices k ad, we deote with a k the differece a k+ a k betwee two elemets of a sequece with differece step size. More geerally, for suitable r N 0, deote { a k, if r = 0, r a k := r a k, otherwise, the Forward- Differece of r Order with icremet step step size. Propositio..0 Derivatives of the Berstei Polyomial. Let 0 r. The the r th derivative B f r of the th Berstei polyomial has the form: B f r r k = r r f b r,k, k=0 where r =... r + is the r th decreasig factorial of terms. Corollary... We obtai for the Berstei polyomial of the first derivative i the edpoits: a B f 0 = f f 0 b B f = f f.. Kig operators P. P. Korovki [7] itroduced i 960 a result statig that if L is a sequece of positive liear operators o C[a, b], the for each f C[a, b] holds, if ad oly if lim L f x = f x lim L e i x = e i x for the three fuctios e i x = x i, i = 0,,. There are a lot of well-kow operators, like the Berstei, the Mirakja-Favard-Szász ad the Baskakov operators, that preserve e 0 ad e see [67]. However, these operators do ot reproduce e. We are ow iterested i a o-trivial sequece of positive liear operators L defied o C[0, ], that preserve e 0 ad e : L e 0 x = e 0 x ad L e x = e x, = 0,,,.... I [67] J. P. Kig defied the followig operators. Defiitio... Let r x be a sequece of cotiuos fuctios with 0 r x. Let V : C[0, ] C[0, ] be give by: k V f ; x = r x k r x k f k = k=0 v,k x f k=0 k, for f C[0, ], 0 x. v,k are the fudametal fuctios of the V operator. 9

21 Prelimiaries Remark..3. For r x = x, N, the positive liear operators V give above reduce to the Berstei operator. Propositio..4 Properties of V.. V e 0 = ad V e ; x = r x;. V e ; x = r x + r x ; 3. lim V f ; x = f x for each f C[0, ], x [0, ], if ad oly if lim r x = x. 4. The secod momet i the geeral case is give by V e x ; x := r x + r x xr x + x = r x r x + r x x,.. where 0 r x are cotiuous fuctios. For special "right" choices of r x = r x, J. P. Kig showed i [67] that the followig theorem holds. Theorem..5. see Theorem.3. i [50] Let V N be the sequece of operators defied before with { r x = x, for =, r x := r x = + x + 4, for =, 3,... The we get V e ; x = x, for N, x [0, ] ad V e x ; x = e x. V is ot a polyomial operator. The fudametal fuctios of this operator, amely v,k x = r k x k rx k, satisfy k=0 v,k x =, for =,,.... Propositio..6 Properties of r. i 0 r x, for =,,..., ad 0 x. ii lim r x = x for 0 x. The secod momet of the special Kig-type operators V is give by so we discrimiate betwee two cases. V e x ; x = xx r x, 0

22 . Positive liear operators The first case is =, so r x = x ad the secod momet is V e x ; x = x x. For the secod case, =, 3,..., we have rx = + so the secod momet is see Theorem..4: x + 4, V e x ; x = xx r x = xx V e ; x. The iterest is ow i fidig r, such that the secod momet is miimal. Such a approach was give i the thesis of P. Piţul [93]. There it was prove, that if the fuctio 0, x [0, r mi x := x, x [, ], x, ] is give, the the miimum value of the secod momet is obtaied. For the miimal secod momets of V mi, the followig represetatio was give x, x [0, V mi e x ; x := x x 4, x [, ] x, x, ]..3 The Berstei-Stacu operator The followig geeralizatio of the classical Berstei operators was itroduced by D.D. Stacu i 97 see [09], [49]. For α, β, γ positive umbers with α 0 ad 0 β γ, the defiitio of the Berstei-Stacu positive liear operators S <α,β,γ> : C[0, ] Π is: S <α,β,γ> f ; x := s α k + β,k x f, x [0, ],.. + γ k=0 where s α,k x are the fudametal polyomials s α x,k x := [k, α] x [ k, α] k [, α], x [0, ], k = 0,...,. Here x [k, α] is the factorial power of order k with step α of x, i.e., x [0, α] =, x [k, α] = x x + α... x + k α, k N. Whe α = β = γ = 0, we obtai the defiitio for the Berstei operators. This is the reaso why they are called "Berstei-Stacu"-type operators.

23 Prelimiaries We are iterested i the case α = 0. The the operators S <0,β,γ> accordig to.. as S <0,β,γ> f ; x = b,k x f k=0 k + β, + γ ca be writte for b,k the fudametal Berstei polyomials. Oe importat result that we will use later o ivolves powers of the operator S <0,β,γ>. For a detailed proof see the proof of Theorem 4.3 i [93]. Theorem..7. If N is fixed, the for all f C[0, ], x [0, ] [ lim m S <0,β,γ> ] m f ; x = b0 e 0 x, where b 0 = b 0 f is a covex combiatio of the values of the fuctio f that appear i the operator s defiitio, amely j + β b 0 = d j f, + γ j=0 with suitable d j R...4 The Berstei-Durrmeyer operator with Jacobi weights The Durrmeyer operators, which were itroduced by J.L. Durrmeyer i 967 i his thesis [34], were give o L [0, ] as a modificatio of the Berstei operators. They were the geeralized as follows: We cosider the Jacobi weight o 0, to be w α,β x = x α x β, α, β >, ad deote L 0, the space of Lebesgue-measurable fuctios f o 0,, such w α,β that the orm f w α,β := f xw α,β xdx is fiite. The operators D α,β : L 0, C[0, ] are defied by w α,β D α,β f ; x := b,k x k=0 0 0 b,k t f tw α,β tdt 0 b,k tw α,β tdt where b,k are the Berstei fudametal fuctios, ad they are called the geeralized Durrmeyer operators w.r.t. the Jacobi weight w α,β. They are also called Berstei-Jacobi operators because for ay fuctio f C[0, ], D α,β f ca be writte as a liear combiatio of Jacobi polyomials. These,

24 . Positive liear operators operators have the properties of beig self-adjoit ad commutative, properties iherited from the classical Durrmeyer operators. More details ad properties ca be foud i [90], [3], [4]. I order to ivestigate the behaviour of the over-iterates of the Berstei-Durrmeyer operator, we recall the followig theorem that was prove i [93]. Theorem..8. see Theorem 4.40 i [93] If N is fixed, the for all f itegrable o [0, ] ad x [0, ], we have [ lim m D α,β ] m f ; x = 0 f t t α t β dt e 0 x. As oe ca coclude from the above result, the over-iterates of the operator ted toward a costat fuctio...5 The Hermite-Fejér iterpolatio operator L. Fejér see [38] gave oe proof of Weierstrass s approximatio theorem by meas of iterpolatio polyomials. We recall here his result as follows. The classical Hermite-Fejér iterpolatio operator is a positive liear operator defied by T x H f ; x := f x k x x k, x x k= k where f C[, ] ad x k = cos k π, k, are the zeroes of T x = cos arccosx, x, the th Chebyshev polyomial of the first kid. Remark..9 Properties of H, see [38], [59]. H f, x is the uique polyomial of degree such that H f ; x k = f x k, for k =,,...,, ad H f ; x k = 0, k =,,...,. It is well-kow that, for all x [, ], T x x x k 0 x x k ad T x x x k =. x x k= k The result of L. Fejér is preseted ext. Theorem..0. L. Fejér, [38] If f C[, ] the lim H f f = 0, where deotes the uiform orm o the space C[, ]. 3

25 Prelimiaries Next, let x j be the ode earest to x, for x. If two such odes exist, let x j be either of them. Aother result we will eed i the sequel is a lemma give by O. Kiš see [68], p. 30. Lemma... O. Kiš Let x = cos θ, x k = cos θ k, θ k = k π, for k =,,..., ad x j be the ode closest to x. The θ θj π cos θ. The secod momet of this operator is give as follows. H e x ; x = x k x T x x x k x x k= k = T x = T x. k= x x k } {{ } =..6 The quasi-hermite-fejér operator The quasi Hermite-Fejér operators were first cosidered by P. Szász i [4]. The iterest was to fid a uiquely defied polyomial of degree less tha or equal to +, that satisfies the coditios L f ; x v = f x v, for 0 v +, L f ; x v = 0, for v, for the fudametal odes x,..., x,, x 0 =, x + =. The quasi-hermite-fejér iterpolatio operator Q : C[, ] Π + with arbitrary odes has the form: for wx = c Q f ; x := f v= + v= x w wx + f f x v x xv [ + c v x x v ] x x v, c = 0 ad + x w wx + wx w x v x x v, c v = x v x v w x v w x v, v. For all x [, ], + c v x x v 0. From this iequality we ca say that Q is a positive liear operator. All of this holds especially for the zeroes of a Jacobi-Polyomial P α,β, for 0 α, β. 4

26 . Positive liear operators We are iterested i the approximatio behaviour of a special case of kots, meaig Chebyshev kots of the secod kid. For the zeroes of the Jacobi-Polyomials P, for α = β =, the quasi-hermite-fejér iterpolatio polyomial has the form Q f ; x := + f x v F,v x Ux, v=0 where U x is the th Chebyshev polyomial of the secod kid with roots x v = cos v + π, for v ad with x, for v = 0, + x F,v x := x v x, for + x x v v, +x, for v = +. + Q is a positive liear operator, for all. It holds si + arccos x U x =, siarccos x ad we have the relatios: U ± = + ad x v U x v = v+ +. This operator also reproduces costat fuctios. The secod momet of the operator is give by Q e x ; x = x U x +, while the first momet is Q e x; x = x U x {T + + x x U x}, for T + x = cos + arccos x...7 The almost-hermite-fejér operator The so-called almost-hermite-fejér iterpolatio was studied by may authors. For referece, we recall a paper i which a survey presetig results i this settig, icludig our particular case, is give, paper writte by H. Goska i 98 [43]. Let us cosider a r, s Hermite-Fejér iterpolatio operator F r,s; : C[, ] Π +r+s ad the image of a fuctio f C[, ] uder such a operator. The we get the uiquely determied algebraic polyomial that, for a fixed sequece of odes = x 0 > x >... > x > x + =, 5

27 Prelimiaries satisfies the + r + s coditios F r,s; f ; x v = f x v, F r,s; f x v = 0, for v, F r,s; f ; = f for r, F r,s; f ρ = 0, for ρ r, F r,s; f ; = f for s, F r,s; f σ = 0, for σ s. Give the fact that the odes x, x,..., x, are distributed like this, it is atural to aalyse r, s Hermite-Fejér formulas based o the edpoits ± ad the roots x,..., x of the Jacobi polyomials P α,β, α, β >. This process was treated by may authors see [43], [7], [69], [8] ad the refereces i these papers. The the correspodig operators are F α,β r,s;. We cosider the particular case r, s =, 0 ad the correspodig operators F α,β,0; are the almost-hermite-fejér-iterpolatio operators. They are give by the formula F α,β wx,0; f ; x := f w + f x v x [ + c x vx x v ] l v x, v= v where l v deotes the v th Lagrage fudametal polyomial ad Furthermore, wx = x x v. v= c v = w x v x v w x v. If + c vx x v 0 for all x [, ], the the above operator is positive ad liear. If the odes x,..., x are the zeroes of a Jacobi polyomial P α,β, the this is the case for all if ad oly if α, β [0, ], 0]. Oe iterestig case that we ow cover is for α, β =,, i which case the correspodig operators are positive ad the property of uiform covergece for every f C[, ] holds. For this choice of α, β the operators have the followig form see [5]: where F,,0; f ; x := f wx w + f x v x xx v x v= v xv l v x, wx = si + arccos x si arccos x, x v = cos v π, v, + ad l v is the vth Lagrage fudametal polyomial. For the above positive operators it holds F,,0; e 0 ; x =. For these operators F,,0; based upo the roots of the Jacobi polyomials P, ad the edpoit, we have for all that the first absolute momet see [43] ad [73] is give by F,,0; e x ; x c + x l, + 6

28 . Positive liear operators for a suitable costat c. The secod momet is give by the followig equality see [4]: F,,0; e x x wx ; x =, 3 while the absolute value of the first momet see [4] satisfies F,,0; e x; x = + x wx [ x w x + x wx] 4 x wx,...8 Covolutio-type operators These types of operators were treated by may authors, like J. -D. Cao, H. Goska ad H. -J. Wez see []. The followig cocepts, as well as the give applicatios, ca also be foud i [0]. Oe of the first authors to give the followig defiitio was H. G. Lehhoff i [74]: Defiitio... For the case X = [, ], give a fuctio f CX ad ay atural umber, the covolutio operator G m is give by G m f ; x := π π π f cosarccosx + υ K m υdυ, where the kerel K m is a positive ad eve trigoometric polyomial of degree m satisfyig π K m υdυ = π, π meaig that G m e 0 ; x =, for x X. It is clear that G m f ; is a algebraic polyomial of degree m ad the kerel K m has the followig form: K m υ = m + ρ k;m coskυ, k= for υ [ π, π]. We also eed aother result that goes back to H.G. Lehhoff [74]: Lemma..3. For x X, the equality G m e x ; x = x { 3 ρ ;m + ρ ;m holds. Here e deotes the first moomial give by e t = t for t. } { + x } ρ ;m The first momet of the covolutio-type operator see [0] is give by: G m e x; x = x [ρ ;m ]. The above lemma gives the secod momet of the covolutio-type operator, which, alog with the first momet, will be eeded i the sequel. Furthermore, we take ito accout differet degrees m ad differet covolutio operators, respectively. 7

29 Prelimiaries..8. Covolutio operators with Fejér-Korovki kerel If we cosider degree m =, for N, the Fejér-Korovki kerel is give by K υ = si π + cos + υ + cosυ cos π + with π ρ ; = cos, ρ + ; = π + cos Usig the latter relatios, we get G e x ; x 3 ρ ; + ρ ; + ρ ; 3 π cos π + cos π + cos + π π π = Covolutio operators with de La Vallée Poussi kerel We ow have degree m = N 0 ad we defie the de La Vallée Poussi kerel by V υ =! υ! cos, with ρ ; = +, ρ ; = + +. Usig the two relatios, we have for the secod momet: G e x ; x We also kow G e ; x = ρ ; x = + x, 8

30 . Positive liear operators which implies that G e ; x x = + x x = x + = x Covolutio operators with Jackso kerel Fially, the last operator we cosider is of degree m =, with N. For this, the Jackso kerel has the form with J υ = ad the secod momet satisfyig 3 si υ 4 + si υ, ρ ; = +, ρ ; = , G e x ; x Shepard-type operators We preset some Shepard-type operators defied i the geeral settig. A example of such operators goes back to the work of I. K. Crai, B. K. Bhattacharyya [8] ad D. Shepard [04] ad was first ivestigated by W. J. Gordo ad J. A. Wixom [60]. Other importat refereces are, e.g., the Habilitatiosschrift [45] ad the paper [44], both of H. Goska. I both of the latter refereces, we have the followig: Defiitio..4. see Defiitio 3.. i [00] Let X, d be a metric space ad let x,..., x be a fiite collectio of distict poits i X. We further suppose that for each -tuple x,..., x we have a fiite give sequece µ,..., µ of real umbers µ i > 0. The the Crai-Bhattacharyya-Shepard CBS operator is give by S f ; x := S µ,...,µ x,...,x f ; x := f x i i= f x i dx,x i µ i dx,x l µ l l= Here x X ad f is a real-valued fuctio defied o X., x {x,..., x },, otherwise. 9

31 Prelimiaries Remark..5. see Remark 3.3. i [00] From the above defiitio, we ca state that S is a positive liear operator o CX that satisfies S X ; x =, for all x X. Also it holds that S f ; x i = f x i, for all x i, i. If we restrict ourselves to the simpler case µ = µ =... = µ, we deote the correspodig operator by S µ. This looks like dx,x f x i i µ S µ, x {x,..., x }, f ; x := i= dx,x l µ l= f x i, otherwise, f x i s µ i = x, x {x,..., x }, i= f x i, otherwise,..3 while the secod momet of this CBS operator ca be writte as dx,x i µ S d µ, x {x,..., x },, x; x = i= dx,x l µ l= 0, otherwise...4 For a secod special case, we take X = [0, ] ad the metric dx, y := x y, for x, y X. The we get the CBS operator S µ + based o + equidistat poits x i = i, 0 i, give by S µ + f ; x := f x i i=0 f x i x i µ l=0 x l µ, x {x 0,..., x }, otherwise A piecewise liear iterpolatio operator S We cosider the operator S : C[0, ] C[0, ] see [46] iterpolatig the fuctio at the poits 0,,..., k,...,,, which ca be explicitely described as S f ; x = [ k, k, k + ] k ; α x f, α k=0 where [a, b, c; f ] = [a, b, c; f α] α deotes the divided differece of a fuctio f : D R o the distict kots {a, b, c} D, D R, w.r.t. α. Propositio..6 Properties of S. i S is a positive, liear operator preservig liear fuctios. ii S preserves mootoicity ad covexity/cocavity. iii S f ; 0 = 0, S f ; = f. iv If f C[0, ] is covex, the S f is also covex ad we have: f S f. 30

32 . Positive liear operators The operator S ca also be give as follows: k S f x := f u,k x, k=0 for f C[0, ] ad x [0, ], where u,k C[0, ] are piecewise liear ad cotiuous fuctios, such that l u,k = δ kl, k, l = 0,...,. [ ] k, k, we have We ow give the secod momet of the operator. For x S e x ; x = x k [ k k k x x = x k k x, which is maximal whe x = k. This implies S e x ; x 4... The BLaC operator: Defiitios ad Properties ] x The idea to examie BLaC operators comes from the BLaC-wavelets Bledig of Liear ad Costat wavelets, itroduced aroud 996 by G. P. Boeau, S. Hahma ad G. Nielso see [6]. They preset a multiresolutio aalysis that implies a fuctio represetatio at multiple levels of detail. This is a tool for hadlig large sets of data. The wavelet coefficiets are the oes who store the loss of detail i each level of represetatio. The wavelets are basis fuctios ecodig the differece betwee two succesive levels. Throughout their work, they discrimiate amog Haar ad liear wavelets. The Haar wavelets are ot cotiuous, but have perfect locality, while the liear oes are cotiuous, but the regularity they possess ca be a drawback. A compromise betwee the locality of the aalysis ad the regularity of the approximatio is desired. This compromise is obtaied by usig a bledig parameter 0 <. We ow itroduce the operator. The results that appear i the sequel are also preset i [0]. Defiitio..7. see [47] For f C[0, ] ad x [0, ], the BLaC operator is give by BL f ; x := f ηk ϕ k x. k= Now we explai the defiitio. For the real bledig parameter, the scalig fuctios ϕ : R [0, ] are give by x, for 0 x <,, for x <, ϕ := x, for x < +, 0, else. 3

33 Prelimiaries Remark..8. For =, ϕ reduces to B-Splie fuctios of first order or hatfuctios, while for the case 0 the piecewise costat fuctios are obtaied. That s why we choose 0, ]. For the idex k =,...,, N, by dilatatio ad traslatio of ϕ we obtai the family of fudametal fuctios: ϕ k x := ϕ x k, x [0, ]. The midpoits η k of the support lie of each fudametal fuctio ϕ k are give by η k := k + +, for k = 0,...,. For k {, } let η := 0 ad η :=. Propositio..9. Properties of the BLaC operator, see [47] i BL : C[0, ] C[0, ] is positive ad liear; ii BL is a modificatio of the piecewise liear iterpolatio operator S,; iii BL iterpolates f at η k, k =,..., also at the edpoits 0 ad ; iv BL reproduces costat fuctios; v The first absolute momet of the BLaC operator is: BL e x ; x = vi The secod momet of the BLaC operator is give by: BL e x ; x = ηk x ϕ k x, for all x [0, ]; k= k=.. The Mirakja-Favard-Szász operator ηk x ϕ k x, for all x [0, ]. The Mirakja-Favard-Szász operators see e.g. [6] were idepedetly itroduced by G.M. Mirakja see [8] i 94 ad by J. Favard ad O. Szász see [37] ad [3]. The classical th Mirakja-Favard-Szász operator M is defied by M f ; x := e x x k f k! k=0 k,..6 for f E, x [0, R ad N. E is the Baach lattice { } f x E := f C[0, : is coverget as x, + x f x edowed with the orm f := sup. +x x 0 The series o the right-had side of..6 is absolutely coverget ad E is isomorphic to C[0, ]; see [6], Sect