Best multi Approximation of Unbounded Functions by Using Modulus of Smoothness

We present an estimate of the degree of best multi approximation of unbounded function on [−1, 1] ! by algebraic polynomials in weighted space. The studied of the relation between the best approximation of derivatives functions in weighted space and the best approximation of unbounded functions in the same space.


INTRODUCTION
This interest originated from the study of a certain class of integro-differential equations and applications in error estimations for singular integral equations.Following the initial works of Kalandiya [1] and Pr¨ossdorf [2], some problems of approximation in H¨older spaces have been studied by Bloom and Elliott [3], Ioakimidis [4], Nevertheless, most interesting and sharp results have been obtained for approximation of algebraic functions (see, for example, [5]).Also, the approximation problems of the unbounded functions by algebraic polynomials in the weighted spaces have been investigated by several authors.In particular, some direct and inverse theorems in weighted and nonweighted Lebesgue spaces with variable exponent have been obtained in [6,7, 8, 9, 10, 11,12 and 13].
) of all unbounded functions of several variables, with  ∈  ",℘ ( ! ) given norm defined by We consider the following linear operators :  +,5 () , ^ (7) , , Where In the terms of Poisson integrals, one can give the following interpretation of the derivative  (7) : Assume that 0 ≤ δ < 1, then ^ (7) , ,•_ The purpose of this paper is to investigate the operators  +,5 () and (, ,•) as the linear methods of approximation of functions in the weighted spaces.In this case, our attention is drawn to the relationship of the approximative properties of the sums  +,5 () and (, ,•) with the differential properties of the function , In addition, we study the relationship between the derivatives of an algebraic polynomial of best approximation and the best approximation of unbounded functions of several variables in weighted space.

1. AUXILIARY RESULTS
We begin with the following lemma which needs it is in our main results Lemma 2.1.[11].For  ∈ ℝ , let I.
& + 2 7  = + ⋯ +  7  0 + ⋯ Be two series in a Banach space (B, ‖•‖) For some c > o if and only if there exists R ∈ B such that Where c and C are constants that depend only on one another.
Lemma 2.4.Let  0 ∈  0 , and let  ∈ ℝ , then there exists a constant  > 0 independent of  and such that We get From lemma 2.2 we have Constructed on that, we have By using lemma 2.1 (with = 0 ) to series for every real  it follows from relation (10) in [13] and the last inequality that 57 Lemma 2.5.Let  0 ∈  0 be the best approximation of, ∝ ∈ ℝ , and  = 0,1,2, … then Where the constant  > 0 depends only on  .

MAIN RESULTS
This section of our work can be formulated as follows: Theorem 3.1.
We need to show 2 ⟹ 1 The proof is complete.

CONCLUSIONS
We understand from this functional mixture between the two topics, the importance of finding the best multiapproximation of unbounded function by algebraic polynomials in weighted space and the derivative of these functions.From now on, more research exploration continues For more useful to finding best multi-approximate of unbounded functions by some positive linear operators.