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 مـن L,p (1  p < )تم في هذا البحث ایجاد تقدیر افضـل القتـراب الـدوال المقیـدة القابلـة للقیـاس فـي الفضـاءات .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. SCI. VOL.23 (2) 2010 Direct and Inverse Inequalities for Jackson Polynomials of 2- Periodic Bounded Measurable Functions in Locally Clobal Norms S.K.Jassim, N.J.Mohamed Department of Mathematics,College of Science, University of Al-Mustansirya Department of Mathematics-Ibn-Al-Haitham, College of Education, University of Baghdad Abstract Convergence properties of Jackson polynomials have been considered by Zugmund [1,ch.X] in (1959) and J.Szbados [2], (p=) while in (1983) V.A.Popov and J.Szabados [3] (1  p  ) have proved a direct inequality for Jackson polynomials in Lp-space of 2- periodic bounded Riemann integrable functions (f  R) in terms of some modulus of continuity . In 1991 S.K.Jassim proved direct and inverse inequality for Jackson polynomials in locally global norms (L,p) of 2-periodic bounded measurable functions (f  L) in terms of suitable Peetre K-functional [4]. Now the aim of our paper is to proved direct and inverse inequalities for Jackson polynomials of (f  L) in (L,p) in terms of the average modulus of continuity . Introduction We denote the set of 2-periodic 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 the following:- IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (2) 2010 n 1 sin(n )u 2D (u) u 2sin( ) 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 the 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 polynomial of function f  L. Now we will use the so called average modulus of continuity to solve the problem 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 kth 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 kth modulus of smoothness for fLp, k is defined by: 9. k k 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 kth 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 positive 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 proved to be very useful in some approximation theoretical problems 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.  k k 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. SCI. VOL.23 (2) 2010 13. p p L L t (f , ) sup f ( t) f ( )          for  constant and 14.  k k h p (f , ) sup f ( ) : h      also for  constant, be the kth 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 property 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 equipped with semi norm ,p  . By Tn we denote the set of all trigonometric polynomials in R of degree not greater than n. The best approximation to a given continuous function with trigonometric polynomials from Tn on the interval X is given by: 16.  T n nE (f : X) inf f ( ) T( ) : T T       . While the best approximation of a function fLp(X) with trigonometric polynomials from Tn in the metric of the space Lp is given by: 17.  T n p np E (f ) inf f ( ) T( ) : T T     . We also define the best approximation of a function fL(X) with trigonometric polynomials from Tn in the metric of the spaces Lp or L,p are respectively given by:- 18.  T n ,p n,p (X ) E (f ) inf f ( ) T( ) : T T       . The best one sided approximation of a function fL(X) with trigonometric polynomials from Tn in the metric of the space Lp or L,p are respectively given by 19.  T n p n n np(X) E (f ) inf T ( ) T ( ) : T T ,T (x) f (x) T (x), x X           . 20.  T n ,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 1 p p pp p K(f ,t, L ,w ) inf f g t g ' : g w    , 23.  1 1 t,p p pt,p p K(f ;t, L , w ) inf f g t g ' : g w    , 24.  1 1 1 1 p p p p pp p p K(f ,t, L ,w , w ) inf f g t g ' t g : g w w       , 25.  1 1 1 1 t,p p p p pt ,p p p K(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. SCI. 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 Theorem: [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 Theorem: [3] For 2-periodic 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 Theorem: [4] For f  L, then 41. 1 1 1 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 1 1 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 Theorem: [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 paper is to find a better estimation for the rate of convergence in L,p space of (f  L) by Jackson polynomials in terms of some modulus of functions. We shall prove direct and inverse theorems of (f  L) by Jackson polynomials 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. SCI. VOL.23 (2) 2010 2.1 Theorem: (Direct Theorem) Let f  L, then 1 p 1 p 1n ,p n 1 p 1 1 c (f, ) c (f , ) , p 1, n nf J (f ) 1 c (f, ) , 1 p . n                 2.2 Theorem: (Inverse Theorem) 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 T n 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 polynomial 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 1p f h A  and 2 p h ' A then we get T n n p 1 2p 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. SCI. VOL.23 (2) 2010 1 2 2n 1 1 p n 10 2n 2 1 p n 1 p 0 1 2n 2 n p 1 2n 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. SCI. VOL.23 (2) 2010 Let t > 0, f  L, we have 1p 47. 1 1 1 1 p 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 1 p 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                                                   Tn 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. SCI. VOL.23 (2) 2010 Proof of Theorem 2.1 (Direct Theorem) Let nT Tn such that T n p n n p E (f ) T T   then by using linearity of Jackson polynomials, (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 nC 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 nC 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 nC 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 nC E (f ) 1 C E (f ) C (f, ) , 1 p n C E (f )                                                 1 p 8 1 p p 10 1 p 1 p 1 p 1 p 1 1 (f, ) C (f , ) , p 1, n nC 1 (f, ) , 1 p n 1 1 (f , ) (f , ) , p 1, n nC 1 (f , ) , 1 p n                               IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (2) 2010 Proof of Theorem 2.2 Let g 1 pW 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 1 pW  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 Polynomials, Acta. Math., Sci. Hung., 24,399-406. 3. Popov, V.A. and Szabados, (1984), On the Convergence and Saturation of the Jackson on Polynomials in Lp-Spaces Approximation Theory and it's Appl., 1, 1-10. 4. Jassim, S.K. (1990), Direct and Inverse Inequalities for Some Discrete of Bounded Measurable Functions, Serdica. Bulgaria Mathematical Publications, 17. 5. Hristov, V.H. (1989), Best One Sided Approximation and Mean Approximations by Interpolation Polynomials of Periodic Functions, Math. Balkanica, New Series, 3,( 3-4): 418-429. 6. Popov, V.A. and Sendov, B. (1988), The Average Modulus of Smoothness Wiley and Sons. 7. Jassim, S.K., (1991), One-Sided Approximations and Approximations with Discrete Operators, Sofya University. 8. Jassim, S.K. (1990), Best One Sided Approximation with Algebraic Polynomials Serdica Bulgaria Mathematics Publications, 16, :263-269. 9. Timan, A.F. (1960), Approximation Theory of Function, Moscow in Russian Language. IHJPAS