Derivation of New Degrees for Best COCUNP Weighted Approximation: II

Approximation Theory is a branch of analysis and applied mathematics requiring that the approximation process preserves certain f -shaped properties deﬁned at a ﬁnite interval [ a , b ] , such as convexity in all or parts of the interval. The (Co)convex and Unconstrained Polynomial (COCUNP) approximation is one of the key estimations of the approximation theory that Kopotun has recently raised for ten years. Numerous studies have been conducted on modern methods of weighted approximation to construct the best degree of approximation. In developing COCUNP a novel technique, the Lebesgue Stieltjes integral-i technique is used to resolve certain disadvantages, such as Riemann’s integrable functions, which do not have the degree of the best approximation in norm space. In order to achieve the main goal, Derivation of New Degree (DOND) of the best COCUNP approximation was constructions. The theoretical results revealed that, in general, the new degrees of best approximation were able to smaller errors compared to the existing literature in the same estimating. In conclu-sion, this study has successfully developed DOND for the best (Co)convex Polynomial (COCP) weighted approximation.


Introduction
The L p space is one of the most important and interesting concepts in approximation theory. In other words, a large number of theorems of best approximation can be regarded depend on L p space, so that a normed space is probably the most important kind of spaces in functional analysis, at least from the field of approximation. 1 p is norm for p ≥ 1, and quasi norm for 0 < p < 1. Also, f ∞ , which is called the essential supremum of f , is defined by The following notation was introduced by [2]. For α, β ∈ J p , let us denote J p = ( −1 p , 0) if 0 < p < ∞, and J p = [0, ∞) if p = ∞. The following is definition of L α,β p,r and L α,β p spaces.
and for convenience denote L α,β The following is the definition of the knots of Chebyshev partition.

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Derivation of New Degrees for Best COCUNP Weighted Approximation: II and t j 's as the nots of Chebyshev partition and note that t j , 1 ≤ j ≤ η −1 are the extremum points of the Chebyshev polynomial of the first kind of degree ≤ n. If η > 1, the set t j = t j,η = − cos( jΠ η ), j = 0, · · · , η is called the Chebyshev partition of [−1, 1].
We will use the following notations below which was defined by [4]. Let θ N be a partition of [−1, 1] which have at least k intervals, that is, Let i, 0 ≤ i ≤ N be a fixed, and denote j(i) = max{0, i − k + 1}, then we write and the length of the largest interval in that partition.
Let D = [−1, 1] be measurable subset of R and P = {D j } j∈N be a family of finite subsets of D. We have Lebesgue partition P of D, if D j are measurable sets, ∪ j∈N D j = D and D j ∩ D ι = / 0, for j = ι. Now, the following definition is referred to as Lebesgue Stieltjes integral-i, a term that will be used extensively throughout this paper.
The class of all functions of Lebesgue Stieltjes integral-i is defined as follows.
Let I f be the class of all functions of Lebesgue Stieltjes integral-i of f that satisfying Definition 1.4, i.e.,

COCUNP and Auxiliary Results
In this section, we will mention the concepts COCP and Unconstrained Polynomial (UNP). The definition of convex function was given as follows.
1 is given in general abstract space. Since the real line (or an interval) is a convex set the definition of CP simply can be formulated as follows.
Next, we recall the definition of coconvex functions. If the polynomial p n preserves the shape (or curve) of the function f , it is then known as a constrained polynomial. Otherwise, it is an unconstrained polynomial.
[8] Let s, r ∈ N • , α, β ∈ J p and 0 < p ≤ ∞. Let P be a Lebesgue partition of D, andT η be a Chebyshev partition , then there is a constant c depend on η and J j,η such that This study aims to complement the DOND of best COCUNP approximation in our previous works to estimate the function, but based on the Chebyshev partition and the Lebesgue partition. In addition, we aim to develop the degree of best coconvex and unconstrained polynomial approximation of f , that has one inflection point.

Main Results
This section introduces a new definition called the degrees of best COCP and UNP weighted approximations of f of , for α, β ∈ J p respectively, the degrees of best (co)convex and unconstrained polynomial weighted approximations of f are denoted by We are starting with the following main result.
Then, for 1 ≤ η ≤ r,T n Chebyshev partition and P Lebesgue partition of D, we have where c is a constant depend on δ • , η and φ . In particular, if P ∩T n = / 0, then, Proof. We first prove our theorem by virtue of Lebesgue Stieltjes integral-i properties in Definition 1.4.
Note that Definition 3.1, implies that such that p n ∈ π n ∩ ∆ (2) (Y s ) ∩ I f and the function f : D → R belong to ∆ (2) (Y s ) ∩ Φ p (w α,β ), then, there exist L i : D → R are nondecreasing functions and i ∈ N. Thus, we have Lebesgue partition P of D. Now, suppose thatT η is Chebyshev partition such that D ∩T η = / 0 and 1 ≤ η ≤ r. Then, Now, by virtue of Theorems 2.7 and 2.8, we will put |D| ≤ δ • and |φ | ≤ δ • , then If P ∩T n = / 0, by use proof of Theorem 2.8, then, we lead to (5).
The ensuing complements for Theorems 2.7 and 2.8 are given follows.
(6) In particular, ifT η is Chebyshev partition and P is Lebesgue partition of D, 2 and P ∩T n = / 0.

Conclusions
In this work, the DOND for the best COCUNP weighted approximation was considered with the effect of the Chebyshev and Lebesgue partitions. Cocovex functions space have been used to convert Jackson type approximations into weighted approximations. The resultant approximations were converted by employing the Lebesgue Stieltjes integral-i technique to prove it. Also, we obtained the following: Suppose that Y s ∈ Y s , σ , s, n ∈ N and σ = 4. If f ∈ ∆ (2) (Y s ) ∩ Φ p,r (w α,β ), then,