[ ) so that the function obtained (denoted, as before, by

Transcript

1 Ukrinin Mhemicl Journl, Vol. 56, No. 9, 4 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS OF SUMMATION OF THEIR FOURIER INTEGRALS Yu. I. Khrkeych nd T. V. Zhyhllo UD 57.5 We obin sympoic equliies for upper bounds of he deiions of operors genered by - mehods (defined by collecion Λ ( of funcions coninuous on ; { } [ nd depending on rel prmeer on clsses of (, β -differenible funcions defined on he rel xis.. Auxiliry Asserions nd Semen of he Problem For mny yers, Sepnes nd his followers he inesiged he pproximion properies of he clsses L β nd ˆLβ defined by he propery h he generlized ( β-deriies, of heir elemens belong o cerin se. For numerous resuls concerning hese problems, see [ 9]. According o [3] (hp. IX, he clsses ˆLβ re defined s follows: Le L p, p, be he se of π-periodic funcions ϕ( wih finie norm ϕ p, where ϕ p π / p p ϕ d ( for p [ ; nd ϕ ϕ M ess sup ϕ(, so h L M. The spces ˆLp, p, re inroduced s he ses of (no necessrily periodic funcions ϕ( defined on he enire rel xis R nd hing finie norm ϕ ˆp, where ϕ pˆ + π / p p sup ϕ( R d for p [, nd ϕ ˆ ess sup ϕ(. I is obious h, for ll p, he inclusion Lp L ˆ p is lwys rue. Le denoe he se of funcions ( conex downwrd for ll nd such h lim (. [ so h he funcion obined (denoed, s before, by We exend eery funcion o he segmen, ( is coninuous for ll, (, nd is deriie ( ( + hs smll riion on he segmen [,. Denoe he se of hese funcions by. The subse of funcions for which Volyn Uniersiy, Lus k. Trnsled from Ukrins kyi Memychnyi Zhurnl, Vol. 56, No. 9, pp. 67 8, Sepember, 4. Originl ricle submied Noember, /4/ Springer Science+Business Medi, Inc. 59

2 5 YU. I. KHARKEVYH AND T. V. ZHYHALLO ( d < is denoed by F. We se ˆ ( ˆ ( ( cos π β + d, where F nd β is cerin fixed number. If F, hen, s shown in [4], for ny β R he rnsformion ˆ ( is summble on he enire xis: ˆ ( d <. Le ˆLβ denoe he se of funcions f( x Lˆ h, for lmos ll x R, cn be represened in he form f( x A ( x ˆ + ϕ + ( d, ( where A is cerin consn, ϕ( Lˆ, nd he inegrl is undersood s he limi of inegrls oer symmericlly expnding inerls. If f( Lˆ β nd, in ddiion,, where is cerin subse of coninuous funcions from ˆL, hen we ssume h f( Lˆ β. The subses of coninuous funcions from ˆLβ ˆLβ ( re denoed by Ĉ β ( Ĉ β. If coincides wih he se of funcions ϕ( sisfying he condiion ess sup ϕ (, hen he clss Ĉ β is denoed by ˆ β,. If f Lˆβ nd f β, hen we sy h f L ˆ ˆ β,. In [3] (hp. IX, i ws shown h if ϕ( is π-periodic summble funcion, hen he ses ˆLβ, ˆ L β,, nd ˆ β, rnsform ino he clsses L β, L β,, nd β,, respeciely. In he periodic cse where relion ( holds, we he ϕ( f β ( lmos eerywhere. In his connecion, ny funcion equilen o he funcion ϕ( in relion ( is clled, s in he periodic cse [see, e.g., [] (hp. I nd [4] (hp. III], he ( β-deriie, of f ( nd is denoed by f β (. As menioned boe, he clsses ˆLβ were inroduced by Sepnes. He lso considered he problem of he pproximion of funcions from he clsses ˆLβ by using he so-clled Fourier operors, which, in he periodic cse, re Fourier sums of order [ ] ; in he generl cse, hey re enire funcions of exponenil ype (see [4, 5]. In hese works, Sepnes obined represenion on he clsses ˆLβ for he deiions of he operors U( f, x,, which re inegrl nlogs of he polynomil operors genered by ringulr - mehods of summion of Fourier series. These resuls were pplied in [6 9] o he problem of pproximion of funcions from he clsses ˆLβ by he operors of Zygmund, Seklo, de l Vllée-Poussin, ec.

3 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 5 U The im of he presen pper is o sudy he deiions (on he clsses ˆ L β, nd ˆ β, of he operors ( f, x, genered by -mehods (defined by collecion Λ ( { } of funcions coninuous on [, nd depending on rel prmeer of summion of Fourier inegrls. In he periodic cse, for ( r, r >, he mos complee resuls in his direcion were obined in []; for funcions decresing o zero, he mos complee resuls were obined in []. Le Λ { } be collecion of funcions coninuous for ll nd depending on rel prmeer. We ssocie eery funcion f L wih he expression ˆβ U ( Λ U( f, x, Λ A + f x + + β d d π ( ( cos. ( Furher, we ssume h he funcions ( nd re such h he rnsformions $ π ( cos + d re summble on he enire number xis. Then, using relions ( nd (, for eery funcion f( ˆ β we obin f( x U ( f, x, Λ + f x r + β ( cos dd, (3 π where, for, r (, (4 nd, on he segmen, he funcion r is rbirrily defined so h i is coninuous for ll, equl o zero he origin, nd such h is Fourier rnsform is summble on he enire number xis. In he presen pper, we inesige he quniies rˆ ( r cos + d π ( β, f ˆ β, ˆ, U ( Λ sup f( x U ( f, x, Λ, (5

4 5 YU. I. KHARKEVYH AND T. V. ZHYHALLO ( β, ˆ ˆ f Lˆ β, L ˆ, U ( Λ sup f( x U ( f, x, Λ where U( f, x, Λ re he operors defined by (. Firs, we presen seerl uxiliry definiions nd semens necessry for wh follows., (6 Definiion []. Le funcion τ( be defined on [,, bsoluely coninuous, nd such h τ(. We sy h τ( if he deriie τ ( cn be defined he poins where i does no exis so h, for cerin, he following inegrls exis: / dτ (, dτ (. / Le K nd K i denoe consns h re, generlly speking, differen in differen relions. Lemm []. If τ(, hen τ( τ( + τ( + / dτ ( + dτ ( : H( τ. (7 / Theorem []. Suppose h τ( nd sin τ(. In order h he inegrl A( τ τ( cos + d d π (8 be conergen, i is necessry nd sufficien h he inegrls τ( sin d, τ( τ( + d be conergen. In his cse, he following esime is rue: 4 d A( τ ξsin τ (, j [ τ ( τ ( + ] KH( τ π, (9 where ξ( AB, is he funcion defined s follows []: ξ( AB, π A, B A, A A rcsin + B A, B > A, B (

5 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 53 j, < <,,. ( Le. Following [, pp. 59, 6], we se ( η( :, µ ( : η(, { : < µ (, K }, { } c : < K < µ (, K. If nd, moreoer, or for, hen, following [4, p. ], we wrie or, respeciely. Theorem [, p. 6]. A funcion belongs o if nd only if he quniy α( (, ( : ( +, ( sisfies he condiion α( K >. Theorem 3 [, p. 75]. In order h funcion belong o, i is necessry nd sufficien h here exis consn K such h, for ll, he following inequliy is rue: ( ( c K, where c is n rbirry consn h sisfies he condiion c >. ( β,. Asympoic Relions for ˆ, U ( Λ For conenience, performing chnge of ribles, we rewrie relions (3 nd (4 in he form f( x U ( f, x, Λ π τ ( β + f x ( cos + dd, ( ( τ( τ( ( (, (, (3

6 54 YU. I. KHARKEVYH AND T. V. ZHYHALLO where, s before, he funcion τ ( is rbirrily defined on he segmen,, equl o zero he origin, nd such h is Fourier rnsform so h i is coninuous for ll is summble on he enire number xis. Then he following heorem is rue: Theorem. Suppose h he following condiions re sisfied: (i ( F ; (ii τ( ; τˆ( ( cos π τ + d (4 (iii sin τ ( ; (i he following inegrls conerge: τ sin ( d, ( ( + d. (5 Then he funcion τ( τ ( ( ( (, (, ( ( ( (,, (6 sisfies he following relion: ( β, ˆ, U ( Λ 4 π ξ ( sin τ (, j [ τ ( τ ( + ] d + O( ( H(,, (7 τ where H( τ, ξ( AB,, nd j re defined by (7, (, nd (, respeciely. Proof. Using Theorem, we show h he inegrl A( τ conerges, nd, hence, by irue of Lemm in [8], he following relion holds s : ( β, ˆ, U( Λ ( A ( τ. (8

7 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 55 One of he condiions of Theorem is he conergence of he inegrl τ( τ( + d, (9 wheres one of he condiions of Theorem is he conergence of he inegrl ( ( + d. ( Le us show h if, hen τ( τ( + d ( ( ( + ( d+ H( τ O(, ( where O( is quniy uniformly bounded in. Therefore, he conergence of inegrl ( yields he conergence of inegrl (9. Using relion (6, we ge τ τ ( ( + ( ( (, (, ( ( ( (,, ( ( ( (, ( +, ( ( + ( ( +,. ( ( (3 Firs, we consider he cse > nd represen relion (9 s sum of wo inegrls: τ( τ( + d τ( τ( + d + τ( τ( + d. (4 Le us esime he firs erm on he righ-hnd side of (4. To his end, we dd nd subrc he quniy ( ( ( + ( ( under he modulus sign in he inegrnd. As resul, we obin

8 56 YU. I. KHARKEVYH AND T. V. ZHYHALLO τ( τ( + d ( ( ( + ( d + O ( τ ( ( + + ( ( ( + τ ( d. (5 Since relions ( nd (3 re rue, for, we ge ( ( τ( ( ( nd ( ( + τ( +. ( ( + Then ( τ( τ( + + ( ( + ( ( d ( τ ( ( ( d + ( τ ( + ( ( + d. (6 Since τ (, ccording o Lemm we ge ( τ ( ( ( d + ( τ ( + ( ( + d H( τ O ( ( ( ( ( d + ( ( + ( ( ( + d. (7

9 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 57 Le us show h, s, I, : I, : ( ( ( ( ( ( ( + ( ( ( + d O(, (8 d O(, (9 where he quniy O( is uniformly bounded in. Indeed, he funcion ( / ( ( for ll δ,, < δ <, nd, furhermore, by irue of Theorem, we he is bounded ( / ( ( ( lim ( K. Therefore, I, O( s. Pssing o he esimion of he inegrl I,, we noe h I, < ( ( ( ( + d. Performing he chnge of ribles u ( +, we ge I, < ( ( ( u du < u ( ( ( u du. u Using Lemm.5 from [3] nd Theorem 3 for he righ-hnd side of he ls inequliy, we obin I, < K( K( K3. ( ( Thus, equliies (8 nd (9 re rue. ombining relions (5 (9, we ge τ( τ( + d ( ( ( + ( d+ H( τ O(. (3 Le us esime he second erm on he righ-hnd side of (4. I is obious h

10 58 YU. I. KHARKEVYH AND T. V. ZHYHALLO τ( τ( + d ( ( ( + ( d + O ( τ( τ( + + ( ( ( + ( d. (3 Using relions ( nd (3, for, we ge ( ( τ( ( (3 nd Hence, by irue of Lemm, we obin ( ( + τ( +. ( ( + ( τ( τ( + + ( ( + ( ( d ( τ ( τ ( + ( ( ( + ( d H( τ O ( ( ( ( + ( d + ( ( ( + d. (33 We esime he righ-hnd side of (33 s follows: ( ( ( d - ln O(. (34 ( (

11 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 59 By nlogy wih he proof of relion (9, we ge ( ( + ( ( ( + d ( ( u ( u du ( ( u ( u du ( ( O(. (35 Using relions (3 (35, we obin τ( τ( + d ( ( ( + ( d + H( τ O(. (36 ombining relions (36 nd (3, we rrie equliy (. By nlogy wih he proof of relion (, for > <. For <, relion (3 is rue nd Then ( ( + τ ( +. ( one cn show h equliy ( is lso lid for τ( τ( + d ( ( ( ( + d. The conergence of inegrl ( yields he conergence of inegrl (9. Thus, for, ll condiions of Theorem re sisfied. Then, subsiuing relion (9 ino equliy (8, we ge (7. Theorem is proed. Noe h n nlogous heorem ws proed in [] in he periodic cse where ( [] for he clsses β,,. orollry. Suppose h he condiions of Theorem re sisfied. If r, r >, nd in τ ( τ ( + sin τ (, [, ], >, (37 hen

12 5 YU. I. KHARKEVYH AND T. V. ZHYHALLO ˆ ( β,, U( Λ π τ ( ( sin d + O ( sin π τ( d ( ( + + O( ( d + O( ( H( τ,. (38 If sin τ ( < τ ( τ ( +, [, ], >. (39 hen ˆ ( β,, U( Λ 4 π ( ( + ( d + O( π j ( ( + d + O τ ( sin ( d O( ( H( τ,. (4 Proof. Relion (38 follows direcly from equliy (7 nd he definiion of he funcion ξ( AB., To proe relion (4, noe h, by irue of (39 nd (4, he following equliies re rue: ξ sin τ [ τ τ + ] (, j ( ( d sin τ ( sin ( rcsin τ τ ( τ ( + + ( τ( τ( + sin τ ( d sin τ ( sin τ ( τ( τ( + ( d + O sin τ ( d,. (4

13 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 5 Since relion (39 is rue, we conclude h sin τ ( ( τ ( τ ( + [, ] if [, ]. Using relion (4 nd he expnsion of he funcion, [, ], in power series, we obin ξ sin τ [ τ τ + ] (, j ( ( d d τ ( sin τ ( + Osin d,. (4 Subsiuing (4 ino (7 nd using relion (, we ge (4. orollry is proed. orollry. Le Λ{ nk, }, where n, k,, nd n, for ll n, be recngulr numericl mrix h ssocies eery funcion f ˆβ wih series (. Suppose h he mrix Λ is such h series ( is he Fourier series of cerin coninuous funcion denoed by Un( f, x, Λ. Also ssume h k he mrix Λ is deermined by sequence of funcions n ( u, u <, such h nk, n n nd n, for ll n. The sympoic equliies for he quniies ( ˆ β,, U n ( Λ sup f( x U (,, n f x Λ ˆ f β, cn be obined by seing n, n N, in relions (7, (38, nd (4, proided h ll condiions of Theorem re sisfied. We ge ( ˆ β,, U n ( Λ 4 π ξ ( n d sin τ n (, j u τ n ( τ n ( [ + ] + O ( ( H ( τn,, where he funcions τ n (, n,,, re defined by he equliies τ n ( ( ( ( (, n n, ( n ( ( (, n n. If

14 5 YU. I. KHARKEVYH AND T. V. ZHYHALLO hen τ ( τ ( + sin τ ( n n n, [, ], >, ( ˆ β,, U n ( Λ π τ ( ( n sin n nπ τ d + O ( n sin n( d + O n n( ( n( + ( d + O( ( n H( τ n,. If sin τ n( < τ n( τ n( +, [, ], >, hen ( ˆ β,, U n ( Λ 4 π ( ( ( n n n + d + O( n nπ j n( n( + d ( β, Λ ˆ 3. Asympoic Relions for L ˆ, U ( τ + O ( n sin n( d + O( ( n H( τ n, n. In his secion, we sudy he behior of upper bounds of (6. Firs, we gie seerl definiions nd uxiliry resuls. Assume h f L ˆβ, F, nd he funcion τ ( is defined by (3 nd such h is Fourier rnsform τˆ ( ˆ ( (4 is summble on R. Then, lmos eery poin x R, we he τ f( x U ( f, x, ( f x + ( cos + β d d π τ Λ. (43 Noe h he funcion τ ( cn be chosen so h i is coninuous for nd is Fourier rnsform τˆ ( is summble on R. Using relion (43, we esblish he following semen:

15 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 53 Lemm. Suppose h F, he funcion τ ( is defined by (3 nd coninuous for ll, nd inegrl (8 conerges. Then he following relion holds s : ( + β, ˆ ˆ f Lˆ β, L ˆ ; U ( Λ sup f( x U ( f, x, Λ ( A( τγ ( (, (44 where γ ( nd Proof. Tking ino ccoun h γ ( O τ( cos + d d. (45 π f x + d τˆ ( ˆ π sup f x + + ˆ ( ddx τ f ˆ ˆ τ ( d R π nd using equliies (6 nd (43, we ge ( sup f( x U( f, x, β, ˆ L ˆ, U ( Λ ˆ f Lβ, On he oher hnd, by irue of Proposiion. in [3, p. 69], we he Λ ˆ ( τˆ ( d ( A ( τ. (46 Lˆ L β (, π Lβ β Lβ he se of π-periodic funcions wih men lue zero on ( π,. Therefore, Lˆ,,. Hence,, where L (, π is sup f x ˆ + ( d τ f Lˆ β, ˆ sup f x ˆ + ( d τ f L β,. (47 Moreoer, i is shown in [, p. 4] h sup ( f ˆ x + ( d ( A( + ( ( τ τγ, (48 f L β, where γ ( nd relion (45 is rue. Using (46 (48, we obin relion (44. The lemm is proed. In he periodic cse, n nlogous lemm ws proed in [] for he clsses L β,.

16 54 YU. I. KHARKEVYH AND T. V. ZHYHALLO ( nd β, ˆ ompring he lemm proed wih Lemm in [8], we conclude h he quniies L ˆ, U ( Λ ( ˆ β,, U( Λ my differ only by quniy h does no exceed γ ( in order, i.e., he following relion holds s : ( ( + ( Lˆ, U ( Λ ˆ, U ( Λ O ( γ β, ˆ β,. Using he ls equliy, we cn proe n nlog of Theorem for funcions of he clsses ˆ L β,. Theorem. Suppose h he following condiions re sisfied: (i ( F ; (ii τ ( ; τ (iii sin ( ; (i inegrls (5 conerge. Then, for he funcion τ( ( defined by (6, he following sympoic equliy is rue: τ ( β, ˆ L ˆ, U ( Λ 4 π ξ ( sin τ (, j [ τ ( τ ( + ] d π + O ( sin τ ( + j [ τ ( τ ( + ] d + O( ( H(,, where H( τ, ξ( AB,, nd j re defined by (7, (, nd (, respeciely. If, in ddiion, inequliy (37 is rue, hen τ ( L ˆ, β, U ( Λ ˆ ( sin π τ ( d + O ( sin π τ( d + O ( If inequliy (39 is rue, hen ( ( + d + O( ( H( τ,.

17 APPROXIMATION OF FUNTIONS DEFINED ON THE REAL AXIS BY OPERATORS GENERATED BY -METHODS 55 ( L ˆ, ( β, U Λ ˆ 4 π ( ( + ( d + O( π j ( ( + d + O τ ( sin ( d + O( ( H( τ,. REFERENES F. A. I. Sepnes, lssificion nd Approximion of Periodic Funcions [in Russin], Nuko Dumk, Kie (987.. A. I. Sepnes, Mehods of Approximion Theory [in Russin], Vol., Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (. 3. A. I. Sepnes, Mehods of Approximion Theory [in Russin], Vol., Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (. 4. A. I. Sepnes, lsses of funcions defined on he rel xis nd heir pproximion by enire funcions. I, Ukr. M. Zh., 4, No., ( A. I. Sepnes, lsses of funcions defined on he rel xis nd heir pproximion by enire funcions. II, Ukr. M. Zh., 4, No., ( M. G. Dzimisrishili, Approximion of lsses of oninuous Funcions by Zygmund Operors [in Russin], Preprin No. 89.5, Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (989, pp M. G. Dzimisrishili, On he Behior of Upper Bounds of Deiions of Seklo Operors [in Russin], Preprin No. 9.5, Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (99, pp V. I. Rukso, Approximion of funcions defined on he rel xis by de l Vllée-Poussin operors, Ukr. M. Zh., 44, No. 5, ( L. A. Repe, Approximion of funcions of he clsses ˆ ϕ, β, by operors of he form U, in: Fourier Series: Theory nd Applicions [in Russin], Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (99, pp L. I. Buso, Liner mehods of summion of Fourier series wih gien recngulr mrices. I, Iz. Vyssh. Uchebn. Zed., 46, No. 3, 5 3 (965.. A. K. Noiko, On pproximion of funcions in he spces nd L, in: Problems of Summion of Fourier Series [in Russin], Preprin No. 85.6, Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (985, pp S. A. Telykoskii, On he norms of rigonomeric polynomils nd pproximion of differenible funcions by liner mens of heir Fourier series. II, Iz. Akd. Nuk SSSR, Ser. M., 7, No., 53 7 ( A. I. Sepnes, Approximion of funcions wih slowly rying Fourier coefficiens by Fourier sums, in: Approximion of Periodic Funcions by Fourier Sums [in Russin], Preprin No , Insiue of Mhemics, Ukrinin Acdemy of Sciences, Kie (984, pp. 3 5.