Microsoft Word - 271-281 Mathematics | 271 2016) عام 2العدد ( 29مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 On the Degree of Best Approximation of Unbounded Functions by Algebraic Polynomial Alaa A. Auad Dept. of Mathematics / College of Education for Pure Science/University of Al- Anbar Received in:27/March/2016,Accepted in:22/May/2016 Abstract In this paper we introduce a new class of degree of best algebraic approximation polynomial Α , , for unbounded functions in weighted space Lp,α(X), 1 ∞ .We shall prove direct and converse theorems for best algebraic approximation in terms modulus of smoothness in weighted space. Keyword: degree of best approximation, unbounded functions, weight space, modulus of smoothness. Mathematics | 272 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 1. Introduction Many papers have recently appeared connected with best approximation to functions of one variable see [1,2,3] and the references there. In this paper we shall consider the degree of best algebraic approximation of unbounded functions. Indeed, in terms of the modulus of smoothness direct ( Jackson – type ) as well as inverse ( Bernstein – type ) theorems are established so that a constructive theory of unbounded functions may developed in weighted space Lp,α(X). Let ∞,∞ , by‖.‖ we denote the Lp(X)- norm, 1 p ˂ ∞ and ‖ ‖ | | ∞ ………………………….(1) and define for a suitable set , of all weight functions on open interval , such that | | , where M is positive real number and : → weight function. We shall denote by Lp,α(X) the space of all unbounded functions on , which are equipped with the following norm ‖ ‖ , | | ∞. ……..(2) For ∈ , and ∈ , we define the local modulus of order of in point as follows , , , | | ∆ ∶ , ∈ , ∩ ………...(3) where ∆ ∑ 1 , ∈ 0 ……….(4). Also the modulus of smoothness of order of function is the following function of : , , | | ∆ . , …………………………….…(5). Let be the set of natural numbers and ℙ the set of all algebraic polynomials of degree less than or equal to ∈ . The degree of best algebraic approximation of order of the function ∈ , is given by , ‖ ‖ , ; ∈ ℙ …………………….……..(6). The most essential consequence of the proved here is Corollary : For 0 and 1 ∞, we have , , Ο if and only if , Ο . Mathematics | 273 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 If and ,are positive integer and 0, then , Ο equivalently , , Ο is valid if and only if has a derivative of order satisfying , ∞ and , , Ο . Further details and elementary properties of , , see [4,5]. With the aid of these concepts one may now work out a constructive theory of functions for the weight space , , parallel to classical theory of Jackson-Bernstein for , , see sections 2 and 3 for detail. We formulate and prove our Jackson-type theorem and in section 3, we formulate and prove our Bernstein-type theorem. The corollary announced above follows from direct theorem (1) of section 2 combined with converse theorem (2) of section 3. 2. The Direct Algebraic Approximation Theorem: Let ∈ , ,1 ∞, natural number and 0. We define Α , ∑ 1 ………………….…(7). We set | | √ 0 | | √ and √ , Α , , . It is clear that ∈ ℙ . Lemma 1 [4]: We have for every 1 ∞, 0 1 and every positive function Α , ∆ and Α , , | | ∆ , 1,2,…, . Prove of this lemma see [4]. Lemma 2 : For every ∈ , and 1 ∞, we have Mathematics | 274 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 ‖ ‖ , ,√ , and , ,√ , . Proof : From theorem of Marchoud see [5], we have , , ‖ ‖ , , , since for | | √ , we obtain | | for ∈ ∞,∞ . Consequently ‖ ‖ , | | ∆ ∆ . , ,√ , ; √ . Now ‖ ‖ , ‖ ‖ , ∆ . , ‖ ‖ , ,√ , since ‖ ‖ , ,√ , so ‖ ‖ , ,√ , . We have Α , , , for ∈ and by lemma 1, , ∑ Α , , Mathematics | 275 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 ∑ ∆ , . As consequence of 0 for | | √ , hence ∆ 0 for | | √ , so , ,√ , ; √ . ■ Theorem 1 : Let ∈ , and 1 ∞. For every natural there exists constant depending on , such that , ,√ , . Proof : From definition of degree of best algebraic approximation, we have , Α , , . , , where Α , ∈ ℙ . Hence from lemma 2, we obtain , ,√ , ■ 3. The Converse Approximation Theorem : We need the following lemmas to prove converse theorem Lemma 3 [6] : We have for every ∈ ℙ , 1 ∞ and n=1,2,… / . Lemma 4 [7] : We have for every ∈ ℙ , 1 ∞ and n=1,2,… , ‖ ‖ , . Lemma 5 : Let ∈ , , 1 ∞ , ∈ ℙ and ‖ ‖ , . Mathematics | 276 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 We have ,√ , ‖ ‖ , ∑ 1 , . Proof : Let ∈ ℙ be a polynomial satisfying ‖ ‖ , 2 , . By using lemma 4, we obtain , , ‖ ‖ , | | , , , We have , from properties of modulus of continuity and lemma 4 , , , , , , ‖ ‖ , ‖ ‖ , | | | | ‖ ‖ , 2‖ ‖ , ‖ ‖ , , ‖ ‖ , so , , , , ‖ ‖ , also, we can prove Mathematics | 277 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 , , , 2 / , ‖ ‖ , ∑ 1 , ‖ ‖ , …(8). Let 2 2 , , ‖ ‖ , | | | | ‖ ‖ , ‖ ‖ , …………………..(9) From (8) and (9), we get ,√ , ‖ ‖ , ∑ 1 , . ■ Theorem 2 : Let ∈ , ,1 ∞. For every natural number there exist a constant depending on , such that , √ , √ ‖ ‖ , ∑ 1 , , 1,2,…. . Proof : We insert in lemma 5 , , √ and take in consideration that by the monotony of , , ∑ , So, , √ , √ ‖ ‖ , ∑ 1 , √ ‖ ‖ , ∑ 1 , . ■ Mathematics | 278 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 Theorem 3 : Let ∈ , ,1 ∞ and ∑ 1 , ∞. Then has derivative of order , ∈ , and , , ∑ 1 , Proof : Let ∈ ℙ satisfy ‖ ‖ , 2 , , we have in consequence of lemma 3 , , , , , the sequence is convergent in weighted space , , we denote it is limit by ∈ , . Since , → 0, is derivative of order of . Moreover , we have , , ∑ 2 / , 2 / , ∑ 1 , ……(10) Let 2 2 , form lemma 3 we have , ‖ ‖ , , …………………..(11) From (10) and (11), we obtain , , , ∑ 1 , ■ Theorem 4 : We have for every natural and m every ∈ , Mathematics | 279 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 , √ , , √ ∑ 1 , ∑ 1∞ , . Proof : From theorem 2 and theorem 3 , we have , √ , , √ , ∑ 1 , , , ∑ 1 , ..(12) As well as from (10) and take 0 , ∑ 1 , ……………………….(13) new, from (12) and (13), we obtain , √ , , √ ∑ 1 , ∑ 1 , ∑ 1 , , √ ∑ 1 , ∑ 1 , ∑ 1 ∑ 1 , , √ ∑ 1 , ∑ ∑ 1 1 , , √ ∑ 1 , ∑ 1 , ■ Acknowledgments I am grateful to Prof. S. Jassim , from whom I learnt the approximation of unbounded functions played in development of the nation of weighted spaces. Mathematics | 280 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 References 1. Adell, A., and Bustamante J.,(2014) , Polynomial operators for one-sided Lp- approximation to Riemann integrable functions, Journal of inequalities and applications, 2014/1/494. 2. Draganov, R. ,(2011),On the approximation by convolution operators in homogeneous Banach spaces of periodic functions, Mathematica Balkanica, new series, Vol. 25, fasc., 1-2, 39-59. 3. Jassim S. and Alaa A.,(2014) ,Direct and inverse theorems in weighted space Lp,w(X)” , International journal of scientific and engineering research, Vol. 5, Iss. 4, 833-839. 4. Popov V. A., (1983) ,The Averaged modli and their Function Spaces, In. Constructive function theory 81,published House Bulg. Acad. Sci. , Sofia. 5. Sendov B. and Popov V. A., (1988), The Averaged modli of Smoothness, Wiley, New York. 6. Freud G.,(1972), On Direct and Converse Theorems in the Theory of Weighted Polynomial Approximation, Published by Springer- verlag, Math. Z., 126, 123-136. 7. Devore A. and Lornentz G., (1993), Constructive Approximation ,Springer-verlag, New York. Mathematics | 281 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 دراسة درجة أفضل تقريب للدوال الغير مقيدة بواسطة متعددات الحدود الجبرية عالء عدنان عواد جامعة االنبارقسم الرياضيات / كلية التربية للعلوم الصرفة/ 2016أيار//22،قبل في:2016اذار//27استلم في: الخالصة افضل تقريب للدوال الغير مقيدة بواسطة متعددات الحدود الجبرية في في هذا البحث عرضنا نوع جديد من درجة )فضاء الوزن , 1)و ( . سوف نبرهن المبرهنات المباشرة والمعكوسة ألفضل تقريب لهذه الدوال ∞ بواسطة متعددات الحدود الجبرية بشروط مقياس النعومة في فضاء الوزن. : درجة أفضل تقريب , الدوال الغير مقيدة , فضاء الوزن , مقياس النعومة. الكلمات المفتاحية