ON SOME PROPERTIES OF THE PICARD OPERATORS. Lucyna Rempulska and Karolina Tomczak

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1 ACHIVUM MATHEMATICUM BNO Tomus 45 9, 5 35 ON SOME POPETIES OF THE PICAD OPEATOS Lucya empulska ad Karolia Tomczak Abstract. We cosider the Picard operators P ad P ;r i expoetial weighted spaces. We give some elemetary ad approximatio properties of these operators.. Itroductio.. The Picard operators P f; x := fx te t dt = fx te t dt, x ad N, N = {,,...}, =, are ivestigated for fuctios f : from various classes i may moographs ad papers e.g. [ [8 [,. G. H. Kirov i the paper [9 itroduced the geeralized Berstei polyomials B ;r for r-times differetiable fuctios f C r [, ad he showed that B ;r have better approximatio properties tha classical Berstei polyomials B. The Kirov method was used i [ to the geeralized Picard operators 3 P ;r f; x := P F r t, x; x, x, N, r N, f j t F r t, x := x t j, j! N = N {} of r-times differetiable fuctios f :. Obviously P ; f P f. I this paper we examie the Picard operators P i Sectio ad P ;r i Sectio 3 for fuctios f belogig to the expoetial weighted spaces L p ad which defiitio is give below. We preset some elemetary properties, the orders of approximatio ad the Voroovskaya type theorems for these operators. L p,r Mathematics Subject Classificatio: primary 4A5. Key words ad phrases: Picard operator, expoetial weighted space, degree of approximatio, Voroovskaya type theorem. eceived August 7, 8, revised March 9. Editor O. Došlý.

2 6 L. EMPULSKA AND K. TOMCZAK.. Let > ad p be fixed, 4 v x := e x for x, ad let L p L p be the space of all fuctios f : for which v f is Lebesgue itegrable with p-th power over if p < ad uiformly cotiuous ad bouded o if p =. The orm i L p is defied by /p v xfx dx p if p <, 5 f p, f p, := v x fx if p =. sup x Moreover, let r N ad L p,r L p,r be the class of all r-times differetiable fuctios f L p havig the derivatives f k L p, k r. The orm i L p,r is give by 5. L p, L p. The spaces L p ad L p,r are called expoetial weighted spaces [. As usual, for f L p ad k N we defie the k-th modulus of smoothess: 6 7 ω k f; L p ; t := sup k hf p, for t, h t k k hfx := k j The above ω k has the followig properties: 8 9 k j fx jh. ω k f; L p ; t ω k f; L p ; t for t < t, ω k f; L p ; λt λ k e kλt ω k f; L p ; t for λ, t, lim ω kf; L p ; t =, t for every f L p ad k N see [6, Chapter 6 ad [3, Chapter 3. By ω k we defie the Lipschitz class Lip k M L p ; α := { f L p : ω k f; L p ; t Mt α for t } for fixed umbers: p, >, k N, M > ad < α k.. Some properties of P.. By elemetary calculatios ca be obtaied the followig two lemmas. Lemma. The euality there holds for every r N ad s >. t r e st dt = r! s r Lemma. Let e x =, e x = x ad let ϕ x t = t x for x, t. The 3 P e i ; x = e i x for x, N, i =,,

3 ON SOME POPETIES OF THE PICAD OPEATOS 7 ad 4 5 P ϕ k x t; x k k! = k, P ϕ x t k k! exp ϕ x t ; x = k, for x, ad k N. Usig the above results ad arguig aalogously to the proof of Lemma i [ we ca obtai the followig basic lemma. Lemma 3. Let f L p with fixed p ad >. The 6 P f p, f p, for. The formula ad 6 show that P,, is a positive liear operator actig from the space L p to L p... By 6, 7, ad 6 ca be derived the followig geometric properties of P give by. Theorem. Let f L p with fixed p ad > ad let N. The i if f is o-decreasig o-icreasig o, the P f is also o-decreasig o-icreasig o, ii if f is covex cocave o, the P f is also covex cocave o, iii for every k N there holds the ieuality ω k P f; L p ; t ω k f; L p ; t, t, iv if f Lip k M L p ; α with fixed k N, < α k ad M >, the also P f Lip k M L p ; α with the same k ad α ad M = M, v If f L,r with a fixed r N, the P f L,r ad for derivatives of P f there holds P k f, = P f k, f k, Proof. For example we prove iii. From the formulas ad 7 there results that k hp f; x = P k h f; x for x, h, k N. Next, by 5 ad 6, we have k hp f; p, = P k h f, p, k hf p, for h ad, ad usig 6, we get the statemet iii.

4 8 L. EMPULSKA AND K. TOMCZAK.3. Arguig similarly to [5 ad [ ad applyig 6 9, ad 6 we ca prove the followig approximatio theorem. Theorem. Suppose that f L p with fixed p ad >. The P f f p, 5 33 ω f; L p ; for every 3. From Theorem ad 8, ad there results the followig Corollary. If f L p, p, >, the 7 lim P f f p, =. I particular, if f Lip M L p ; α with fixed < α ad M >, the P f f p, = O α as. Applyig Corollary, we shall prove the Voroovskaya-type theorem for P. Theorem 3. Let f L, with a fixed >. The 8 lim [P f; x fx = f x for every x. Proof. Choose f L, ad x. The, by the Taylor formula, we have ft = fx f xt x f xt x ψt; xt x for t, where ψt ψt, x is a fuctio belogig to L ad lim t x ψt; x = ψx =. Usig operator P,, ad 3 ad 4, we get 9 P ft; x = fx f x P ψtϕ x t; x ad by the Hölder ieuality ad 4: P ψtϕ x t; x P ψ t; x P ϕ 4 x t; x / = 4P ψ t; x /. From properties of ψ ad 7 there results that lim ψ t; x = ψ x =. Coseuetly, P lim P ψtϕ x t; x = ad by 9 ad follows 8. Now we estimate the rate of covergece give by 8. Theorem 4. Let f L, with a fixed >. The [P f f f, 4 4 ω f ; L ; for.

5 Proof. For f L, where ON SOME POPETIES OF THE PICAD OPEATOS 9 ad x, t there holds the Taylor-type formula ft = fx f xt x f xt x t x It, x, It, x := u [f x ut x f x du. Usig operator P,, ad 3-5, we get P ft; x = fx f x P ϕ x tit, x; x, which implies that 3 [P f; x fx f x P ϕ x t It, x ; x, for x ad. Now, applyig 6, 8 ad 9, we get from : It, x uω f ; L ; u t x e x du ω f ; L ; t x e x ω f ; L ; t x e x t x. ad ext by 4 ad 5 we ca write v xp ϕ x t It, x ; x ω f ; L ; { P t x e t x ; x P t x 3 e t x ; x } = ω f ; L ; ω f ; L ; for x,. Now the estimate is obvious by 3, the last ieuality ad 5. Theorem 5. Suppose that f L,r with fixed > ad r N. The 4 P r f f r, 5 33 ω f r ; L ; for 3. Proof. If f L,r, the for the r-th derivative of P f we have by Theorem, 3 ad 7: P r f; x f r x = = [ f r x t f r x e t dt [ t f r x t e t dt.

6 3 L. EMPULSKA AND K. TOMCZAK From this ad by 6, 9 ad we deduce that P r f f r, ω f r ; L ; t e t dt ω f r ; L ; = ω f r ; L ; { t e 3t dt } 3 3 for 3, which yields the estimate Some properties of P ;r 3.. The formulas 3 show that the operators P ;r, r N, geeralize P ad P ; f P f for f L p,. By this fact ad Sectio, we shall cosider P ;r for r N oly. Lemma 4. Let p, > ad k N be fixed umbers. The for every f L p,r ad there holds 5 P ;r f p, f j p,. The formulas 3 ad the ieuality 4 show that P ;r,, is a liear operator actig from L p,r to L p. Proof. Let p <. The, by 3, the Mikowski ieuality ad, we get for f L p,r ad : P ;r f p, = j! P f j tϕ j xt; p, j! j! e x j! f j p, t j f j x te t dt p dx /p t j e t e x f j x t p /pdt dx t j e t dt f j p, j f j p,. The proof of 5 for p = is similar.

7 ON SOME POPETIES OF THE PICAD OPEATOS First we shall prove a aalogy of Theorem. Theorem 6. Suppose that f L p,r with fixed p, > ad r N. The 6 P ;r f f p, M r ω f r ; L p ; for every, where M = r r. Proof. For every f L p,r formula: 7 fx = where 8 I r t, x := From 7, 8 ad 3 there results that ad x, t there holds the followig Taylor-type f j t x t j x tr j! r! I rt, x, [ u r f r t ux t f r t du. F r t, x = fx ad ext by, 3 ad 7 it follows that x tr r! I rt, x, P ;r f; x fx = r r! P t x r I r t, x; x = r 9 t r u r r! utf r x tdu e t dt for x ad. If p <, the usig the Mikowski ieuality ad 5 9 ad, we get from 9: P ;r f f p, = e x t r e t u r r! utf r x t du t r e t u r r! utf r p, du dt t r e t u r ω f r ; L p r! ; u t du dt t r e t ω f r ; L p r! ; t dt r! ω f r ; L p ; = ω f r ; L p ; t r te t dt r r r for, which implies 6 for p <. /p dt p dx

8 3 L. EMPULSKA AND K. TOMCZAK The proof of 6 for f L,r is aalogous. From Theorem 6 we ca derive the followig Corollary. If f L p,r, p, > ad r N, the lim r P ;r f f p, =. Moreover, if f r Lip M L p ; α with some < α ad M > the P ;r f f p, = O r α as. Arguig aalogously to the proof of Theorem give i paper [ ad applyig Corollary, we ca obtai the followig Voroovskaya-type theorem for operators P ;r. Theorem 7. Let f L α,r with fixed r N ad >. The P ;r f; x fx = r r f r x r [ r r f r x o r as, at every x. I particular, if r is eve umber, the 3 lim r [P ;r f; x fx = r f r x at every x. Similarly to Theorem 4 ow we shall estimate the rate of covergece give by 3. Theorem 8. Let > ad eve umber r N be fixed. The for every f L,r ad there holds 3 r [P ;r f f r f r p, M ω f r ; L ; where M = r4 r 4. Proof. Similarly to the proof of Theorem 6 we use the Taylor-type formula of f L,r : 3 fx = for x, t, where 33 I t, x := r Aalogously for f r L, 34 f j t x t j j! x tr I t, x, r! [ u r f r t ux t f r t du. ad x, t we have f r t =f r x f r xt x t xi t, x,

9 ON SOME POPETIES OF THE PICAD OPEATOS 33 with 35 I t, x := [ f r x ut x f r x du. By 3 ad 34 the formula 3 ca be rewritte i the form: 36 fx = F r t, x x tr f r x r! r! r! f r xx t r x tr x tr I t, x r! r! x tr I t, x for x, t. r! [ f r t f r x Let ow x be a fixed poit. Usig operator P ad 3 ad 3 5, we get from 36: where fx = P ;r f; x r r f r x 3 T i x for, i= T x := r! P t x r I t, x; x, T x := r! P T 3 x := r! P Coseuetly we have [ t x r f r t f r x ; x, t x r I t, x; x r [P,r f f r f r, r T i,. From 35 ad 6 9 it follows that v x T x e x r! P t x r I t, x ; x r! P r! ω i= t x r ω f r ; L ; t x ; x f r ; L ; [P t x r e t x ; x P t x r e t x ; x

10 34 L. EMPULSKA AND K. TOMCZAK ad further by 5 we have 38 T, r! ω f r ; L ; [ r! r3 r 3! r4. Aalogously, by 6 9 there results that f r t f r x ω f r ; L ; t x e x e x t x t x ω f r ; L ; ad from 33: I t, x u r ω f r ; L ; u t x e t du e t ω f r ; L ; t x u r du r e t t x t x ω f r ; L ;. Usig the above ieualities ad 5, we deduce that 39 T, ω f r ; L ; [ r 3 r3 r4 ad 4 T 3, ω f r ; L ; [ r 3 r3 r4, for. Summarizig 37 4, we immediately obtai the desired ieuality 3. emarks. Theorem 6 shows that the order of approximatio of fuctio f L p,r by P ;r f is depedet o r ad it improves if r grows. Moreover, Theorem 6 ad Theorem show that the operators P ;r with r have better approximatio properties tha P for f L p,r. We metio also that the similar theorems ca be obtaied for the Gauss-Weierstrass operators W f; x := /π fx te t dt, x, N, defied i expoetial weighted spaces L p with the weighted fuctio v x = e x, >.

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