On Co-Positive Approximation of Unbounded Functions in Weighted Spaces

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1 On Co-Positive Approximation of Unbounded Functions in Weighted Spaces Alaa A Auad 1 and Alaa M FAL Jumaili 2 1 Department of Maematics, College of Education for pure Science, University of Anbar, Al-Ramadi City, Iraq 2 Department of Maematics, College of Education for pure Science, University of Anbar, Al-Ramadi City, Iraq Orcid : Abstract The aim of present is article is to introduce a new meod for establishing polynomials and spline approximation to unbounded functions in weighted space in terms of averaged modulus, modulus of smooness and Ditzian-Totik modulus Keywords: Weighted space, unbounded functions, polynomials and spline Maematics Subject Classifications: 41A05 and 41A15 INTRODUCTION on, such at be a set of measurable functions by (4) Also an averaged modulus of degree defined for every functions in by (5) We denoted degree Ditzian-Totik modulus in e via (6) (1) be a set of each weighted functions on (s t) where and be a weight function, be e space of each unbounded functions which e following rule has And be e set of all functions such at for (ie) Every e sign will be change at e points in and is non-negative proximate of 1 (2) Now remember several definitions of modulus which are needs rough our article The difference of is known by In specific, if, en be e set of all non-negative functions on A function is said to be copositive wi function if We are concerned about approximation e functions from by polynomials belong to of degree less an or equal and spline wi less an knots at are copositive wi, if is is also called positive approximation Then e modulus smooness of (3) is defined For, let (7) 11392

2 Is e degree of best unconstraint approximation of unbounded function of to polynomial and let: And For every, (8) be e degree of co-positive polynomial approximation of In specific We shall define e operator by: (9) Such at is e degree of best co-positive approximation By e side even if has only signal change, following its signal is not so easy and order of approximation fails It was publicized by Zhou [1] ere exists, such at Hu and Yu [2] wi [3], [4], [5], [6],[7] and [8] showed at and Kopotun [9] shows at, on standardized so at Hence preserver linearity at is for any THE MAIN THEOREMS We summarize all e result in is paper by e following eorems The first eorem shows at co-positive spline approximation of unbounded functions in terms e modulus of smooness of order 2 for, e second eorem shows at Ditzian-Totik modulus, is indeed easy to get to, us belong e order of copositive polynomials approximation of unbounded functions, also e ird eorem construct e inequality in terms averaged modulus smooness of co-positive spline approximation to NOTATION AND DEFINITION be an integer and Theorem 31: of e order Then spline (natural number) on e knots sequence satisfies be e divider of And Define e so support knots see [10] by (10) is a constant depends on and Proof: By using Holder inequality, we have, Denote, en for Since, Thus (11) 11393

3 Also from de Boor [11] and Devore [10], spline wi knots satisfying (13), Then such at, And by supposition, (15) (12) be e best approximation to on Then is e best approximation on each and (see [12]), we have (13) From (11), (12) and (13), we obtain We define, where a polynomial which is co-positive wi It is clear at We need to estimate and, By (15), we have Thus (10) is proved Theorem 32: be given and If Such at (14) is constant Proof: Suppose at (14) is hold for each function wi From Kopotun [9], e lemma (35) in [13] and e properties of Ditzian-Totik modulus, We obtain, by proposition in [13], 11394

4 We have Thus, And for any at least one interval at is not contaminated Note at dose not change sign amid & If, at least two non-contaminated intervals amid & Set two polynomials and such at Because of, And is a co-positive wi and is co-positive wi and satisfies We make a local polynomial by interpolation on, take e interpolation of at & by polynomial, its co-positive wi on Used for its rate of approximation, we take two polynomials & are exists such at, its holds Thus, (14) followed true at & Since Theorem 33: If, change its sign term at,, where, denote and, en, ere exists quadratic spline wi knots at copositive wi And satisfies (16) is constant Proof: contaminated, if of sign change of, We have one in every of suitability, we correspondingly denote and That is amid and 11395

5 Now, we have construct local polynomials which are copositive wi and have estimate of order ree, we now composite em for a spline approximation wi e similar estimate order If is a non- contaminated interval, & similarity on, en must be noncontaminated also, or ere would be no at very Construct spline on ese knots at connected wi & at respectively, Furermore, e display of mendacities amid ese, hence is too co-positive wi and satisfies polynomials, Amer Ma ci, 88, pp, , 1983 [8] I V Smazhenko, on e degree of copositive approximation, Journal Cocr Appl Ma, 3, pp, , 2005 [9] K A Kopotun, on copositive approximation by algebraic polynomial, Anal Ma, 21, pp, , 1995 [10] R A Devore, and G G Lorentz, Constructive approximation, Springer-Verlage, Berline, 1993 [11] C de Boor, Partical Guid to splines, Springer- Verlage, New Yark, 1987 [12] A D Ronald, and A Popov, Interpolation of Besove spaces, Amer Ma ci, 305(1), 1988 [13] Y K Hu, K Kopotun and X M Yu, On positive and co-positive polynomial and spline approximation in, Journal approximation eory, 96, pp, , 1996 Thus, ACKNOWLEDGEMENTS: I would like to express my sincere gratitude to e referees for eir valuable suggestions and comments which improved e paper REFERENCES [1] S P, Zhou, on copositive approximation, Journal approx eory appl, 9 (2), pp , 1993 [2] Y K Hu and X M Yu, The degree of cp-positive approximation and computer algoriem, SIAM, Journal Numer Anal, 32(5), pp, , 1995 [3] G A Dzybenko and J Gilewiez, Co-positive approximation of periodic functions, Acta Ma Hunger, 4, pp, , 2008 [4] Y K, Hu, D Leviatan, and X MYu, Co-positive polynomial approximation in C[0,1], Journal Anal, pp, 85-90, 1993 [5] Y K, Hu, D Leviatan and X U Yu, Co-positive polynomial and spline approximation, approx, Theory, 80, pp, , 1995 [6] Y K Hu, Positive and co-positive spline approximation in, Comput Ma Appl, 30, pp, , 1995 [7] Leviatan, The degree of co-positive approximation by 11396