IBN AL- HAITHAM J. FO R PURE & APPL. SC I . VO L.23 (2) 2010 On the Riesz Means of Expansion by Riesz Bases Formed by Eigen Functions for the Ordinary Differential Ope rator of 2m- th Order A. A. Aswhad Departme nt of Mathematics, College of Education Ibn-Al-Haitham , Unive rsity of Baghdad Abstract: The aim of this p aper is to p rove a theorem on the Riesz means of exp ansions with resp ect to Riesz bases, which extends the p revious results of [1] and [2] on the Schrödinger op erator and the ordinary differential op erator of 4-th order to the op erator of order 2m by using the eigen functions of t he ordinary differential op erator. S ome S ymbols that used in the paper: the uniform norm. <,> t he inner product in L 2 . G t he set of all boundary elements of G. û the dual function of u. 1. Introduction The theory of non-self adjoint differential op erators has gr eat imp ortance in several applications, many mathematicians worked on the equiconvergence theorem for this op erators like Ilin, Joó, Komornik and Tahir for the Schrödinger op erator see [3], [4] and for Laguerre functions see [5]. By using the method and r esults of this p apers, we shall p rove un equiconvergence theorem for the ordinary differential op erator of order 2m. Let G  R be an arb itrary finite open interval q(x)L 1 (G) an arbitaray comp lex function and consider the op erator Lur:= (2m ) ru +q(x)ur, where m . Given a complex nu mber , the function u – 1:GC, u – 10 is called an eigenfunction of order – 1 of the op erator L with the eigenvalue . A function u r:GC, u r0 (r=0,1,…) is said to be an eigenfunction of order r of the op erator L with the eigenvalue  if ur together with its derivative is absolutely continuous on ev ery comp act subinterval of G and if for almost all xG the equation Lur(x)=ur(x) –ur – 1(x) ho lds, where ur – 1(x) is an eigenfunction of order (r – 1) with the same . IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 Let us now give a Riesz bases (ur(x))L 2 (G) of the op erator L. Let r (resp .0r) denote the eigenvalue (resp . the order) of ur and assume that t he following conditions are satisfied : r sup 0r< ...(1) In case 0r > 0, ur – Lur = ur – 1 …(2) Sup p ose the biorthogonal sy stem (vr) of the sy stem (ur) have the prop erty r 2 r r L ( G ) 1 v V         …(3) Now, consider the Riesz mean of the biorthogonal ser ies r 2 s sr r r 2 ( f , x) : f , v u (x )(1 )          …(4) (fL 1 (G), xG, >0, 0 s <1/2), where (vr) is the du al sy st em of (ur), i.e., (vr)L 2 (G) and =rj . Given any comp act interval KG, denote by R an arbitrary number from the interval (0,dist(K,G)), where dist(K,G)=inf{d(a,b), aK,bG}. Now fix xK arb itrary and define sRW :GR by s 1/ 21 / 2 s s s 1 / 2 R a(s) t J ( t ) if t R, W ( t) 0 otherwise           …(5) where a(s):=2 s (2) – 1/2 (s+1). M oreover, for any function fL 1 (G), x  RG def ine x R s s R x R S (f , x ) : W (t)( y x)f ( y)dy     …(6) Denote by  s (x,y ,) the sp ectral function of the Riesz means (i.e.) r 2 s sr r r 2 ( x, y, ) : u ( x) v ( y)(1 )         …(7) where x,y  G. Introduce the operation 0R D :L 1 (G) R 0 0 0 R R R0 2 2 D [f ] : f (R )dR R   …(8) where R0(0,dist(K,G)). IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 2. Main Results Here we will prove the following theorem 2.1 The orem: Given any comp act interval KG, for all 0s<1/2, >0 and fL1(G) t he est imate s s s( f , x) S (f , x) O(1)     …(9) Holds uniformly on the compact interval KG. For the p roof of this theorem we shall choose the 2m-th roots r,i (i=1,2,…,2m) of r such that Re r,1  Re r,2… Re r,m 0 Re r,m +1 … Re r,2m and put r = r,m , r=Re r, r=Im r . Now, we have from ([3]) r r r r r r r 1 r 1 r 1 r 2 r 2 r 2 r m r m r m r ˆ ˆ ˆ ˆ ˆ ˆ ˆu ( x) u (x t) u (x t) u (x 2 R) u ( x 2R ) u ( x m) u ( x m) 1 2ch t 2ch 2 R 2ch m R D 1 2ch t 2ch 2 r 2ch m R 1 2ch t 2ch 2 R 2ch m R                                      By exp anding this determinant according to the first row with t he definition of rû we get t he followin g equation [ur(x – t) +ur(x+t) – 2ur(x) ch rt]d(r,R)= x mR k r r r r 0 k m x mR k 1 d ( , R , t)[u ( x kR ) u ( x kR ) D( , R , t, x )Q( )d                …(10) where m m p p r p p r k r k r2m 1 2 m 1 2 k m p 1 p 1r r m p p r r k r 2m 1 2 k m p 1 r p p r k r 2m 1 r s h (kR x ) sh (t x ) d ( , R, t ) d ( , R) , if x m m s h (kR x ) D( R, t, x ) d ( , R, t ) , if t x 2 m sh (k R x ) d ( , R, t) , i f ( j 1)R x m                                                                 m j k m p 1 jR, 3 j                 We want to p rove the following estimate s s s R( x, y, ) W ( y x ) O(1)      …(11) By using equation (10), we count t he Fourier coefficients of the function sRW (y – x) with resp ect to the sy stem (ur): IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 0 0 0 R s s r R R R R r r 0 R s k r R R r r r r 0 k m r0 k 1 x mR r r x mR u , D W D W ( t)[u (x t) u ( x t)]dt d ( , R, t) D W ( t)[ 2u ch t {u ( x kR ) u ( x kR )} d( , R ) 1 D( , R, t, x )Q( )d ]dt d ( , R )                             Then 0 0 0 0 R R s s s k r r R R R R r r R R r r 0 k m r0 0 k 1 R x mR s R R r r0 x mR d ( , R, t) u , D W D W (t ) 2u (x )ch tdt D W (t) {u (x kR) u (x kR)}dt d( , R) 1 D W (t) D( , R, t, x )Q( )d dt ...(12) d( , R)                            Now, we want to find  s (x,y ,) – sRW (y –x), by using the definition of  s (x,y ,) and the relations 0 s s s R r r R R r W ( t) v (y) u , D W O(1)    and 2 s sr R r 2 r0 W ( t) cos dt (1 )       , see ([1],[2]). We have the followin g  s (x,y ,) – sRW (y –x)= s r r r O(1) u ( x) v ( y)  2 sr 2 r (1 )     0 s r r R R r v ( y) u , D W   s (x,y ,) – sRW (y –x)= 0 R s s r r R R r r 0 O(1) u (x)v (y) D W (t ) cos tdt     0 0 0 R R s s k r r r R R r r R R r r r 0 k m r0 0 k 1 R x mR s r r R R r r r0 x mR d ( , R , t) 2 v ( y)u (x )D W ( t)ch tdt v ( y) D W (t) {u ( x kR) d( , R ) 1 u ( x kR )}dt v (y ) D W ( t) D( , R , t, x )Q( )d dt ...(13) d ( , R )                            By using the interval transformation R 0 0 R       , we have 0 0 0 0 s s s R r r R R r r R R R s s k r R R r r r R R r r r 0 k m r0 0 k 1 R s r R R r r r0 x (x, y, ) w ( y x ) O(1) u (x) v (y)[D w (t ) cos tdt d ( , R, t ) D w (t )(cos t 2ch t)dt] v (y)D w (t ) {u (x kR) u ( x kR )}dt d( , R ) 1 v ( y)D w (t ) D( , R, t, x d( , R )                                      x mR mR )Q( )d dt     Now, we want to find the estimates of the integrals in the right hand side, so we will denote to this integrals by A1,…,A4 resp ectively. First ly , we know from [1] IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 s 0 1 2 r r c(R , s) A 1       0 R s 2 R R r r 0 A D W ( t)(cos t 2ch t)dt   By the relations in [4] r r r t r 1 1 0 t cos t 1 e t              and r r r r t r 1 c t 0 t c h t 1 ce t               We have the followin g inequ ality r r r r r t r 1 1 c t 0 t cos t 2ch t 1 ce t                  Then r r 0 r r r r r 1 R ts s 2 R R r R 10 Rs s s 1 s 0 1 2 31 1 1 0 t 0 t t R A D W ( t) (1 c t) dt c W (t) e dt c c max t c max t c e max t                                      Since RR0, we get r 0 r r r 0 r 0 Rs s 1 s 2 0 31 1 0 t t R Rs s 1 s 0 0 3 0 R s 0 A c c max t c e max t c [cR c e R ] c(R , s) e                                  0 R s k r 3 R R r r 0 k m r0 k 1 d ( , R , t) A D W (t) {u ( x kR ) u ( x kR )}dt d( , R )         By using the following inequality in [4] r r, m 1Re( 2 )R0 r r d ( , R , t) ce d ( , R)     …(14) and IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 r, m 1 r, m 1 r, m k 1 r, m k 2 r ,m 2 r ,m 1 r, m 1 r ,1 r, 2 r, m 2 Re ( t 2 R ) Re t Re( ... ) Rk r r r Re ( t 2 R ) Re( .. . ) s ce if k 2 d ( , R , t) min(1, t) ce if 2 k m d( , R ) ce if k m                                           …(15) where r,1 r, m 1 Re( m .. . ) R rQ( , R ) e        and d(r,R)=Q(r,R). Hence 0 R s k r R R 0 k m r0 k 1 d ( , R, t) D W ( t) dt d( , R )      0 R 1 1 s s k r2 2 R 1 s 0 k m r20 k 1 d ( , R , t) D a(s) t J ( t ) dt d( , R )              r 0 r 1 1 1 s s k r2 2 R 1 s 0 k m r0 2 k 1 R 1 1 s s k r2 2 1 s 0 k m1 r2 k 1 d ( , R, t) D a(s) t J ( ( t)) dt d ( , R ) d ( , R , t) a(s) t J ( (t)) dt d( , R)                                      0 R s k r R R 0 k m r0 k 1 d ( , R, t) D W ( t) dt d( , R )      r r r 13 ss ts 1 s22 1 1 0 t t R c(s) max t 3c(s) e max t               r r r 13 ss ts 1 s22 1 1 0 t t R c(s) max t 3c(s) e max t               From (14), (15) r r r r 0 r 0 r 0 3 1 ts s 1 s2 2 1 1 0 t t R 3 1 Rs s 1 s2 2 0 0 3 1 Rs s 2 2 0 0 Rs 0 c(s) ( max t 3 e max t ) c(s) ( R 3 e R ) c(s) R ( R 3 e ) c '( R , s) e                                       Then r 0 R 0 Rs 3 0 r L ( k ) A c '( R , s) e u   IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 To find the estimate of A4 0 R x mR s 4 R R r r0 x mR 1 A D W ( t) D( , R , t, x )Q( ) d dt d( , R )           0 r 0 R x m R s 4 R R r r0 x m R R x mR 1 2m Rs R R r r 0 x mR 1 A D W (t) D( , R, t, x )Q( ) d dt d( , R ) D W (t) c min{1, t }e Q( ) d dt                      From [5]. By using the argument in [6],   2s 2s x 2 s L ( K ) L (K ) x 2 s Q( ) d c u u *        We have 4A  r0 mR mR R 1 2 m Rs R R r r r rL ( K ) L ( K ) 0 ˆD W ( t) min{1, t }e u u dt         r r 0 0mR mR r 1 R 1 2 m R s s r r r R R r R RL ( K ) L ( K ) 10 ˆc e u u D W ( t) tdt D W ( t) dt                            r mR mR r r r mR mR r 0 mR mR 3 1 1 2 m s sR s 1 s2 2 4 r rL ( K ) L ( K ) 1 1 0 t t R 3 1 Rs s 1 s2 2 r r 0 0L (K ) L ( K ) Rs 0 r rL ( K ) L ( K ) ˆA c '(s) e u u max t max t ˆc '(s) e u u R R ˆc '( R , s) e u u                                                     Now, from (3) and ([7],lemma (3)) and Cauchy -Schwartz inequality we obtain the followin g estimates s r r i r v (y ) u (x ) A O(1) (i=1,2) s r i r v (y ) A O(1) (i=3,4) From this we obtain  s (x,y ,) – sRW (y –x)= sO(1) …(16) Proof of the The orem (2.1) Consider the op erator s s sL (f , x ) [ (f , x) S (f , x )]     IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 From L 1 (G) into C(K) for any comp act KG we know that r 2 s sr r r 2 ( f , x) f , v u (x )(1 )          s s r r R r S (f , x ) f , v u , W     Then s s s s s s R r r R r ( f , x) S (f , x) f , W f , v u , W            By using Cauchy -Schwartz inequality and the p revious result in (16), we obtain C(K) L (f , x) M  For any fixed KG. Furt her there exists HL 1 (G) s.t . H =L 1 (G) and the relation L(h,x)0 holds un iformly in x on the K for any hH. Hence the result follows by the Banach-Steinhaus theorem. Re ferences 1. Loi, N.H.,(1987), On The Riesz M eans of Exp ansion by Riesz Bases Formed by Eigenfunctions of the Schrödinger Op erator, Periodica, M ath., Hung, 18(1):59-71. 2. Tahir, M .B., (1993), On Equiconvergence of Riesz M eans of Eigenfunction Exp ansions, Acta. M ath.,Hung., 30: 314-328. 3. Komornik, V., (1984), On the Equiconvergence of Exp ansions by Riesz Bases Formed by Eigen Functions of a Linear Differential Op erator of Order 2n, Acta. M ath.Hung. 44(3-4): 311-325. 4. Tahir, M .B., (1992), On The Convergence of Some Eigenfunction Exp ansions, Ph.D. Thesis, Budapest, Hung. 5. Stemp ak، K. (1996), Equiconvergence for Laguerre Function Series, Studia M athematica, 118 (3). 6. Joo', I. and Komarnik,V., (1983), On t he Equiconvergence of Exp ansions by Riesz Bases Formed by Eigenfunctions of The Schrödinger op erator, Acta. Sci.M ath.,46: 357-375. 7. Horva'th, M ., Joo', I. and Komarnik, V., An Equiconvergence Theorem, Ann. Univ.Sci. Budapest, Sect., M ath., 31: 19-26. IHJPAS 2) 2( 23مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 010 حول متوسطات ریس للتوسیعات بواساطة قواعد ریس المصاغة باستخدام 2mالدوال الذاتیة للمؤثر التفاضلي االعتیادي من الرتبة د عصوادأسماء عب جامعة بغداد،ابن الھیثم -كلیة التربیة ،قسم الریاضیات الخالصة ھدف ھذا البحث برھنة نظریة حول متوسطات ریس للتوسیعات بالنسبة الى قواعد ریس التي توسع النتائج السابقة لـ المتحققة على مؤثر شرودنكر والمؤثر التفاضلي االعتیادي من الرتبة الرابعة الى مؤثر تفاضلي من الرتبة ) لوي وطاھر( 2m العتیاديباستخدام الدوال الذاتیة للمؤثر التفاضلي ا. IHJPAS