ON APPROXIMATION OF FUNCTIONS BY TRIGONOMETRIC POLYNOMIALS WITH INCOMPLETE SPECTRUM IN Lp, 0<p<1. Introduction T

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1 Joural of Mathematical Scieces, Vol. 165, No. 4, 2010 ON APPROXIMAION OF FUNCIONS BY RIGONOMERIC POLYNOMIALS WIH INCOMPLEE SPECRUM IN L, 0<<1 Yu. S. Kolomoitsev UDC Let B be a set of itegers with certai arithmetic roerties. We obtai estimates of the best aroximatio of fuctios i the sace L, 0 <<1, by trigoometric olyomials that are costructed by the system {e ix } Z\B. Bibliograhy: 13 titles. Itroductio Let =( ß; ß] be the uit circle. Deote by L the set of all 2ß-eriodic fuctios f such that ( f := f(x) dx) 1 < : Let A be a roer subset of the set Z. he the trigoometric system {e ix } A is ot comlete i the sace of itegrable fuctios. he situatio is quite differet i the case of the sace L with 0 <<1. It seems that A. A. alalya was the first to show that there exists a ifiite set B Z such that the system {e ix } Z\B is comlete i the sace L, 0 < <1 (see [1]). Various ecessary ad sufficiet coditios of comleteess for trigoometric systems with gas were obtaied i the aers [2 5]. Let us state oe simle coditio of comleteess of the systems cosidered [6]. Proositio A. Let 0 <<1. Assume that a set B = { } Z Z satisfies the followig coditios: ad = for N: (1) he system {e ix } Z\B is comlete i L if ad oly if lim ( +1 )= : (2) he aim of the reset aer is to obtai rates of aroximatio of fuctios by olyomials that are costructed by the system {e ix } Z\B for some sets B that satisfy coditios (1)ad (2). Letusitroduce the ecessary otatio. Let f():= 1 2ß f(x)e ix dx, Z, bethefourier coefficiets of a itegrable fuctio f; let e = e (x) := e ix ; deote by sec f := { Z : f() 0} the sectrum of a fuctio f; let deote the set of trigoometric olyomials of order ot exceedig. We deoteby C ositive costats deedig o the idicated arameters (costats C may bedifferet i the same lie). Defie the value of the best aroximatio of a fuctio f L by olyomials whose sectrum belogs to a set A as follows: E (f; A) := if f : : sec A We deote E (f) := E (f;z). I the sace L, 0 < < 1, first estimates of the value of the best aroximatio E (f) (Jacso-tye theorems) were obtaied by E. A. Storozheo, V. G. Krotov, ad P. Oswald i [7] ad, ideedetly, by V. I. Ivaov i [8]. I the case where A Z, the value E (f; A) deeds, to a greater extet, ot o the smoothess of f but o the arithmetical structure of the set A ad of the sectrum of the aroximated fuctio f if the fuctio f belogs to L (see Proositio 1 ad Corollary 1 below). I this aer, we study the behavior of the value E (f; A) for sets A with certai arithmetical roerties. Istitute of Alied Mathematics of NAS, Doets, Uraie, olomus1@mail.ru. raslated from Zaisi Nauchyh Semiarov POMI, Vol. 366, 2008, Origial article submitted November 27, /10/ c 2010 Sriger Sciece+Busiess Media, Ic. 463

2 Itroduce the class of fuctios where E 1 (f) := f. We estimate the values H1; ff := { f : f 1 + su ff E 2 (f) 1}; 1 E (H ff 1; ;A) := su E (f; A) f H1; ff i the followig cases: (1) A = Z \ ( m; m), where m N; (2) A = { Z : 0; = q ; Z + }, where q N, q 2; (3) A = { Z : s ; Z + },wheres N, s 2. I articular, i this aer we show that, i ay of cases (1) (3), E (H ff 1; ;A) 1 ff for ff>0 small eough; thus, the rate of the best aroximatio of fuctios by olyomials whose sectrum belogs to the set A does ot differ i essece from the rate of the best aroximatio by olyomials with the full sectrum. If ff is large eough, the situatio is essetially differet; for examle, i case (2), E (H ff 1; ;A) 1 1 (see heorems 1 3). I the first sectio of the aer, we get auxiliary results of geeral ature. I Sec. 2, we aly these results i estimatio of the values E (f; A) for articular sets A. 1. Auxiliary results he followig statemet idicates a essetial differece betwee the values E (f) ad E (f; A) for A Z. Proositio 1. Let 0 <<1. he (i) if A Z \{0}, the for ay fuctio f L 2 there exists 0 1 such that 1 1 E (f; A) f(0) ; 0 ; (ii) for ay sequece ff = {ff } there exists a set B ff Z that satisfies coditios (1) ad (2) such that for ay f L 2 such that f(0) 0. lim ff E (f;z \ B ff ) = Proof. Let f be a fuctio from L 2 ad let be a olyomial whose sectrum belogs to Z \{q} Z, where q N;q 2. he 464 f = q 1 s=0 2ß q 0 2ß q 0 2ß(s+1) q 2ßs q q 1 { ( f q s=0 = f(x) (x) dx x + 2ßs q ) ( x + 2ßs )} dx q f(q)e iqx dx = q 1 = f(q)e :

3 We aly the Hölder iequality ad Parseval equality to the iequalities above ad coclude that ( ) f 2ßq { } 1 f(0) f(q) 2 2 : (3) o rove statemet (i), we tae q = i iequality (3)ad ote that f() 2 0as. 0 Let us rove statemet (ii).for a give sequece ff we set B ff = {2 j } N 2 j, where {N j } j=1 0 j=1 is a strictly icreasig sequece of atural umbers such that ff N 2 j > 4 j(1= 1), j N. Clearly, the set B ff satisfies coditios (1)ad (2). Let N be large eough. he there exists j N such that2 Nj <2 Nj+1. Note that (Z \ B ff ) ( ; ) {2 j+1 } Z =. hus, it follows from iequality (3)with q =2 j+1 that ( ) ff E (f;z \ B ff ) 2 { } (j 1)(1 ) f(0) f(2 j+1 ) 2 2 for ay fuctio f L 2. Passig i the last iequality to the limit as ad taig ito accout that f(0) 0,we get the desired statemet. Before statig the ricial (hady)lemma of the aer, let us itroduce some otatio. Let A = Z\{ } Z, where Z, 0= 0 < 1 < 2 <::: ad = for N, letn N, ad let ff N = ff N (A):= card{ N \ A : <2N }. We assume that, for a give choice of N, there exists a iteger fi N [0;ff N 1] ad a sequece of atural fin +1 umbers {m } =0 deedig o N ad such that (a) l ± (mod m )for all =0;:::;fi N ad l =0;:::;ff N, l ; (b) l ± (mod m fin +1 )for all = fi N +1;:::;ff N ad l =0;:::;ff N, l. Set m := m, Z +. Deote x j; = 2ßj ad 4+1 f;(t):= 1 4 f (x j; + t) e i(xj;+t) : Lemma 1. Let f L, 0 <<1. I the above otatio, the iequality { [ E 2N (f; A) C E N (f) + fi N ( m 2N ) 1 f ;N m fin +1 +( ) 1 2N ffn f ] 1 } ;N fi N < ff N (4) holds, where the costat C deeds o oly. Proof. Defie a erel of de la Vallée Poussi tye, V (x):= 2 = 2 ( ) g e ix ; where g is a fuctio from C (R)such that g(x) =1for x 1 ad g(x)= 0 for x 2. It is ow (see, for examle, [3])that V C 1 1 ; (5) where C is a costat that does ot deed o. 465

4 Set e ix V [ ] 2N (m x) if [ fi N ;fi N ]; 2m K ;N(x):= e ix V [ ] 2N ff N (mfin +1 x) if [ ff N ;ff N ] \ [ fi N ;fi N ]; 2m fin +1 where [x] is the iteger art of x. Coditios (a)ad (b)imly that sec K ;N (( 2N;2N) A) { }: (6) Further, we aly a family of liear olyomial oerators, W (f; x; t):= 1 4 f(x j; + t)v (x x j; t); 4 +1 itroduced by K. V. Ruovsy (see, for examle, [9]). It is ow (see [9] ad [10, Cha. 4])that f W (f; ;t) dt CE (f) ; N; (7) for ay fuctio f L,0<<1, where C is a costat that deeds o oly. Set N (f; x; t):= W N (f; x; t) ff N ( ) g f ;N(t)K ;N(x): N = ff N Sice K ;N( )=1,itfollows from (6)that sec N ( 2N;2N) A. Further, 2ßE 2N (f; A) f N (f; ;t) dt f W N (f; ;t) dt + ( N We aly iequality (5)to show that ( ) g f ;N(t)K ;N( ) N dt ff N V[ 2N f ;N + C fi N { fi N 2m ] ( 2m 2N V[ 2N ff N 2m fin +1 ff N g ) 1 ( f ;N 2mfiN N ffn We get the desired iequality from relatios (8), (7), ad (9). Corollary 1. Let f L ad 0 <<1. I the above otatio, { where E 2N (f; A) C [ + E N (f) + H N; E N (f) 1 fi N ( m 2N [ H N; := fi N ) f ;N(t)K ;N( ) dt: (8) ] f ;N fi N < ff N ) 1 f } ;N : (9) fi N < ff N ) 1 ( ) 1 ( f( ) mfin +1 ) ] 1 + 2N f( } ) ; (10) ffn fi N < ff N ( m ad C is a costat that deeds o oly. 2N ) 1 ( ) 1 ] 1 mfin +1 + (ff N fi N ) ; 2N ffn Proof. Iequality (10)follows from Lemma 1 ad statemet (i)of Lemma 2, see below. 466

5 Lemma 2. For ay N, the followig statemets hold: (i) If f L, the f ; 1 f() + CE (f) 1 ; ; (11) where C is a absolute ositive costat; (ii) if 0 <<1, ff>0, adf H1; ff, the f ; 1 C( +1) (ff+1 1 )+ ; ; (12) where x + = x if x 0 ad x + =0if x<0, adc is a costat deedig o ff ad oly. Proof. Let 2 1 (x) = 1 f(x t)v (t)dt; 2ß where V is the de la Vallée Poussi erel defied i the roof of Lemma 1. It is ow that f CE (f) 1, where the costat C does ot deed o f ad. he equality f ; (t) = 1 4 {f(x j; + t) 2 1 (x j; + t)} e i(xj;+t) f() imlies that f ; 1 f f() CE (f) 1 + f() : Let us rove statemet (ii). If ff 1 1, the f ; 1 f 1 1. Cosider the case ff> 1 1. For ay itegrable fuctio f, the followig relatio betwee values of the best aroximatio of this fuctio holds: { E (f) 1 C ( +1) 1 1 E (f) E (f) }; N; (13) =+1 where the costat C deeds o oly (see [11]). Iequality (13)imlies that E (f) 1 C 1 1 ff. We also ote that f(±) E 1 (f) 1, N. Alyig the above estimates to iequality (11), we rove iequality (12). Lemma 3. Let f L, 0 <<1, let A Z, leth beaolyomial whose sectrum belogs to (Z\A) [ ; ], ad let H (f; t):= 1 2 f(t x j; )h (x j; ); x j; = 2ßj : he where the costat C deeds o oly. H (f; ) C 1 1 h E (f; A) ; (14) Proof. Let be a arbitrary olyomial whose sectrum belogs to A [ ; ]. he We deduce from equality (15)that It remais to aly the iequality (see, for examle, [12]). H (f; t) = {f(t x j; ) (t x j; )}h (x j; ): (15) H (f; ) (2 +1) f h (x j; ) : 2 h (x j; ) C h 467

6 Lemma 4. Let 0 <<1 ad ff>0. he there exists a fuctio f ff H1; ff such that E (f ff ) 1 ff for all N; (16) where meas a two-sided iequality with ositive costats that do ot deed o. Proof. We tae the fuctio f ff of the form f ff (x):= fl ff =1 2 ff e i2x : he estimate from above i (16)is obvious. We tae the costat fl ff such that f H ff 1;. o get the estimate from below, we aly Lemma 2 i which we set A = [ 2 ; 2 ] Z ad h N (t) = e i2+1t V 2 1(t), where N = 3 2 ad V 2 1 is the de la Vallée Poussi erel (see the roof of Lemma 1). For the fuctio f ff ad the give olyomial h N, H N (f ff ;t)=fl ff =+1 2 ff e i2t ; ( ) where = g l 2 +1 if 2 l (mod (2N + 1)) for 2 <l<3 2 ad 2 1 = 0 otherwise. It is well ow that if sec f [0; ), the f(0) f (see, for examle, [4]). Hece, H N (f ff ; ) = e 2 +1H N (f ff ; ) fl ff 2 ff(+1) : (17) Note that E 2 (f) = E N (f; A). hus, it follows from iequalities (14), (5), ad (17) that E 2 (f ff ) CN 1 1 h N E 2 (f ff ) Cfl ff 2 ff(+1) : he lemma is roved. 2. Aroximatio rates for articular sets A I this sectio, we obtai estimates of the value E (f; A) for articular sets A Let m N. Cosider the set M = Z \ ( m; m). Lemma 5. Let f L, 0 <<1, ad let m; N with m<. he { ( m ) 1 1 E 2 (f; M) C E (f) + f ; }; (18) <m where the costat C deeds o oly. Proof. o rove iequality (18), it is eough to tae N =, fi N = 0, ad m 0 = m 1 =2m i Lemma 1. heorem 1. Let 0 <<1 ad let m; N with m<. he (i) if 0 <ff< 1 1 ad m<1 ff, the E (H ff 1; ;M) 1 ff ; (ii) if 1 1 ff 1, the ( m ) 1 1 ( m ) 1 C 1 E (H1; ff ;M) 1 m C ff ; =1 (iii) if ff> 1, the ( m ) 1 E (H1; ff ;M) 1 ; 468

7 where is a two-sided iequality with ositive costats deedig o ad ff oly, ad C 1 ad C 2 are ositive costats that deed o ad ff. Proof. Let f be a fuctio from H1;. ff We aly the Hölder iequality ad iequality (12) to deduce from Lemma 5 that { ( m ) 1 1 } E 2 (f; M) C E (f) + f ; <m 1 { ( m ) 1 C ff 1 m + (ff+1 }: 1 )+ Now it is easy to get the uer estimates i statemets (i) (iii). he estimate from below i statemet (i)follows from Lemma 4. o get the estimates from below i statemets (ii)ad (iii), we refer to Lemma 3. ae i Lemma 3 f(x) 1 ad h (x) =V [ m 2 ](x), where V [ m 2 ] isaerel of de la Vallée Poussi tye (see the roof of Lemma 1). he we deduce from iequalities (14)ad (5)that ( m =1 ) 1 1 CE (1;M) ; where C is a costat that deeds o oly. his gives us the estimates from below i (ii) ad (iii). he theorem is roved Let q be a atural umber, q 2, ad let (q) = q 1 for N ad (q) = (q) for Z +. Set Q = Z \{ (q) } Z. For sets of this tye the followig lemma is valid. Lemma 6. Let f L, 0 <<1, ad let N. he } E 2q (f; Q) C {E q (f) + q ( 1 1) F q; (f) ; (19) where { [log q ] + F q; (f) := f 0;q + q (1 ) f ±q ;q + 1 =0 =[log q ] ++1 } 1 f ±q ;q ; (here [x] + is the iteger art of x if x>0 ad [x] + =0if x 0), where C is a costat that deeds o ad q oly. Here ad below, the symbol a ± deotes the sum {a + a }. Let m ad q be relatively rime atural umbers; set ord ± m(q):= if { ffi N : q ffi ±1 (mod m) } : he value ord + m(q)is called the order (exoet)of the umber q modulo m. o rove Lemma 6, we eed the followig lemma. Lemma 7. Let m be a rime odd umber that is relatively rime with q 2. he there exists a atural umber 0 3 such that ord ± m (q) m 0 ; = 0 ; 0 +1;:::: Proof. We fid the umber 0 from the coditio that q m 1 1 (mod m )for 0 3. First we show that, uder such achoice of 0, the followig iequality holds: ord + m (q) m 0+1 ; = 0 1; 0 ;:::: (20) We rove iequality (20)by iductio o. Let ord + m ν (q) mν 0+1 for all 0 1 ν. Assume that there exists m 1such that ord + m +1 (q) =m +2 0 m : (21) 469

8 he it follows from the Euler theorem that m (m 1) 0 (mod (m +2 0 m )). his cogruece is oly ossible either if m = m +2 0 m +1orif m 0 (mod m). By the choice of 0, the first case is imossible; thus, i what follows we assume that m = mm 0 for some m 0 N. Hece, (21)imlies the cogruece q m(m 0 +1 m 0) 1 (mod m +1 ), or, what is the same, m 1 (q m 0 +1 m 0 1) q ν(m 0 +1 m 0) 0 (mod m +1 ): By the iductio assumtio, q m 0 +1 m0 1 (mod m ). hus, if we showthat ν=0 m 1 ν=0 q ν(m 0 +1 m 0) 0 (mod m 2 ); (22) the we get a cotradictio. Note that (22) follows from the fact that the cogruece f(x) := m 1 ν=0 x ν 0 (mod m 2 )has o solutios. Ideed, the oly solutio of the cogruece f(x) 0 (mod m)is give by x 1 (mod m)(see, for examle, [13,.163]). Sice f (1) 0 (mod m)ad f(1) 0 (mod m 2 ),we coclude that the set of umbers give by x 1 (mod m)cotais o umbers for which f(x) 0 (mod m 2 )(see, for examle, [13,. 139]). Now we rove that ord m (q) m 0 ; = 0 ; 0 +1;:::: (23) he required iequality follows from (20). Ideed, if we assume that (23)does ot hold, the we ca fid m 1 such that q m 0 m 1 (mod m ); hece, q 2(m 0 m ) 1 (mod m ). he latter relatio cotradicts (20) sice m 0+1 > 2(m 0 m ). he lemma is roved. fin +1 Proof of Lemma 6. I the roof, we aly Lemma 1 with a roer choice of fi N ad the sequece {m }. =0 We tae N = q ; fi N = [log q ] + ; m = q for =1;:::;fi N ; m 0 = m; m fin +1 = m d ; where m isarimeoddumber that is relatively rime with q, d =[log m ] ,ad 0 is the umber give by Lemma 7 for the chose m ad q. Sice q l 0 (mod m)ad q l ±q (mod q +1 ),we coclude that coditio (a)is fulfilled for the sequece { (q) } Z. Let us chec coditio (b). Let l = 1;:::;, ad = 0;:::;l 1. he q l ±q (mod m d ) sice ord ± m d (q) m d 0 >l by Lemma 7. It remais to substitute the chose values ito iequality (4)ad to erform elemetary estimates. Lemma 6 is roved. heorem 2. Let 0 <<1 ad let N. he (i) if (ii) if 0 <ff< 1 1, the 1 1 ff 2 2, the E (H ff 1; ;Q) 1 ff ; C E (H ff 1; ;Q) C 2 log( +1) ; 470 (iii) if ff> 2 2, the E(H ff 1; ;Q) 1 1 ;

9 where is a two-sided iequality with ositive costats that deed o, q, ad ff oly, ad C 1 ad C 2 are ositive costats that deed o ad q. Proof. For a fuctio f H ff 1; cosider the value F q; (f) from Lemma 6. he Hölder iequality ad iequality (12)imly that F q; (f) C {1+ [log q ] + =0 which gives us the followig estimates: q (1 (ff+1 1 )+) + 1 ( =[log q ] + q (ff+1 1 )+ ) }; { 1 if 0 <ff 2 F q; (f) C 2; 2 1 if 2 (24) <ff: It is easy to get the uer estimates i statemets (i) (iii) from iequalities (19) ad (24). he estimate from below i statemet (i)follows from Lemma 4; the corresodig estimates i statemets (ii)ad (iii)ca be obtaied, for examle, from Proositio 1, item (i) Let s be a atural umber, s 2. Cosider the set S = Z \{ (s) } Z, where (s) = s for N ad (s) = (s) for Z +. he followig lemma holds. Lemma 8. Let f L, 0 <<1, ad let N. he E 2 s(f; S) C where C is a costat that deeds o oly. [2 {E 1=s ] s(f) + (s 1)(1 1 ) =0 f ± s ; s }; Proof. We cosider the case where s is odd (for eve s, the roof is similar to the scheme below). Let μ be a rime umber of the form 2s + s 0, where s 0 is a odd umber that is relatively rime with s ad such that s 0 1 is relatively rime with s as well. he, if a s 0 (mod μ), the cogruecy x s a s (mod μ)has a uique solutio x a (mod μ)(see, for examle, [13,. 163]). Further, by the law of distributio of rimes i arithmetic rogressios, there exists a costat deedig o s ad such that, for ay N, there exists a rime umber μ of the above form such that3 μ. hus, l s s (mod μ )for l; = [2 1=s ];:::;[2 1=s ] with l. Now we ca aly Lemma 1 with N = s, fi N = 0, ad m 0 = m 1 = μ. heorem 3. Let 0 <<1 ad let N. he (i) if 0 <ff s,the (ii) if ff> s, the E (H ff 1; ;S) 1 ff ; C E (H ff 1; ;S) C 2 (1 1 )(1 1 s ) ; where is a two-sided iequality with ositive costats that deed o, s, ad ff oly, ad C 1 ad C 2 are ositive costats deedig o ad s. Proof. he roof of heorem 3 is similar to those of heorems 1 ad 2. o rove the estimates from above, it is eough to aly Lemma 8 ad iequality (12). he estimate from below i statemet (i)follows from Lemma 4; the corresodig estimate i statemet (ii)follows from Proositio 1, item (i). Remar. he uer iequality i statemet (ii)of heorem 3 is strict, i a sese. For examle, for the fuctio g ;s (x):= cos( 1) s x it is easy to show thate s(g ;s ;S) (1 1 )(s 1) (see, for examle, [3]). raslated by S. Yu. Pilyugi. 471

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