A generalized Dunkl type modifications of Phillips operators

The main purpose of this present article is to discuss the convergence of Lebesgue measurable functions by providing a Dunkl generalization of Szász type operators known as Phillips operators. To achieve the results of a better way of uniform convergence of the Phillips operators, we study qualitative results in a Korovkin and weighted Korovkin space.

The Dunkl type generalizations is a very recent crucial work of Szász operators to the approximation processes. Our work is to study and find the uniform approximation properties by Dunkl type generalizations to the Phillips operators [24]. The main ideas of our research methodologies include the estimations of degrees of Phillips approximating operators by using the properties of the modulus of continuity, Lipschitz functions, Peetre's K -functional, and second order modulus of continuity. We have used a technique developed in [2,3] and studied several uniform approximation properties of the Phillips operators by Dunkl generalizations; moreover, see also some of the recent papers [25][26][27][28][29]. In the present article the approximation obtained by these operators designed by Dunkl type provides a better generalization depending on υ and an educational platform to the researcher.

Convergence in Korovkin and weighted Korovkin space
The Korovkin' type approximation theory has many useful connections with the classical approximation theory as well as with other branches of mathematics. In the present section the results related to uniform convergence of the operators defined by (2.1) via the well-known Korovkin' and weighted Korovkin' type theorems are obtained.
In the present article, the set of all functions which are bounded and continuous on [0, ∞) = R + , denoted by C B [0, ∞) and C[0, ∞), denotes the set of all continuous functions in C B [0, ∞). Also the linear normed space with the supremum norm is defined as follows: for which the function f (x) 1+x 2 is uniformly convergent as it approaches ∞.
Take σ (x) = 1 + x 2 is a weight function and the functions f ∈ C[0, ∞) are defined in weighted spaces for which where M f depends only on f and is a constant. It should be noted that, for x ∈ [0, ∞), Theorem 3.2 Let P * n,υ (·; ·) be the operators defined by (2.1).
Proof We used the Cauchy-Schwarz inequality and the results defined by (4.1), (4.2). Hence

Rate of convergence
In the present section we use the usual class of Lipschitz functions and obtain the rate of convergence of the sequence of positive linear operators P * n,υ (f ; x) (2.1) for which the operators uniformly converge to the continuous function f on [0, ∞).
The space of all the functions that are continuous and bounded on R + = [0, ∞) is denoted by C B (R + ). Hence one has with the norm defined on C 2 B (R + ), written as where the norm is defined on C B [0, ∞), Theorem 5.2 Let the operators P * n,υ (·; ·) be defined in (2.1). Then, for every ψ ∈ C 2 B (R + ) defined by (5.2), we have where Θ n = 1 n + 1 n 2 and Λ n,x = 1 n (1 + υ e υ (-nx) e υ (nx) )x.
Proof Let ψ ∈ C 2 B (R + ). From the expansion of Taylor series, the generalized mean value theorem, we have A small calculation leads to linearity on P * n,υ , we have which implies that

Convergence properties of some direct theorem
A potential influences work to obtain a well-known functional known as Peetre's Kfunctional, given by J. Peetre in 1968. The conflict of interest for K -functional to investigate the interpolation spaces between two Banach spaces and interactions to the real interpolation is based on K -functional. This well-known functional property, which is known as K -functional, was defined by Peetre as follows: For anyδ > 0, there exists a positive constant C > 0 such that K 2 (f ,δ) ≤ Cω 2 (f ,δ 1 2 ), where the second order modulus of continuity is given by (6.2) , and the operators P * n,υ (·; ·) be defined by (2.1). Then we have where ω 2 (f ;δ) is defined in (6.2) and D is a nonnegative constant.
Proof We use the results obtained in Theorem (5.2) and get By taking infimum over all ψ ∈ C 2 B (R + ) and using the results obtained by (6.1), we get Now, from the article [30] an absolute constant D > 0 exists, so we use here This completes the proof.

Conclusion
The present research article has an ample experience in applying appropriate properties to obtain uniform approximation results and an assessment of research methodologies to the approximation process. We establish a generalized version of the classic Phillips operators [24] by a Dunkl type generalization to the continuous functions connected with an extended exponential function. The point should be noted that in case of υ = 0, the operators (2.1) reduce to the classical Phillips operators given by [24]. The approximation obtained by these operators designed by Dunkl type provides a better generalization and an educational platform to the researcher to obtain the error estimations of the uniform convergence depending on υ.

Funding
Not applicable.