2010) 2( 32المجلد مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة النظریات المباشرة والعكسیة لمتعددات حدود جاكسون للدوال الدوریة المقیدة القابلة للقیاس في الفضاءات المحلیة صاحب كحیط جاسم ، نادیة جاسم محمد المستنصریةالجامعة ، العلوم كلیة،قسم الریاضیات جامعة بغداد ،ابن الهیثم -كلیة التربیة ،قسم الریاضیات الخالصة L,p (1  pتم في هذا البحث ایجاد تقدیر افضـل القتـراب الـدوال المقیـدة القابلـة للقیـاس فـي الفضـاءات < ) مـن .وفپوپالتقدیرات التي وجدها 1كذلك تم ایجاد العالقة بین نموذج القیاس p 1 (f , ) n  والفرق بین الدالةf ومتعددة جاكسون اي وجدنا ان 1 p n p 1 (f , ) f J (f ) n   IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Direct and Inverse Inequalities fo r Jackson Polynomials of 2- Periodic Bounded Measurable Functions in Locally Clobal Norms S.K.Jas sim, N.J.Mohamed Departme nt of Mathematics,College of Science, Unive rsity of Al-Mustansirya Departme nt of Mathematics-Ibn-Al-Haitham, College of Education, Unive rsity of Baghdad Abstract Convergence p rop erties of Jackson p olynomials have been considered by Zugmund [1,ch.X] in (1959) and J.Szbados [2], (p =) while in (1983) V.A.Pop ov and J.Szabados [3] (1  p  ) have p roved a direct inequality for Jackson p olynomials in Lp-sp ace of 2- p eriodic bounded Riemann integrable functions (f  R) in terms of some modulus of continuity . In 1991 S.K.Jassim p roved direct and inverse inequality for Jackson p olynomials in locally global norms (L,p) of 2-p eriodic bounded measurable functions (f  L) in terms of suitable Peetre K-functional [4]. Now the aim of our p aper is to p roved direct and inverse inequalities for Jackson p olynomials of (f  L) in (L,p) in terms of the average modulus of continuity . Introduction We denote the set of 2-p eriodic bounded measurable functions with usual sup -norm by L  1.  L ( X ) L ( X )L ( X) f : f sup{ f ( x) x X} , f f          and the Lp-norm (1  p < ) of f  Lp by Lp f  2. p p p 1 p p p L (X ) L ( X) X L (X) f : f ( f (x ) dx) ; f f              . and the direct norm is defined by : 3. p n 1pn p p n k ,n( x ) k 0 1 ( ) f : f ( f ( x ) ) , n 1               where p p n n k ,n ( x ) 2k (x ) , (k 0,1, 2, ..., n), f f n 1        . Now let us consider the Dirich let kernel of degree n 4. n n v 1 1 D ( u) cos(vu ) 2      uR, n=0,1,… then it is easy to get t he following:- IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 n 1 sin(n )u 2D ( u) u 2 sin( ) 2   and n 1 D ( u) 1      . 5. let n 0 1 n 1 K (u ) [D ( u) D (u ) ... D (u )] n 1      be the Fejer kernel of degree not grater than n, also it is easy to get t he following:- 2 n n n 2 k 1 u sin (n 1) 1 1 12K (u ) , K (u) (n k 1) cos(ku ) u2(n 1) 2 n 1 sin ( ) 2           and nK (u) du 1    . 6. n n k ,n n k ,n k 0 2 J (f , x ) f (x )K ( x x ) n 2      is the so called Jackson p olynomial of function f  L. Now we will use the so called average modulus of continuity to solve the p roblem in locally global norm (L,p). The locally modulus of continuity for (f  L) is defined by 7. n n h h (f , x, h ) sup f (x ') f (x ), x ', x x , x 2 2               , h is a constant number while k th average modulus of smoothness for fLp is defined by: 8. p p K L K L (f , ) (f , , )      1  p  ,  > 0 and k . where the k th modulus of smoothness for fLp, k is defined by : 9. kk h k k (f , x , ) sup f ( t) : t, t kh x , x X 2 2                    . Now we set k k m k m 0h k ( 1) f ( x mh) if x or x h X f ( x) m 0 otherwise.                    Then we introduce the definition of k th local modulus of Lp-continuity for fLp, (1p) and k . 10. 1 ( x ) p p k k p h ( x ) 1 (f , x , ( x)) f (x ) dv 2 ( x)                where (x) is an arbitrary p ositive function of x, but here we shall consider only the case (x) is constant. The k th average modulus of smoothness was first introduced by B.Sendov in 1983 and p roved to be very useful in some app roximation theoretical p roblems where the ordinary modulus of continuity (f,) is defined by the following:- 11. (f,)=sup {f(x') – f(x''), x' – x" , x', x"X}. while the k th ordinary modulus of continuity is defined by the following:- 12.  kk h(f , ) sup f (x ) : h , x, x kh X       . Now let us consider the definition of ordinary Lp-modulus of continuity ((f,)p). IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 13. p p L L t (f , ) sup f ( t) f ( )          for  constant and 14.  kk h p(f , ) sup f ( ) : h      also for  constant, be the k th ordinary Lp-modulus of continuity of fLp. Now for fLp instead of usual sup -norm, let us consider the family of semi norm (locally global norm) for  > 0, (1p). 15. 1 p p ,p X f f (x ) dx         ,where f(X)=sup {f(t):tU(,x)} and U(,x)={y X:x–y}. The main p rop erty of these semi norm is that they do not necessarily vanish if f = 0 Lebesgue almost every where. Let us denoted by L,p the set of functions from L which equip p ed with semi norm ,p  . By Tn we denote the set of all trigonometric p olynomials in R of degree not greater than n. The best app roximation to a given continuous function with trigonometric p olynomials from Tn on the interval X is given by : 16.  Tn nE (f : X) inf f ( ) T( ) : T T     . While the best app roximation of a function fLp(X) with trigonometric p olynomials from Tn in the metric of the sp ace Lp is given by : 17.  Tn p npE (f ) inf f ( ) T( ) : T T     . We also define the best app roximation of a function fL(X) with trigonometric p olynomials from Tn in the metric of the sp aces Lp or L,p are resp ectively given by :- 18.  Tn ,p n,p ( X )E (f ) inf f ( ) T( ) : T T      . The best one sided app roximation of a function fL(X) with t rigonometric p olynomials from Tn in the metric of the sp ace Lp or L,p are resp ectively given by 19.  Tn p n n np ( X)E (f ) inf T ( ) T ( ) : T T , T (x ) f (x ) T ( x), x X           . 20.  Tn ,p n n n,p ( X)E (f ) inf T ( ) T ( ) : T T , T ( x ) f (x ) T ( x), x X            . Now let f be a function defined on a domain D then the Stecklov transformation is given by 21. 1 / n n 1 / n n f (x ) f ( x t)dt 2    . Let f and g be two functions then we say that f(x) = O{g(x)} if f(x) 0), (1  p  ). 22.  1 1p p pp pK(f , t, L , w ) inf f g t g ' : g w    , 23.  1 1t,p p pt,p pK(f ;t, L , w ) inf f g t g ' : g w    , 24.  1 1 1 1p p p p pp p pK(f , t, L , w , w ) inf f g t g ' t g : g w w       , 25.  1 1 1 1t,p p p p pt ,p p pK(f , t, L , w , w ) inf f g t g ' t g ' : g w w       . Next Bernstein inequalities are given since we use them further on 26. 'n n n npp T n T , T T ,1 p .     27. 'n n n npp T n T , T T ,1 p .     1- Assertions 1.1 Lemma: [3] Let f  Lp, then (1p) 28. p n 1 n pp n kp k 0 1 J (f , ) O(1) f ( t ) O f n            . 1.2 Lemma: [5] Let T  Tn, then (1p) 29. 1 p ,p ( ) p ( ) T C(1 n ) T       . 1.3 Lemma: [3] For f, f ' Lp we have (1p) 30. p p L (f , h ) O(h ) f '  , h is constant. 1.4 Lemma: [6] Let f  L, then we have (1p) 31. 1 p 1 p 1(f , ) (f , ) (f , )        . 1.5 Lemma: [7] Let f  L, then we have (1p) 32. p m 1 ,p m f C f  , (m = 1, 2, 3, …). 1.6 Lemma: [8] For f  L, we have 33. p ( ) ,p ( ) , ( ) ( ) f f f f           . 1.7 Lemma: [6] Let f  L and (1p) we have IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 34. p k 1 k p k L(f , ) ( 2( 1)) (f ; ) , 0         . 1.8 Lemma: [6] For g, g'  Lp, we have (1p) 35. 1 p p(g, ) , g '   . 1.9 Lemma: [5] For f  L and (1p), we have 36. n p n 1 n p ,p n E (f ) E (f ) cE (f )   . 1.10 The orem: [6] For f  L(), we have 37. n p k k p 1 E (f ) c (f , ) n  , 1p. 38. n k k 1 k p k v p v 0 1 (f , ) c n ( v 1) E (f ) n        . 1.11 The orem: [3] For 2-p eriodic bounded Reimann integrable functions we have the following:- 39. n 1 p 1 pp 1 1 f J (f , x) O(1) (f , ) W ( f , ) , p 1, n n             40. n 1 pp 1 f J (f , x) O(1) (f , ) ,1 p . n      where f ( x ) is the conjugate function to f(x). 1.12 The orem: [4] For f  L, then 41. 1 11 p p 1 p 1 ,p n 1 1 1 K(f , , L , W , W ) c (f , ) W (f , ) , p 1, n n n            . 42. 1 11 p 1 p p 1 p ,p n 1 1 1 (f , ) K(f ; ;L , W , W ) c (f , ) 1 p . n n n       1.13 The orem: [9] For f  Lp, we have 43. n k 1 K p v pk v 0 1 C( k) W (f , ) (1 v) E (f ) n n     . 2- Main Results The aim of his p aper is to find a bett er estimation for the rate of convergence in L,p sp ace of (f  L) by Jackson p olynomials in terms of some modulus of functions. We shall p rove direct and inverse theorems of (f  L) by Jackson p olynomials in locally global norms in terms of ordinary Lp-modulus of continuity and average modulus of continuity . IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 2.1 The orem: (Direct The orem) Let f  L, then 1 p 1 p 1n ,p n 1 p 1 1 c (f , ) c (f , ) , p 1, n n f J (f ) 1 c (f , ) , 1 p . n                 2.2 The orem: (Inverse The orem) Let f  L, then n n s(1/ s 1),p (1/ s 1 ),p 1 p s 0 s (1/ s 1),p f J (f ) f J ( f ) , p 1,1 C (f , ) n n f J (f ) , 1 p .                     2.3 Lemma: Let n nT   such that Tn p n n p E (f ) T T   , f  L and n nT ( x) f (x ) T (x )    then 44. n n 1 p 1 pp 1 1 T ' C ( f , ) (f , ) n n           , 1p. Proof: Let n 1 2 f (x ) n f ( x u) du    be the Stecklov transformation n n 1 h( x) f ( x)D ( t) dt       be a trigonometric p olynomial of degree n, such that Dn(t) is the Dirichlet kernel, then by using Bernstein inequality , we get n n pp p n pp n n pp p T ' n T h h ' n T f n f h h ' n T T n f h h '                      Now let us assume that 1pf h A  and 2p h ' A then we get T n n p 1 2 p 1 n np p 1 2 n n n 1 2 n p 1 2n n np 1 2n T ' n (f ) nA A 1 A f h [f ( ) f ( )]D ( t)dt 1 n [f ( ) f ( u)]duD (t )dt 1 n f ( ) f ( u) duD ( t)dt                                        This by using definition of average modulus of continuity and by (34), we get IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 1 2 2n 1 1 p n 10 2n 2 1 p n 1 p 0 1 2 n 2 n p 1 2 n p n 1 1 A n (f , ) duD ( t)dt 2n 1 1 1 (f , ) D (t )dt C (f , ) 2n 2n 1 A h ' n f '( u )duD (t )dt 1 1 1 n f '( ) f ( ) D (t )dt 2n 2n                                        Then by using definition of ordinary Lp-modulus of continuity we get 2 2 1 p 1 A C n (f , ) n   Thus T n n p 1 1 p 2 pp 1 1 T ' n (f ) nC (f , ) C n (f , ) n n         By using (37) n n 1 p 1 pp 1 1 T ' C ( f , ) (f , ) n n           . 2.4 Lemma: Let f  L then we have 1p 45. 1 1n ,p , p n n J (f ) C f . Proof: By using (29), (28) and (32), we get p n 1 p 1n 1 n,p p n n 2 k k 0 3 14 ,p n 1 J (f ) C (1 n ) J (f ) n 1 C O(1) f ( t ) n C f C f         2.5 Lemma: Let f  L then we have 1p 46. k p p(f , ) C f .   Proof: By using definition of average modulus of continuity , we get k k p k hp p k i 0 p p k k (f , ) (f , ) sup f (t) ; t, t kh x , x X 2 2 k k k sup f ( ih) ; t, t kh x , x X i 2 2 C f                                              2.6 Lemma: IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Let t > 0, f  L, we have 1p 47. 1 1 1 1p p p t ,p p pK(f ;t; L , W , W ) K(f ; t;L , W , W )   Proof: By using definition of Peetre K-functional and (33), we get     1 1 1 1 p p p p pp p p 1 1 p pt, p p p 1 1 t, p p p K(f ; t; L , W , W ) inf f g t g ' t g ' ,g W W inf f g t g ' t g ' , g W W K(f ; t; L , W , W )                  2.7 Lemma: Let f, g be two function define on the same domain then for 1p, we have 48. k p k p k p 1 1 1 (f , ) (f g, ) (g , ) n n n      . Proof: The proof follows from the definition of k(f,)p. 2.8 Lemma: For t > 0, f  L, we have 49. 1 1 1p p p p pK(f ;t; L , W ) K(f ; t; L , W , W )  Proof: By using definition of Peetre K-functional     1 1 p p pp p 1 1 p pp p p 1 1 p p p K(f ;t; L , W ) inf f g t g ' ; g W inf f g t g ' t g ' ; g W W K(f ; t;L , W , W )              2.9 Lemma: Let f  L, 1p, then we have 50. T T n S (1/ s 1),p S (1/ s 1 ),p1 1 p p p T s 0 S (1/ s 1),p (f ) (f ) , p 1,1 C K(f ; ;L , W , W ) n n (f ) , 1 p                   Proof: By using (49), (41), (49), (38), (43), (36) and (33), we get 1 1 1 1 p p p (1 / n),p p p 1 1 p 1 p 2 1 p ' 'n T T1 1 1 S p 1 S p S 0 ' n T1 2 S p S 0 T S (1/ s 1 1 K(f ; ; L , W , W ) K(f ; ; L , W , W ) n n 1 1 C (f , ) (f , ) , p 1, n n 1 C (f , ) , 1 p . n C C C E (f ) C E (f ) , p 1, n n C C E (f ) , 1 p . n (f )C n                                                   T n 1), p S (1 / s 1),p T s 0 S (1/ s 1), p (f ) , p 1, (f ) , 1 p                IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Proof of The orem 2.1 (Direct The orem) Let nT  Tn such that T n p n n p E (f ) T T   then by using linearity of Jackson p olynomials, (45), (29), (39), (40), (30), (44) and (46) n n n n n n n(1/ n ),p (1/ n ) ,p ( 1 / n ),p (1/ n ),p n n n n n 1 n n( 1 / n ), p (1/ n ),p (1/ n ),p T 2 n p 3 n n n p 1 n p 1 n p T 2 n p 3 1 n f J (f ) f T T J (T ) J (T f ) T T T J ( T ) C T T C E (f ) C T J ( T ) 1 1 O(1) ( T , ) (T , ) , p 1, n n C E (f ) C 1 O(1) ( T ,                                           p 4 1 n p 4 n p T 2 n p 4 1 n p 4 4 1 n p 4 1 p T 2 n p 5 1 p 1 p n p 4 1 n p 4 1 p ) , 1 p n 1 1 C (T , ) C O( ) T , p 1, n n C E (f ) 1 C (T , ) , 1 p n C1 1 C (T f , ) C (f , ) O( ) n n n 1 1 C E (f ) C (f , ) (f , ) T , p 1, n n 1 1 C (T f , ) C (f , ) , 1 n n                                                      6 n 7 1 p 8 1 pp T 2 n p 6 n 4 1 pp p 1 1 C T f C (f , ) C ( f , ) , p 1, n n C E (f ) 1 C T f C (f , ) , 1 p n                                   6 n n 7 1 p 8 1 ppT n 2 n p(1/ n ),p 6 n n 4 1 pp T 6 n p 7 1 p 8 1 p T 2 n p T 6 n p 4 1 p T 9 n 1 1 C T T C (f , ) C (f , ) , p 1, n n f J (f ) C E (f ) 1 C T T C (f , ) , 1 p n 1 1 C E (f ) C (f , ) C (f , ) , p 1, n n C E (f ) 1 C E (f ) C (f , ) , 1 p n C E (f )                                                 1 p 8 1 p p 1 0 1 p 1 p 1 p 1 p 1 1 (f , ) C (f , ) , p 1, n n C 1 (f , ) , 1 p n 1 1 (f , ) (f , ) , p 1, n n C 1 (f , ) , 1 p n                               IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Proof of The orem 2.2 Let g 1pW then by using the following (48), (46), (35), (47) and (50), we get 1 p 1 p 1 p 1 p p p 1 1 1 (f , ) ((f g ), ) (f , ) n n n 1 1 C f g g ' g ' n n                Now if we take the infimum of g 1pW  1 pW  , we obtain 1 1 1 p 1 p p p 1 1 1 (1/ n ),p p p T T n S (1/ S 1),p S (1 / S 1 ),p2 1 T S 0 S (1/ S 1),p S S(1/ S 1),p (1/ S 1),p S (1/ S 1),p 1 1 (f , ) C K(f ; , L , W , W ) n n 1 C K(f ; , L , W , W ) n E (f ) E (f ) , p 1,C C n E (f ) , 1 p f J ( f ) f J (f ) , p 1,C n f J ( f ) , 1 p .                                  n S 0      References 1. Zugmund, A. (1959), Trigonometric Series, I, II Cambridge. 2. Zabados, J.S. (1973), On the Convergence and Saturation Problem of the Jackson Poly nomials, Acta. M ath., Sci. Hung., 24,399-406. 3. Pop ov, V.A. and Szabados, (1984), On the Convergence and Saturation of the Jackson on Poly nomials in Lp-Sp aces Ap p roximation Theory and it's Ap p l., 1, 1-10. 4. Jassim, S.K. (1990), Direct and Inverse Inequalities for Some Discrete of Bounded M easurable Functions, Serdica. Bulgaria M athematical Publications, 17. 5. Hrist ov, V.H. (1989), Best One Sided Ap p roximation and M ean Ap p roximations by Interpolation Poly nomials of Periodic Functions, M ath. Balkanica, New Series, 3,( 3-4): 418-429. 6. Pop ov, V.A. and Sendov, B. (1988), The Average M odulus of Smoothness Wiley and Sons. 7. Jassim, S.K., (1991), One-Sided Ap p roximations and Ap p roximations with Discrete Op erators, Sofya University . 8. Jassim, S.K. (1990), Best One Sided Ap p roximation with Algebraic Poly nomials Serdica Bulgaria M athematics Publications, 16, :263-269. 9. Timan, A.F. (1960), Ap p roximation Theory of Function, M oscow in Russian Language. IHJPAS