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 Operator of 2m- th Order A. A. Aswhad Department of Mathematics, College of Education Ibn-Al-Haitham , University of Baghdad Abstract: The aim of this paper is to prove a theorem on the Riesz means of expansions with respect to Riesz bases, which extends the previous results of [1] and [2] on the Schrödinger operator and the ordinary differential operator of 4-th order to the operator of order 2m by using the eigen functions of the ordinary differential operator. Some Symbols that used in the paper: the uniform norm. <,> the inner product in L2. G the set of all boundary elements of G. û the dual function of u. 1. Introduction The theory of non-self adjoint differential operators has great importance in several applications, many mathematicians worked on the equiconvergence theorem for this operators like Ilin, Joó, Komornik and Tahir for the Schrödinger operator see [3], [4] and for Laguerre functions see [5]. By using the method and results of this papers, we shall prove un equiconvergence theorem for the ordinary differential operator of order 2m. Let G  R be an arbitrary finite open interval q(x)L 1(G) an arbitaray complex function and consider the operator Lur:= (2m) ru +q(x)ur, where m . Given a complex number , the function u – 1:GC, u – 10 is called an eigenfunction of order – 1 of the operator 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 operator L with the eigenvalue  if ur together with its derivative is absolutely continuous on every compact subinterval of G and if for almost all xG the equation Lur(x)=ur(x) –ur – 1(x) holds, 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 operator L. Let r (resp.0r) denote the eigenvalue (resp . the order) of ur and assume that the following conditions are satisfied: r sup 0r< ...(1) In case 0r > 0, ur – Lur = ur – 1 …(2) Suppose the biorthogonal system (vr) of the system (ur) have the property r 2 r r L (G ) 1 v V         …(3) Now, consider the Riesz mean of the biorthogonal series 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 dual system of (ur), i.e., (vr)L 2 (G) and =rj . Given any compact 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 arbitrary and define s RW :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):=2s(2) – 1/2(s+1). Moreover, for any function fL1 (G), x  RG define x R s s R x R S (f ,x) : W (t)(y x)f (y)dy     …(6) Denote by  s(x,y,) the spectral 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 0RD :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 Theorem: Given any compact interval KG, for all 0s<1/2, >0 and fL 1 (G) the estimate s s s(f, x) S (f , x) O(1)    …(9) Holds uniformly on the compact interval KG. For the proof 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 2R) 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 expanding this determinant according to the first row with the definition of rû we get the following 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 sh (kR x ) sh (t x ) d ( ,R, t) d ( ,R) , if x m m sh (kR x ) D( R, t, x ) d ( ,R, t) , if t x 2 m sh (k R x ) d ( ,R, t) , if ( j 1)R x m                                                                 m j k m p 1 jR, 3 j                We want to prove the following estimate s s s R(x, y, ) W ( y x ) O(1)     …(11) By using equation (10), we count the Fourier coefficients of the function s RW (y – x) with respect to the system (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,) – s RW (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 following s(x,y,) – s RW (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,) – s RW (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) 2v (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 respectively. Firstly, 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 following inequality 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 2R ) Re t Re( ... )Rk r r r Re (t 2R ) 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 1Re( 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 mR s 4 R R r r0 x mR 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 2s L (K ) L (K ) x 2s Q( ) d c u u *        We have 4A  r 0 mR mR R 1 2m 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 2m 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 2m 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 following 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,) – s RW (y–x)= sO(1) …(16) Proof of the Theorem (2.1) Consider the operator 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 compact 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 previous result in (16), we obtain C(K) L (f , x) M  For any fixed KG. Further there exists HL1(G) s.t . H =L1(G) and the relation L(h,x)0 holds uniformly in x on the K for any hH. Hence the result follows by the Banach-Steinhaus theorem. References 1. Loi, N.H.,(1987), On The Riesz Means of Expansion by Riesz Bases Formed by Eigenfunctions of the Schrödinger Operator, Periodica, Math., Hung, 18(1):59-71. 2. Tahir, M.B., (1993), On Equiconvergence of Riesz Means of Eigenfunction Expansions, Acta. Math.,Hung., 30: 314-328. 3. Komornik, V., (1984), On the Equiconvergence of Expansions by Riesz Bases Formed by Eigen Functions of a Linear Differential Operator of Order 2n, Acta. Math.Hung. 44(3-4): 311-325. 4. Tahir, M.B., (1992), On The Convergence of Some Eigenfunction Expansions, Ph.D. Thesis, Budapest, Hung. 5. Stempak، K. (1996), Equiconvergence for Laguerre Function Series, Studia Mathematica, 118 (3). 6. Joo', I. and Komarnik,V., (1983), On the Equiconvergence of Expansions by Riesz Bases Formed by Eigenfunctions of The Schrödinger operator, Acta. Sci.Math.,46: 357-375. 7. Horva'th, M., Joo', I. and Komarnik, V., An Equiconvergence Theorem, Ann. Univ.Sci. Budapest, Sect., Math., 31: 19-26. IHJPAS 2010) 2( 23مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 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