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 : 0000-0002-9363-2562 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
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
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
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
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, 101-105, 1983 [8] I V Smazhenko, on e degree of copositive approximation, Journal Cocr Appl Ma, 3, pp, 221-239, 2005 [9] K A Kopotun, on copositive approximation by algebraic polynomial, Anal Ma, 21, pp, 269-283, 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, 320-334, 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 104-110, 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, 1428-1435, 1995 [3] G A Dzybenko and J Gilewiez, Co-positive approximation of periodic functions, Acta Ma Hunger, 4, pp, 301-314, 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, 204-218, 1995 [6] Y K Hu, Positive and co-positive spline approximation in, Comput Ma Appl, 30, pp, 137-146, 1995 [7] Leviatan, The degree of co-positive approximation by 11396