On wavelets Kantorovich ( p , q ) -Baskakov operators and approximation properties

In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the ( p , q )-Baskakov Kantorovich operators


Introduction and preliminaries
It has been three decades since Alexandru Lupas [18] for the first time introduced the notion of quantum calculus in the field of approximation theory.Since there, the area has become more active in research due to its application in different fields of science, engineering, and mathematics.Many researchers work on the extension of operators (see in Aral et al. [5]), the extended operators were known as exponential-type operators, which include Baskakov operators, Szász-Mirakyan operators, Meyer-König-Zeller operators, Picard operators, Weierstrass operators, and Bleiman, Butzer and Hahn operators.Moreover, q-analogue of standard integral operators of Kantorovich-and Durrmeyer-type was introduced.
The classical Baskakov operators for functions that are continuous on [0, ∞) were established by Baskakov [6], and the Baskakov-Kantorovich operators using integration were constructed by Ditzian and Totik [10].In 2009, Zhang and Zhu [27] investigated some preservation properties, including monotonicity, smoothness, and convexity of Baskakov-Kantorovich operators.Aral and Gupta [3] and Radu [26] generalized the Baskakov operators using a q-integer.With the use of q-integration Gupta and Radu [12] proposed the following Kantorovich variant of the q-Baskakov operators: [b+1]q [n]q q[b]q [n]q h q 1-b y d q y, ( where Recently, many researchers have established and investigated the approximation properties of positive linear operators by employing the techniques of post-quantum calculus.Several operators have been defined, and their approximation properties are discussed in [14][15][16][17][20][21][22][23][24]. (p, q)-calculus play a vital role in differential equations, physical sciences, hypergeometric series, and oscillator algebra.For example, Burban [8] uses the concept of (p, q)-calculus to present the (p, q)-analogue of two-dimensional conformal field theory based on the (p, q)-deformation of the su(1, 1) subalgebra of the Virasoro algebra.

Definition 1.2
The (p, q)-power basis is also known as (p, q)-binomial expansion, that is, for n ∈ N, we have: Definition 1.3 The (p, q)-derivative of the function f : R → R is denoted as D p,q f and is defined as: If f is differentiable at y = 0, then (D p,q f )(0) = f (0) holds.The assertions below hold true for n ≥ 1, and D p,q (b ⊕ y) 0 p,q = 0.The formula for product and quotient (p, q)-derivative is: p i q i+1 when q p < 1. (1.5) The integral in Equation (1.5) is not always positive unless is assumed that h is nondecreasing function.Therefore, Acar et al. [1] introduced the following (p, q)-integration to avoid some technical error during the construction of the Kantorovich modification of various operators: q n p n+1 when p n q n+1 when p q < 1. (1.6)

Construction of operators
Refer to some critical facts on wavelets as defined by Meyer [19] and Graps [11].The wavelets formed by dilation and translation of a single function (known as basic wavelets or mother wavelets) are the set of functions that take the form xλ η , for η > 0 and λ ∈ R.
If a and b are integers, then in the Franklin-Stromberg theory, we replace the constant η by 2 a , and 2 a b replaces λ.Given an arbitrary function h ∈ L 2 in analysis of this function, the wavelets will take the significant part of orthonormal basis, and the function h is defined as: where, An orthonormal basis for L 2 (R) defined in the form 2 , where a and b are integers, and ω is the positive integer, was constructed by Daubechies [9] with [0, 2ω +1] as ω support.If αω is the order of continuous derivatives of ω and α is a positive constant, then for any 0 ≤ b ≤ ω, where ω is a natural number, we have: Now, if we put ω = 0, the system reduces to a Haar system.Using wavelets in the construction of Baskakov-type operators, Agratini [2] established the following condition for ∈ L ∞ , (a) there exists a finite constant γ such that γ > 0 with the property With the use of the Haar basis, Agratini [2] constructed wavelet Baskakov operators and defined them as: (2. 2) The operators defined by Equation (2.2) extend the Baskakov-Kantorovich operators defined by Ditzian and Totik [10].Furthermore, for ⊂ [0, γ ], Agratini [2] expressed the operators μ n,b h in the form of: In construction of the q-Baskakov-type operators, Nasiruzzaman et al. [25] introduced other conditions.Let a positive constant be γ , and let ω (x) be any continuous derivatives of order γ ω.Also, for 0 ≤ b ≤ ω, such that ω ∈ N and q > 0, we have: Note that if we put q = 1, Equation (2.4) reduces to Equation (2.1), and if we take ω = 0 and q = 1, the system goes to the Haar basis.So, Nasiruzzaman et al. [25] provide the following conditions ∀ ∈ L ∞ (a) there exists a finite constant γ such that γ > 0 with the property ⊂ [0, γ ], (b) its first ω moment vanishes; that is, for The following is the q-analogy for Baskakov-Kantorovich-type wavelet operators introduced by Nasiruzzaman et al. [25] ( The operators defined by Equation (2.5) extended the Kantorovich q-Baskakov operators defined by Radu [26].That is, if we choose ω = 0 and Haar basis, Equation (2.5) reduces to operators defined by Equation (1.1).Additionally, by choosing ω = 0, q = 1, and Haar basis, we get the Kantorovich modification of Baskakov operators defined by Ditzian and Totik [10].For ⊂ [0, γ ], operators defined by Equation (2.5) can be rewritten as: Taking q = 1, Equation (2.6) reduces to classical Baskakov-Kantorovich wavelet operators defined by Equation (2.3).
In this section, the Kantorovich (p, q)-Baskakov operators is constructed with the help of Daubechies compactly-supported wavelets.Since the modified Kantorovich (p, q)-Baskakov operators defined by Equation (1.4) do not generalize the Kantorovich q-Baskakov operators expressed by Equation (1.1), that is, for q = 1, Equation (1.4) is not equal to Equation (1.1).Therefore, Equation (1.4) must be rewritten.Now, to achieve our objective of constructing the Kantorovich (p, q)-Baskakov operators, we define the new operators, generalizing the operators defined by Equation (1.1), as follows: We see that by choosing p = 1, Equation (2.7) reduces to Equation (1.1).Hence, Equation (1.1) is generalized by Equation (2.7).

Lemma 2.1
For n ∈ N and 0 < q < p ≤ 1.The following holds .
Proof Using Equation (1.6), we have that .
The two remaining parts can be proved similarly as the first part.Note that for some simple calculation, we have [b + 1] p,q = p b + q[b] p,q and ∞ n=0 ( q p ) n = p p-q .
Though we have some conditions that are considered in the construction of the operators, we have to introduce other conditions to make the wavelets useful in our study.Let a positive constant be γ and ω be a continuous derivative of order γ ω; on top of that, suppose that for any 0 ≤ b ≤ ω such that ω ∈ N and q > 0, we have: Choosing q = 1, the system reduces to Equation (2.4), and for p = q = 1, the system reduces to Equation (2.1).Equation (2.8) becomes a Haar system by choosing ω = 0 an p = q = 1.Now, we present the following conditions.For every ∈ L ∞ , (a) there exists a finite constant γ such that γ > 0 with the property ⊂ [0, γ ], (b) its first ω moment vanishes; that is, for 1 ≤ b ≤ ω, we have that R y b (y) d p,q y = 0, (c) R (y) d p,q y = 1.For x ∈ [0, ∞) and 0 < q < p ≤ 1, below is the (p, q)-analogue of the Baskakov-Kantorovichtype wavelet operators: [n] p,q q b-1 p n-1 y -[b] p,q d p,q y. (2.9) The operators ϒ n,b,p,q extend Kantorovich (p, q)-Baskakov operators defined by Equation (2.7), that is, for the choice of ω = 0 and Haar basis, Equation (2.9) reduces to Equation (2.7), in addition to that, for p = 1, Equation (2.9) reduces to Kantorovich q-Baskakov operators defined by Equation (1.1).Furthermore, for the choice of ω = 0, p = q = 1 and Haar basis, Equation (2.9) reduces to classical Kantorovich-Baskakov operators defined by Ditzian and Totik [10].Now, for ⊂ [0, γ ], Equation (2.9) is rewritten in the form: [n] p,q q b-1 (y) d p,q y. (2.10) Note that clearly for the choice of p = 1, Equations (2.9) and (2.10) reduce to q-Baskakov Kantorovich wavelet operators defined by Equations (2.5) and (2.6), respectively.Similarly, for the choice of p = q = 1, the Equations (2.9) and (2.10) reduce to classical Baskakov-Kantorovich wavelet operators defined by the Equations (2.3) and (2.4), respectively.
3 Main results
Proof To prove this theorem, refer to Inequality (2.10).Now, Equation (2.10) can be written as: p,q (y) d p,q y = 0, we have And from condition (c) above, given that R (y) d p,q y = 1, we have: [b] p,q p n-1 [n] p,q q b-1 j = ( n,b,p,q, , e j )(x).
Remark 3.1 Theorem (3.1) implies that the moments of the operators ϒ n,b,p,q, defined by Equation (2.9) is the same as that of the operators n,b,p,q, defined by Aral and Gupta [4].So, we have the following Lemma proved by Aral and Gupta [4].

Characterization of second-order Lipschitz functions
In this section, we shall present some Bernstein-Markov types of inequality of Kantorovich (p, q)-Baskakov operators, which will be used as our preliminary result to state our main result of this section.The following facts are also needed: Peetre's K-functional defined as: , the K-functional defined by equality (3.1) is equivalent to the modulus of smoothness.Johnen [13] gives the following relation: for some constant > 0 and any ν > 0, we have: where and 0 o t h e r w i s e .

Theorem 3.2 For all h
Proof Take into consideration that [n] p,q q b-1 (y) d p,q y = J h (n, b, p, q).

Thus, for every
From Equation (2.10), we have Employing the definition of (p, q)-derivative to prove inequalities (ii) and (iii).That is, in υ p,q b,n (x), we have that Here, . By making some simple computations, we have Therefore, using the facts above, we have that For every b ∈ N, we have the following equality: Hence, we get that In a similar way, we have that Therefore, we have that Theorem 3.3 Given that 0 < q < p ≤ 1 and K 2 is the Peetre's K-functional, then for all h ∈ C [0, ∞) ∩ C B [0, ∞), we have the following ), the Taylor series expansion of a function h is given as Therefore, using Equation (1.3) and Theorem 3.1, we have Using Theorem 3.2 (i) and taking infimum over all g ∈ C [0, ∞) ∩ C B [0, ∞), we have the following: That is, From Theorem 3.3, let N = γ ∞ , ξ = γ 2 3 and ϕ n,b,p,q = x(1 + p q x).The following corollary holds for the operators (ϒ n,b,p,q h).

Corollary 3.1 For any
where K 2 defined by Equality (3.1).
Furthermore, if Theorem 3.3 and Inequality (3.2) are well known, we easily obtain the following results explained by corollary below.
Remark 3.2 Corollary 3.2 gives the main result of this section.The function ϕ control the rate of convergence of the operators ϒ n,b,p,q .

The norm of the operators ϒ n,b,p,q in L p
In this section, we present the norm of ϒ n,b,p,q in L p space.We shall use r to avoid confusion of using p as its already used in different definition and facts of (p, q)-calculus.Therefore, we shall write L r instead of L p .In this section, we shall use Theorem 3.2 and the Riesz-Thorin theorem given below (also see in the book [7]).
Theorem 3.4 (Riesz-Thorin) Suppose that 0 < r < ∞ and 1 < s < ∞, also let T : S 0 (dμ; C) → L loc (dν; C) be a linear operator.Now, if in the domain of T the following inequalities are valid: then T extends by continuity to an operator acting from L r(ϑ) (dμ; C) into L s(ϑ) (dμ; C).The norm of this operator does not exceed N 1-ϑ 0 N ϑ 1 .
Now, we state the main theorem of this section.
where C r is a constant.[n] p,q q b-1 ∞ d p,q y d p,q x ≤ ∞ ∞ b=0 γ 0 h p n-1 (y + [b] p,q ) [n] p,q q b-1 d p,q y ∞/A 0 υ p,q b,n (x) d p,q x.However, by doing simple calculations in (p, q)-calculus, we have the following facts; β p,q (b + 1, n -1).
This implies that ∞/A 0 υ p,q b,n (x) d p,q x = 1 [n -1] p,q q b p n .
Hence we have that ϒ n,b,p,q h 1 = [n] p,q [n -1] p,q ∞ ∞ b=0 Is the same as writing Now, let us calculate the value of C r , which is the upper bound of the operator's norm.