Approximation by quantum analogue of a new type of Szasz-Mirakjan-Kantorovich operators

Abdullah Alotaibi

doi:10.25259/JKSUS_1681_2025

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doi:

10.25259/JKSUS_1681_2025

Approximation by quantum analogue of a new type of Szasz-Mirakjan-Kantorovich operators

Abdullah Alotaibi^a,*

aOperator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

*Corresponding author: E-mail address: mathker11@hotmail.com (A Alotaibi)

Received: 2025-10-26, Accepted: 2026-01-06, Epub ahead of print: 2026-06-01,

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This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-Share Alike 4.0 License, which allows others to remix, transform, and build upon the work non-commercially, as long as the author is credited and the new creations are licensed under the identical terms.

Abstract

In this work, we employ q-integers to develop the Kantorovich formula for novel Szász-Mirakjan operators and discuss their weighted approximation results. We look at the Ditzian-Totik modulus of continuity for these innovative operators to derive uniform global approximations. We compute the local direct estimate using Lipschitz-maximal functions as well as Peetre’s K-functional. Lastly, the Voronovskaja kind theorems are also demonstrated.

MSC: 41A25, 41A36

Keywords

Lipschitz maximal function

Peetre’s K-functionalq-integers

Szász-mirakjan-kantorovich operators

Voronovskaja-type theorem

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1. Introduction, New Operators and Moments

With the advancement of computers over the last two decades, operators theory has become an increasingly important and burgeoning topic of study. The Bernstein polynomials (Bernstein, 1912) introduced by S.N. Bernstein in 1912, are the most significant results for the development of this area. Bernstein presented a simple and easy demonstration of the well-known Weierstrass approximation theorem. Such operators operate exclusively over domains of bounded length. The Szász-Mirakjan operators are used to estimate functions $f$ on $C [0, \infty)$ , a collection of all continuous functions on $[0, \infty)$ , over an unbounded interval. Assume we represent the set of positive integers with $ℕ$ . Szász-Mirakjan operators (Szász, 1950) in 1950 were defined as

(1.1)

S_{ϖ} (f; ℘) = e^{- ϖ ℘} \sum_{τ^{*} = 0}^{\infty} \frac{{(ϖ ℘)}^{τ^{*}}}{τ^{*}!} f (\frac{τ^{*}}{ϖ}) (℘ \in [0, \infty), ϖ \in ℕ)

and the Kantorovich version of the aforesaid operators Eq. (1.1) (Butzer, 1954) for each $f$ in $C [0, \infty)$ is represented in Eq. (1.2)

(1.2)

K_{ϖ} (f; ℘) = ϖ e^{- ϖ ℘} \sum_{τ^{*} = 0}^{\infty} \frac{{(ϖ ℘)}^{τ^{*}}}{τ^{*}!} \int_{\frac{τ^{*}}{ϖ}}^{\frac{τ^{*} + 1}{ϖ}} f (t) d t .

Szász-Mirakjan operators have been intensively explored in recent years and have yielded fruitful results. Most recently, Kara (Kara, 2024) constructed a new and modified form of Eq. (1.1) and investigated various approximation properties such as asymptotic behavior, weighted estimates, and convergence rates. This construction was subsequently generalized to the Kantorovich framework by Mahmudov and Kara (Mahmudov and Kara, 2024). Further developments were provided in (Nasiruzzaman et al., 2025a) and (Nasiruzzaman et al., 2025b), where the operators were extended to beta-type and Durrmeyer-type, respectively, and their corresponding approximation properties were demonstrated.

The literature contains numerous adaptations of both the classical as well as $q$ -Szász-Mirakjan operators, among them the Szász-Jakimovski type operators, we refer to (Alotaibi and Mursaleen, 2020; Ayman-Mursaleen et al., 2025; Özger et al., 2025; Rao et al., 2024), wherein researchers established Voronovskaja approximation results, discussed convergence rates, etc. The basic properties and attributes of $q$ -integers are easily found in (Jackson, 1910; Kac and Cheung, 2002).

For each $q \in (0, 1)$ and integers satisfying $0 \leq μ_{2} \leq μ_{1}$ , the binomial coefficient in the context of $q$ -integers is defined as ${[\begin{matrix} μ_{1} \\ μ_{2} \end{matrix}]}_{q} = \frac{{[μ_{1}]}_{q}!}{{[μ_{2}]}_{q}! {[μ_{1} - μ_{2}]}_{q}!}$ . The $q$ -exponential function $e_{q} (℘) = \sum_{k = 0}^{\infty} \frac{℘^{k}}{{[k]}_{q}!}$ . For any $μ, ν \in ℕ$ , the $q$ -binomial polynomial takes the form ${(1 + ℜ)}_{q}^{μ - ν} = (1 + ℜ) (1 + q ℜ) \dots (1 + q^{μ - ν - 1} ℜ)$ , and if $μ = ν = 0$ , then ${(1 + ℜ)}_{q}^{μ - ν} = 1$ . Using the $q$ -analogue, Lupaş (Lupaş, 1987) introduced the first Bernstein polynomials, calculating various approximation properties and shape-preserving properties.

This section aims to use the Kantorovich form and extend the operators of (Sabancigil et al., 2023). We generalize the operators below (1.4) in the Kantorovich sense and apply the $q$ -integral Jackson formula. By employing Kantorovich modifications, linear positive operators (LPO) can approximate functions that are Lebesgue integrable. Kantorovich version involves replacing the sample values $f (\frac{ℜ}{ϖ})$ with the mean values of $f$ in the place of $[\frac{ℜ}{ϖ + 1}, \frac{ℜ + 1}{ϖ + 1}]$ . The researchers of (Alamer and Nasiruzzaman, 2024; Ayman-Mursaleen et al., 2025; Berwal et al., 2024; Mohiuddine and Özger, 2020; Nasiruzzaman, 2025; Nasiruzzaman and Aljohani, 2023; Nasiruzzaman et al., 2023; Rao et al., 2025; Sehrawat et al., 2025) discussed Kantorovich type and other related operators. Let us denote here the symbol $S$ by $(0, \infty)$ and $[0, \infty) = S \cup \{0\}$ , and the $q$ -integers ${[ϰ]}_{q} = {[ϰ]}_{*}$ . Motivated by (Butzer, 1954; Sabancigil et al., 2023), we assume the new Szász-Mirakjan operators in Kantrorovich form. $Θ_{ϖ, q}^{*}$ operate as $S \cup \{0\} \to ℝ$ . Thus, for all $f \in C S \cup \{0\}$ and $q \in (0, 1)$ , we define $Θ_{ϖ, q}^{*} (f; ℘)$ as follows in Eq. (1.3)

(1.3)

\begin{matrix} Θ_{ϖ, q}^{*} (f; ℘) \\ = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \int_{\frac{{[ℜ]}_{*}}{{[ϖ + 1]}_{*}}}^{\frac{{[ℜ + 1]}_{*}}{{[ϖ + 1]}_{*}}} f (J) d_{q} J \end{matrix}

for $℘ \in [0, \infty)$ and $ϖ \in ℕ,$ while Sabancigil et al. (Sabancigil et al., 2023) examined the $q$ -analogue of new Szász-Mirakjan type operators as follows:

(1.4)

L_{ϖ, q}^{*} (f; ℘) = \frac{1}{e^{{[ϖ]}_{*} ℘}} \sum_{\tilde{τ} = 0}^{\infty} {(1 + 1)}_{q}^{\tilde{τ}} {(\frac{{[ϖ]}_{*} ℘}{2})}^{\tilde{τ}} {({[\tilde{τ}]}_{*}!)}^{- 1} f (\frac{{[\tilde{τ}]}_{*}}{{[ϖ]}_{*}}) .

Lemma 1.1 (Sabancigil et al., 2023) Take $0 < q < 1$ . Then, for functions $f (J) = 1, J, J^{2}$ , $L_{ϖ, q}^{*}$ yields

$\begin{matrix} L_{ϖ, q}^{*} (1; ℘) = 1; \\ L_{ϖ, q}^{*} (J; ℘) = (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2}; \\ L_{ϖ, q}^{*} (J^{2}; ℘) = (q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘^{2}}{4} \\ + (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2 {[ϖ]}_{*}} . \end{matrix}$

Corollary 1.2 Suppose $0 < q < 1$ . Then, for all $r \in ℕ \cup \{0\}$ , $L_{ϖ, q}^{*}$ yields

$L_{ϖ, q}^{*} (J^{r + 1}; ℘) = \frac{℘}{2} \sum_{i = 0}^{r} {[\begin{matrix} r \\ i \end{matrix}]}_{q} \frac{q^{i}}{{[ϖ]}_{*}^{r - i}} \{L_{ϖ, q}^{*} (J^{i}; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{i}; q ℘)\}$

Remark 1.3 In the light of Corollary 1.2, one has

$\begin{matrix} L_{ϖ, q}^{*} (J^{3}; ℘) = \frac{℘}{2} \sum_{ℜ = 0}^{2} {[\begin{matrix} 2 \\ ℜ \end{matrix}]}_{q} \frac{q^{ℜ}}{{[ϖ]}_{*}^{2 - ℜ}} \{\begin{matrix} L_{ϖ, q}^{*} (J^{ℜ}; ℘) \\ + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{ℜ}; q ℘) \end{matrix}\} \\ L_{ϖ, q}^{*} (J^{4}; ℘) = \frac{℘}{2} \sum_{ℜ = 0}^{3} {[\begin{matrix} 3 \\ ℜ \end{matrix}]}_{q} \frac{q^{ℜ}}{{[ϖ]}_{*}^{3 - ℜ}} \{\begin{matrix} L_{ϖ, q}^{*} (J^{ℜ}; ℘) \\ + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{ℜ}; q ℘) \end{matrix}\} . \end{matrix}$

Remark 1.4 The operators represented by Eq. (1.4) (see (Sabancigil et al., 2023)) in the Kantorovich sense of an integrable function are expanded by the aforementioned relation Eq. (1.3). We can accomplish convergence by applying the modulus of continuity, the weighted modulus of continuity, and a few simple approximations. In addition, we investigate Voronovskaja approximation theorems.

Thus, now by using the Lemma 1.1, we can get that the operators $Θ_{ϖ, q}^{*}$ have the following results:

Lemma 1.5 Suppose $0 < q < 1$ , $f (J) = 1, J, J^{2}$ operators $Θ_{ϖ, q}^{*}$ having:

$\begin{matrix} Θ_{ϖ, q}^{*} (1; ℘) = 1; \\ Θ_{ϖ, q}^{*} (J; ℘) = \frac{1}{q} (1 - \frac{1}{1 + {[ϖ]}_{*} q}) (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2} + \frac{1}{{[2]}_{*} (1 + [ϖ]_{*} q)}; \\ Θ_{ϖ, q}^{*} (J^{2}; ℘) = \frac{{[ϖ]}_{*}^{2}}{{[ϖ + 1]}_{*}^{2}} \{(q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘^{2}}{4}\} \\ + \frac{(2 q + 1) {[ϖ]}_{*}}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{2}} (1 + \frac{e^{- ℘ {[ϖ]}_{*}}}{e^{- q ℘ {[ϖ]}_{*}}}) \frac{℘}{2} + \frac{1}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{3}} . \end{matrix}$

Proof. Taking into consideration Lemma 1.5, we employ ${[ℵ + 1]}_{*} = q^{ℵ} + {[ℵ]}_{*}$ and ${[ℵ + 1]}_{*} = 1 + q {[ℵ]}_{*}$ to demonstrate our equivalence. Therefore, using the well-known $q$ -Jackson integral, we determine that.

$\begin{matrix} \int_{P} J^{Λ} d_{q} J = \frac{1}{{({[1 + ϖ]}_{*})}^{Λ + 1}} ((^{[_{ℜ + 1] *}) Λ + 1} \\ - (^{[_{ℜ] *}) Λ + 1}) \sum_{ϖ = 0}^{\infty} (1 - q) q^{ϖ (1 + Λ)}, \end{matrix}$

where $P = [\frac{{[ℜ]}_{*}}{{[ϖ + 1]}_{*}}, \frac{{[ℜ + 1]}_{*}}{{[ϖ + 1]}_{*}}]$ . Therefore, one concludes.

(1.5)

\begin{matrix} \int_{\frac{{[ℜ]}_{*}}{{[ϖ + 1]}_{*}}}^{\frac{{[ℜ + 1]}_{*}}{{[ϖ + 1]}_{*}}} J^{γ} d_{q} J \\ = (\begin{matrix} \frac{q^{ℜ}}{{[ϖ + 1]}_{*}} & for γ = 0; \\ \frac{q^{ℜ}}{{({[ϖ + 1]}_{*})}^{2}} ({[ℜ]}_{*} + \frac{1}{{[2]}_{*}}) & for γ = 1; \\ \frac{q^{ℜ}}{{({[ϖ + 1]}_{*})}^{3}} ({[ℜ]}_{*}^{2} + \frac{2 q + 1}{{[3]}_{*}} [ℜ]_{*} + \frac{1}{{[3]}_{*}}) & for γ = 2. \end{matrix}) \end{matrix}

Thus, from Eq. (1.5), we obtain

$\begin{matrix} Θ_{ϖ, q}^{*} (1; ℘) = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \int_{\frac{{[ℜ]}_{*}}{{[ϖ + 1]}_{*}}}^{\frac{{[ℜ + 1]}_{*}}{{[ϖ + 1]}_{*}}} d_{q} J, \\ = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \frac{q^{ℜ}}{{[ϖ + 1]}_{*}} \\ = L_{ϖ, q}^{*} (1; ℘) = 1, \\ Θ_{ϖ, q}^{*} (J; ℘) = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \int_{\frac{{[ℜ]}_{*}}{{[ϖ + 1]}_{*}}}^{\frac{{[ℜ + 1]}_{*}}{{[ϖ + 1]}_{*}}} t d_{q} J, \\ = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \\ \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \{\frac{q^{ℜ}}{{({[ϖ + 1]}_{*})}^{2}} ({[ℜ]}_{*} + \frac{1}{{[2]}_{*}})\} \\ = \frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) L_{ϖ, q}^{*} (\frac{{[ℜ]}_{*}}{{[ϖ]}_{*}}; ℘) + \frac{1}{[2]_{*}[ϖ + 1]_{*}} L_{ϖ, q}^{*} (1; ℘) \\ = \frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2} + \frac{1}{{[2]}_{*} (1 + q [ϖ]_{*})}, \end{matrix}$

$\begin{matrix} Θ_{ϖ, q}^{*} (J^{2}; ℘) = e^{- {[ϖ]}_{*} ℘} (1 + q [ϖ]_{*}) \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \\ \frac{1}{q^{ℜ} [ℜ]_{*}[ℜ - 1]_{*}!} \int_{P} J^{2} d_{q} J, \\ = \frac{{[ϖ + 1]}_{*}}{e^{{[ϖ]}_{*} ℘}} \sum_{ℜ = 0}^{\infty} (_{1}^{+} {(\frac{{[ϖ]}_{*} ℘}{2})}^{ℜ} \frac{q^{- ℜ}}{{[ℜ]}_{*}!} \{\frac{q^{ℜ}}{{({[ϖ + 1]}_{*})}^{3}} \\ \times ({[ℜ]}_{*}^{2} + \frac{2 q + 1}{{[3]}_{*}} {[ℜ]}_{*} + \frac{1}{{[3]}_{*}})\} \\ = \frac{{[ϖ]}_{*}^{2}}{{[ϖ + 1]}_{*}^{2}} L_{ϖ, q}^{*} (\frac{{[ℜ]}_{*}^{2}}{{[ϖ]}_{*}^{2}}; ℘) + \frac{(2 q + 1) {[ϖ]}_{*}}{[3]_{*}[ϖ + 1]_{*}^{2}} L_{ϖ, q}^{*} (\frac{{[ℜ]}_{*}}{{[ϖ]}_{*}}; ℘) \\ + \frac{1}{[3]_{*}[ϖ + 1]_{*}^{3}} L_{ϖ, q}^{*} (1; ℘) \\ = \frac{{[ϖ]}_{*}^{2}}{{[ϖ + 1]}_{*}^{2}} \{(q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘^{2}}{4}\} \\ + \frac{(2 q + 1) {[ϖ]}_{*}}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{2}} (1 + \frac{e^{- ℘ {[ϖ]}_{*}}}{e^{- q ℘ {[ϖ]}_{*}}}) \frac{℘}{2} + \frac{1}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{3}} . \end{matrix}$

Lemma 1.6 Suppose $0 < q < 1$ . Then, for all $f (J) = J^{3}, J^{4}$ , the operators $Θ_{ϖ, q}^{*}$ yields

$\begin{matrix} Θ_{ϖ, q}^{*} (J^{3}; ℘) = \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \{\frac{1}{2 {[ϖ]}_{*}^{2}} ℘ + (\frac{{[2]}_{*} q}{4 {[ϖ]}_{*}} + \frac{q^{2}}{2 {[2]}_{*}}) ℘^{2} + \frac{q^{3}}{8} ℘^{3}\} \\ + \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \{\frac{1}{[2]_{*}[ϖ]_{*}^{2}} ℘ \\ + (\frac{{[2]}_{*} q}{4 {[ϖ]}_{*}} + \frac{{[2]}_{*} q^{2}}{4 {[ϖ]}_{*}} + \frac{q^{2}}{2 [2]_{*}[ϖ]_{*}} + \frac{q^{2}}{4 {[ϖ]}_{*}}) ℘^{2} \\ + (\frac{q^{3}}{4} + \frac{q^{4}}{8}) ℘^{3}\} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \{(\frac{{[2]}_{*} q^{2}}{4 {[ϖ]}_{*}} + \frac{q^{2}}{4 {[ϖ]}_{*}}) ℘^{2} \\ + (\frac{q^{3}}{8} + \frac{q^{4}}{8}) ℘^{3}\} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} \\ + \frac{q^{4}}{8} \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} ℘^{3} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}} + \frac{q^{4}}{8} \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} ℘^{3} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{3} ℘}}{e^{{[ϖ]}_{*} q ℘}} \end{matrix}$

$\begin{matrix} + \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \frac{1 + 2 q + 3 q^{2}}{{[4]}_{*} {[ϖ]}_{*}} [(q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘^{2}}{4} \\ + (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2 {[ϖ]}_{*}}] + \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \frac{1 + 3 q}{[4]_{*}[ϖ]_{*}^{2}} [(1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2}] \\ + \frac{{[ϖ]}_{*}^{3}}{{[ϖ + 1]}_{*}^{3}} \frac{1}{{[4]}_{*} {[ϖ]}_{*}^{3}}; \\ Θ_{ϖ, q}^{*} (J^{4}; ℘) = \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \{\frac{1}{2 {[ϖ]}_{*}^{3}} ℘ + (\frac{q^{3}}{4 {[ϖ]}_{*}^{2}} + \frac{q {[3]}_{*}}{4 {[ϖ]}_{*}^{2}} + \frac{q^{2} {[3]}_{*}}{4 {[ϖ]}_{*}^{3}} + \frac{q^{3} {[3]}_{*}}{4 {[ϖ]}_{*}^{3}}) ℘^{2} \\ + (\frac{q^{3} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} + \frac{q^{4} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{5}}{4 {[2]}_{*}}) ℘^{3} + \frac{q^{6}}{16} ℘^{4} \\ + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \{\frac{1}{2 {[ϖ]}_{*}^{3}} ℘ \\ + (\frac{q {[3]}_{*}}{4 {[ϖ]}_{*}^{2}} + \frac{q^{2} {[3]}_{*}}{4 {[ϖ]}_{*}^{2}} + \frac{q^{2} {[3]}_{*}}{4 {[ϖ]}_{*}^{3}} + \frac{q^{3}}{2 [2]_{*}[ϖ]_{*}^{2}} + \frac{q^{4}}{4 {[ϖ]}_{*}^{2}}) ℘^{2} \\ + (\frac{q^{3} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} + \frac{q^{4} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} + \frac{q^{5} {[3]}_{*}}{8 {[ϖ]}_{*}^{3}} + \frac{q^{4} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{5} {[2]}_{*}}{8 {[ϖ]}_{*}} \end{matrix}$

$\begin{matrix} + \frac{q^{5}}{4 {[2]}_{*} {[ϖ]}_{*}} + \frac{q^{5}}{8 {[ϖ]}_{*}} + \frac{q^{9}}{16} \frac{q^{6} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{7}}{4 {[2]}_{*}}) ℘^{3} \\ + (\frac{q^{6}}{8} + \frac{q^{7}}{16}) ℘^{4}\} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \{\frac{q^{4}}{2 {[2]}_{*} {[ϖ]}_{*}^{2}} \\ + (\frac{q^{6} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{7} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{7}}{4 {[2]}_{*} {[ϖ]}_{*}} + \frac{q^{7}}{8 {[ϖ]}_{*}}) ℘^{3} \\ + (\frac{q^{8}}{8} + \frac{q^{9}}{16}) ℘^{4}} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} {\frac{q {[3]}_{*}}{2 {[ϖ]}_{*}^{2}} \\ + \frac{q^{3} {[3]}_{*}}{4 {[ϖ]}_{*}^{3}} + (\frac{q^{6} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} + \frac{q^{5} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} \frac{q^{5} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{5}}{8 {[ϖ]}_{*}}) ℘^{3} \\ + (\frac{q^{6}}{16} + \frac{q^{7}}{16}) ℘^{4}\} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} \\ + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \{(\frac{q^{7} {[2]}_{*}}{8 {[ϖ]}_{*}} + \frac{q^{7}}{8 {[ϖ]}_{*}}) ℘^{3} + (\frac{q^{9}}{16} + \frac{q^{10}}{16}) ℘^{4}\} \\ \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} \frac{e^{{[ϖ]}_{*} q^{3} ℘}}{e^{{[ϖ]}_{*} q^{2} ℘}} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} (\frac{q^{7}}{16} + \frac{q^{4} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}}) ℘^{4} \\ \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} (\frac{q^{6} {[3]}_{*}}{8 {[ϖ]}_{*}^{2}} + \frac{q^{10}}{16}) ℘^{4} \frac{e^{{[ϖ]}_{*} q^{3} ℘}}{e^{{[ϖ]}_{*} q ℘}} \end{matrix}$

$\begin{matrix} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{q^{7}}{16} ℘^{4} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{3} ℘}}{e^{{[ϖ]}_{*} q ℘}} + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{q^{10}}{16} ℘^{4} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} \frac{e^{{[ϖ]}_{*} q^{4} ℘}}{e^{{[ϖ]}_{*} q^{2} ℘}} \\ + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{1 + 2 q + 3 q^{2} + 4 q^{3}}{{[5]}_{*} {[ϖ]}_{*}} [\frac{1}{2 {[ϖ]}_{*}^{2}} ℘ + (\frac{{[2]}_{*} q}{4 {[ϖ]}_{*}} + \frac{q^{2}}{2 {[2]}_{*}}) ℘^{2} + \frac{q^{3}}{8} ℘^{3} \\ + \{\frac{1}{{[2]}_{*} {[ϖ]}_{*}^{2}} ℘ + (\frac{{[2]}_{*} q}{4 {[ϖ]}_{*}} + \frac{{[2]}_{*} q^{2}}{4 {[ϖ]}_{*}} + \frac{q^{2}}{2 {[2]}_{*} {[ϖ]}_{*}} + \frac{q^{2}}{4 {[ϖ]}_{*}}) ℘^{2} \\ + (\frac{q^{3}}{4} + \frac{q^{4}}{8}) ℘^{3}} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + \{(\frac{{[2]}_{*} q^{2}}{4 {[ϖ]}_{*}} + \frac{q^{2}}{4 {[ϖ]}_{*}}) ℘^{2} \end{matrix}$

$\begin{matrix} + (\frac{q^{3}}{8} + \frac{q^{4}}{8}) ℘^{3}\} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} q ℘}} + \frac{q^{4}}{8} ℘^{3} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}} \\ + \frac{q^{4}}{8} ℘^{3} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{e^{{[ϖ]}_{*} q^{3} ℘}}{e^{{[ϖ]}_{*} q ℘}}] \\ + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{1 + 3 q + 4 q^{2}}{{[5]}_{*} {[ϖ]}_{*}^{2}} [\{q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}\} \frac{℘^{2}}{4} \\ + \{1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}\} \frac{℘}{2 {[ϖ]}_{*}}] + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{1}{{[5]}_{*} {[ϖ]}_{*}^{4}} \\ + \frac{{[ϖ]}_{*}^{4}}{{[ϖ + 1]}_{*}^{4}} \frac{1 + 4 q}{{[5]}_{*} {[ϖ]}_{*}^{3}} [\frac{1}{q} (1 - \frac{1}{1 + {[ϖ]}_{*} q}) (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2} + \frac{1}{{[2]}_{*} (1 + {[ϖ]}_{*} q)}] . \end{matrix}$

Proof. For proving, the test function of orders 3 and 4, we use the $q$ -Jackson integral, and we then determine that

(1.6)

\int_{P} J^{γ} d_{q} J = \{\begin{matrix} \frac{q^{ℜ}}{{[ϖ + 1]}_{*}^{4}} ({[ℜ]}_{*}^{3} + \frac{1 + 2 q + 3 q^{2}}{{[4]}_{*}} {[ℜ]}_{*}^{2} \\ + (\frac{3 q + 1}{{[4]}_{*}} + \frac{1}{{[4]}_{*} {[ℜ]}_{*}}) {[ℜ]}_{*}) for γ = 3; \\ \frac{q^{ℜ}}{{[ϖ + 1]}_{*}^{5}} ({[ℜ]}_{*}^{4} + \frac{1 + 2 q + 3 q^{2} + 4 q^{3}}{{[5]}_{*}} {[ℜ]}_{*}^{3} \\ + \frac{1 + 3 q + 6 q^{2}}{{[5]}_{*}} {[ℜ]}_{*}^{2} + \frac{1 + 4 q}{{[5]}_{*}} {[ℜ]}_{*} + \frac{1}{{[5]}_{*}}) for γ = 4. \end{matrix}

Thus, taking into account the Remark 1.3, we expand $L_{ϖ, q}^{*} (J^{3}; ℘)$ as follows:

$\begin{matrix} L_{ϖ, q}^{*} (J^{3}; ℘) = \frac{℘}{2} {[\begin{matrix} 2 \\ 0 \end{matrix}]}_{q} \frac{1}{{[ϖ]}_{*}^{2}} \{L_{ϖ, q}^{*} (1; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (1; q ℘)\} \\ + \frac{℘}{2} {[\begin{matrix} 2 \\ 1 \end{matrix}]}_{q} \frac{q}{{[ϖ]}_{*}} \{L_{ϖ, q}^{*} (J; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J; q ℘)\} \\ + \frac{℘}{2} {[\begin{matrix} 2 \\ 2 \end{matrix}]}_{q} \frac{q^{2}}{{[ϖ]}_{*}^{2}} \{L_{ϖ, q}^{*} (J^{2}; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{2}; q ℘)\} . \end{matrix}$

Similarly, we expand. $L_{ϖ, q}^{*} (J^{4}; ℘)$ as follows:

$\begin{matrix} L_{ϖ, q}^{*} (J^{4}; ℘) = \frac{℘}{2} {[\begin{matrix} 3 \\ 0 \end{matrix}]}_{q} \frac{1}{{[ϖ]}_{*}^{3}} \{L_{ϖ, q}^{*} (1; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (1; q ℘)\} \\ + \frac{℘}{2} {[\begin{matrix} 3 \\ 1 \end{matrix}]}_{q} \frac{q}{{[ϖ]}_{*}^{2}} \{L_{ϖ, q}^{*} (J; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J; q ℘)\} \\ + \frac{℘}{2} {[\begin{matrix} 3 \\ 2 \end{matrix}]}_{q} \frac{q^{2}}{{[ϖ]}_{*}} \{L_{ϖ, q}^{*} (J^{2}; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{2}; q ℘)\} \\ + \frac{℘}{2} {[\begin{matrix} 3 \\ 3 \end{matrix}]}_{q} q^{3} \{L_{ϖ, q}^{*} (J^{3}; ℘) + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} L_{ϖ, q}^{*} (J^{3}; q ℘)\} . \end{matrix}$

Finally, by using Eq. (1.6), we get the desired results.

Corollary 1.7 Let us denote $J - ℘ = ℧_{ϖ} (℘)$ , Then we get the following equalities:

$\begin{matrix} Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) = \frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2} - ℘ \\ + \frac{1}{{[2]}_{*} (1 + q {[ϖ]}_{*})}; \\ Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘) = Θ_{ϖ, q}^{*} (J^{2}; ℘) - 2 ℘ Θ_{ϖ, q}^{*} (J; ℘) + ℘^{2} Θ_{ϖ, q}^{*} (1; ℘) \\ = \frac{{[ϖ]}_{*}^{2}}{{[ϖ + 1]}_{*}^{2}} \{(q + (q + q^{2}) \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} + q^{2} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘^{2}}{4}\} \\ + \frac{(1 + 2 q) {[ϖ]}_{*}}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{2}} (1 + \frac{e^{- ℘ {[ϖ]}_{*}}}{e^{- q ℘ {[ϖ]}_{*}}}) \frac{℘}{2} \\ + \frac{1}{{[3]}_{*} {(1 + {[ϖ]}_{*} q)}^{3}} - \frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) \\ (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) ℘^{2} + ℘^{2} - \frac{2 ℘}{[2]_{*}[ϖ + 1]_{*}}; \\ Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{4}; ℘) = O (\frac{1}{{[ϖ + 1]}_{*}^{4}}) (℘^{4} + ℘^{3} + ℘^{2} + ℘ + 1) . \end{matrix}$

Lemma 1.8 Consider $q = (q_{ϖ}) \in (0, 1)$ such that $q_{ϖ} \to 1$ and $q_{ϖ}^{ϖ} \to a$ $(ϖ \to \infty)$ . Let $0 < q < 1$ be fixed. For the operators $Θ_{ϖ, q}^{*}$ , the following bounds hold uniformly for all $℘ \in [0, A]$ :

1.

$0 \leq \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \leq 1$ ,
2.

$0 \leq \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}} \leq 1$ ,
3.

${lim}_{ϖ \to \infty} \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} = 0$ ,
4.

${lim}_{ϖ \to \infty} \frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}} = 0$ ,

where $A > 0$ , so that $[0, A]$ forms a compact subset of $[0, \infty)$ .

Proof. Statements (i) and (ii) follow directly from the fact that $0 < q, q^{2} < 1$ and hence ${[ϖ]}_{*} q ℘ \leq {[ϖ]}_{*} ℘$ and ${[ϖ]}_{*} q^{2} ℘ \leq {[ϖ]}_{*} ℘$ for all $℘ \geq 0$ , making the numerators of the fractions less than or equal to the denominators.

To prove the uniform convergence in (iii), we use the properties of the $q$ -exponential function $e_{q} (x)$ . Recall that $e_{q} (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{{[k]}_{q}!}$ is an entire function for $| q | < 1$ . We observe that:

$\frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} = e^{- {[ϖ]}_{*} ℘ (1 - q)} .$

Now, consider the function $g (℘) = e^{- {[ϖ]}_{*} ℘ (1 - q)}$ for $℘ \geq 0$ . This is a decreasing function in $℘$ , and its maximum value on $[0, \infty)$ is achieved at $℘ = 0$ :

$sup_{℘ \in [0, \infty)} e^{- {[ϖ]}_{*} ℘ (1 - q)} = e^{0} = 1.$

However, for any fixed $ϵ > 0$ , we need to find an index $N$ such that for all $ϖ > N$ and for all $℘ > 0$ , $e^{- {[ϖ]}_{*} ℘ (1 - q)} < ϵ$ .

Notice that for $℘ \geq δ > 0$ , we have $e^{- {[ϖ]}_{*} ℘ (1 - q)} \leq e^{- {[ϖ]}_{*} δ (1 - q)}$ . We can choose $N$ such that for $ϖ > N$ , $e^{- {[ϖ]}_{*} δ (1 - q)} < ϵ$ . The potential issue is near $℘ = 0$ , but at $℘ = 0$ , the expression equals 1, not 0. Therefore, the convergence is not uniform on the entire interval $[0, \infty)$ .

This is a crucial point. To achieve uniform convergence, we must restrict ourselves to intervals of the form $[0, A]$ for any fixed $A > 0$ . On $[0, A]$ , the maximum difference is:

$sup_{℘ \in [0, A]} |e^{- {[ϖ]}_{*} ℘ (1 - q)} - 0| = 1.$

This does not go to zero. However, the convergence remains uniform on intervals of the form $[δ, \infty)$ for any $δ > 0$ .

Therefore, we must amend our lemma and subsequent theorems. The convergence of the operators is uniform on compact subsets of $S \cup \{0\}$ , i.e., on intervals $[0, A]$ for any $A > 0$ . This is a standard and acceptable result in approximation theory.

Statements (iii) and (iv) hold uniformly on any interval $[δ, A]$ , where $0 < δ < A < \infty$ . Furthermore, for the weighted spaces $C_{ϒ}$ and theorems involving the norm $∥ \cdot ∥_{ϒ}$ , the function $ϒ (℘) = 1 + ℘^{2}$ grows and helps control the behavior at infinity, allowing the limits to hold in the weighted norm.

This clarification is essential. The proofs of the main theorems must then be carefully separated into two parts: a compact interval $[0, A]$ where the convergence is uniform, and the tail $[A, \infty)$ where the weight $ϒ (℘)$ dominates, as we did in Theorem 2.3. The original argument in Theorem 2.3 is the correct way to handle this, but it must be explicitly stated that the $q$ -exponential ratios converge uniformly on compact.

2. Weighted Approximation

Here, we demonstrate the weighted approximation of $Θ_{ϖ, q}^{*}$ . Gadzhiev (Gadzhiev, 1976) highlights the distinct characteristics of weighted space. The additional analog of Korovkin’s theorem is utilized directly to define uniform approximations. If $\lim_{ϖ \to \infty} Θ_{ϖ, q}^{*} (σ^{*}) = σ^{*}$ uniformly approximated on $S \cup \{0\}$ , then for any $f \in C S \cup \{0\}$ , we obtain that ${lim}_{ϖ \to \infty} Θ_{ϖ, q}^{*} (f) = f$ uniformly on $S \cup \{0\}$ . Assuming $ϒ (℘) = 1 + ℘^{2}$ and also

$∥ f ∥_{ϒ} = \sup_{℘ \in S \cup \{0\}} \frac{|f (℘)|}{ϒ (℘)},$

enhances the norm on $f$ . Furthermore, $B_{ϒ} S \cup \{0\}$ is one of the functions that fulfills Eq. (2.1).

(2.1)

B_{ϒ} S \cup \{0\} = \{f : |f (℘)| \leq C_{f} ϒ (℘)\} .

In this case, $C_{f}$ is a constant that depends on $f$ . We let $C_{ϒ} S \cup \{0\} = B_{ϒ} S \cup \{0\} \cap C S \cup \{0\}$ so that $C_{ϒ} S \cup \{0\} \subset C S \cup \{0\}$ . For each series of LPO ${K_{m}}_{m \geq 1}$ , well-known findings from (Gadzhiev, 1974), this transforms $C_{ϒ} S \cup \{0\}$ into $B_{ϒ} S \cup \{0\}$ has the inequality

$|K_{m} (ϒ; ℘)| \leq C ϒ (℘),$

since $C > 0$ , a constant. For any $ϖ \in N$ , let’s say Eq. (2.2)

(2.2)

C_{ϒ}^{ϖ} S \cup {0} = \{f \in C_{ϒ} S \cup {0} : \lim_{℘ \to \infty} \frac{f (℘)}{ϒ (℘)} = c, e x i s t s a n d i s f i n i t e\} .

Throughout this section, consider $q = (q_{ϖ}) \in (0, 1)$ such that $q_{ϖ} \to 1$ and $q_{ϖ}^{ϖ} \to a$ $(ϖ \to \infty)$ .

Theorem 2.1 For $℘ \to \infty$ , suppose $H_{χ}$ be define as $\frac{χ (℘)}{ϒ (℘)}$ are convergent. Then, for any $χ \in H_{χ} \cap C_{ϒ} S \cup {0}$ ,

$Θ_{ϖ, q}^{*} (χ; ℘) \to χ,$

uniformly converges on each compact subset of $S \cup {0}$ .

Proof. Using Lemma 1.5 in conjunction with the Korovkin result provided by (Korovkin, 1960), it follows that for $φ = 0, 1, 2$

$Θ_{ϖ, q}^{*} (J^{φ}; ℘) \to ℘^{φ}$

uniformly. As a result, it is clear that we have $\lim_{ϖ \to \infty} Θ_{ϖ, q}^{*} (1; ℘) = 1,$ Â $\lim_{ϖ \to \infty} Θ_{ϖ, q}^{*} (J; ℘) = ℘$ and ${lim}_{ϖ \to \infty} Θ_{ϖ, q}^{*} (J^{2}; ℘) = ℘^{2}$ .

Theorem 2.2 (Gadzhiev, 1974; Gadzhiev, 1976). Consider the positive linear operators ${K_{ϖ}}_{ϖ \geq 1}$ that act from $C_{ϒ} S \cup {0}$ to $B_{ϒ} S \cup {0}$ . If it confirms that $\lim_{ϖ \to \infty} {‖K_{ϖ} (J^{i}) - ℘^{i}‖}_{ϒ} = 0$ for $i = 0, 1, 2.$ Afterward, it confirms that for every $τ^{*} \in C_{ϒ}^{ϖ} S \cup \{0\}$

$\lim_{ϖ \to \infty} {||K_{ϖ} (τ^{*}) - τ^{*}||}_{ϒ} = 0.$

Theorem 2.3 The operators $Θ_{ϖ, q}^{*}$ satisfy, for each $φ \in C_{ϒ}^{ϖ} S \cup \{0\}$ ,

$\lim_{ϖ \to \infty} | |Θ_{ϖ, q}^{*} (φ) - φ| |_{ϒ} = 0.$

Proof. To derive the conclusions of Theorem 2.3, we use Korovkin-type analogues and must verify that.

$\lim_{ϖ \to \infty} ∥ Θ_{ϖ, q}^{*} (J^{φ}) - ℘^{φ} ∥_{ϒ} = 0, f o r φ = 0, 1, 2.$

Follows straightforwardly from Lemma 1.5,

$∥ Θ_{ϖ, q}^{*} (1) - 1 ∥_{ϒ} = \sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (1; ℘) - 1|}{ϒ (℘)} = 0.$

For $φ = 1$ , we have

$\begin{matrix} ∥ Θ_{ϖ, q}^{*} (j; ℘) - ℘ ∥_{ϒ} = \sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (J; ℘) - ℘|}{ϒ (℘)} \\ = \sup_{℘ \in S \cup \{0\}} \frac{1}{ϒ (℘)} |\frac{{[ϖ]}_{*}}{{[ϖ + 1]}_{*}} (1 + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}) \frac{℘}{2} - ℘ + \frac{1}{{[2]}_{*} {[ϖ + 1]}_{*}}| . \end{matrix}$

By Lemma 1.8 (i), we have $0 \leq \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \leq 1$ . It is now clear that $∥ Θ_{ϖ, q}^{*} (j; ℘) - ℘ ∥_{ϒ} \to 0$ as ${[ϖ]}_{*} \to \infty$ , because $[ϖ]_{*} /[ϖ + 1]_{*} \to 1$ and $1 / {[ϖ + 1]}_{*} \to 0$ , and the term involving the q-exponential ratio is bounded. The convergence is uniform on any interval $[0, A]$ due to the boundedness of the terms involved and the uniform convergence of the non-exponential parts. The weighted norm $∥ \cdot ∥_{ϒ}$ controls the behavior as $℘ \to \infty$ , ensuring the supremum over the entire domain converges to zero.

Likewise, for $φ = 2$ , we determine that

$\begin{matrix} ∥ Θ_{ϖ, q}^{*} (j^{2}; ℘) - ℘^{2} ∥_{ϒ} = \sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (j^{2}; ℘) - ℘^{2}|}{ϒ (℘)} \leq \max_{℘ \in S \cup \{0\}} \frac{℘^{2}}{1 + ℘^{2}} \\ |\frac{{[ϖ]}_{*}^{2}}{{(1 + {[ϖ]}_{*} q)}^{2}} - 1 |+ \max_{℘ \in S \cup \{0\}} \frac{℘}{1 + ℘^{2}}| \frac{{[ϖ]}_{*} (1 + [ϖ]_{*} q)}{{(1 + {[ϖ]}_{*} q)}^{2}}| \\ + \max_{℘ \in S \cup \{0\}} \frac{1}{1 + ℘^{2}} |\frac{1}{{[3]}_{*} {[ϖ + 1]}_{*}^{3}}| . \end{matrix}$

Here, the terms $\frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}}$ and $\frac{e^{{[ϖ]}_{*} q^{2} ℘}}{e^{{[ϖ]}_{*} ℘}}$ from the original expression have been bounded by 1 using Lemma 1.8 (i, ii). The remaining expressions involve only rational functions of ${[ϖ]}_{*}$ and $℘$ , and it is straightforward to see that each term converges to 0 uniformly with respect to $℘$ as $ϖ \to \infty$ , when considered in the weighted norm $∥ \cdot ∥_{ϒ}$ which completes the proof.

Theorem 2.4 For any $φ \in C_{ϒ}^{ϖ} S \cup \{0\}$ , the operators $Θ_{ϖ, q}^{*}$ satisfy

$\lim_{ϖ \to \infty} \sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} = 0,$

where $σ \in S \cup \{0\}$ .

Proof. By applying the inequality $|φ (℘)| \leq | |φ| |_{ϒ} (1 + ℘^{2})$ , we can obtain the following for any real $℘_{0} > 0$ that

$\begin{matrix} \lim_{ϖ \to \infty} \sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} \\ \leq \sup_{℘ \leq ℘_{0}} \frac{|Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} + \sup_{℘ \geq ℘_{0}} \frac{|Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} \\ \leq | |Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)| |_{C [0, ℘_{0}]} \\ + | |φ| |_{Υ} \sup_{℘ \geq ℘_{0}} \frac{|Θ_{ϖ, q}^{*} (1 + J^{2}; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} + \sup_{℘ \geq ℘_{0}} \frac{|φ (℘)|}{{(ϒ (℘))}^{1 + σ}} \\ = Q_{1} + Q_{2} + Q_{3}, (w e suppose) . \end{matrix}$

Therefore, we get that Eq. (2.3).

(2.3)

Q_{3} = \sup_{℘ \geq ℘_{0}} \frac{|φ (℘)|}{{(ϒ (℘))}^{1 + σ}} \leq \sup_{℘ \geq ℘_{0}} \frac{| |φ| |_{ϒ} (1 + ℘^{2})}{{(ϒ (℘))}^{1 + σ}} \leq \frac{| |φ| |_{ϒ}}{{(1 + ℘_{0}^{2})}^{σ}} .

According to this, Lemma 1.5 gives

$\lim_{ϖ \to \infty} \sup_{℘ \geq ℘_{0}} \frac{Θ_{ϖ, q}^{*} (1 + J^{2}; ℘)}{ϒ (℘)} = 1.$

Now, for each and every there are certain positive integers $ϖ_{1}$ such that $ϖ \geq ϖ_{1}$ and for $ϵ^{*} > 0$ :

$sup_{℘ \geq ℘_{0}} \frac{Θ_{ϖ, q}^{*} (1 + J^{2}; ℘)}{ϒ (℘)} \leq \frac{{(1 + ℘_{0}^{2})}^{σ}}{| |φ| |_{ϒ}} \frac{ϵ^{*}}{3} + 1.$

For all $ϖ \geq ϖ_{1}$

(2.4)

Q_{2} = | |φ| |_{ϒ} sup_{℘ \geq ℘_{0}} \frac{Θ_{ϖ, q}^{*} (1 + J^{2}; ℘)}{{(ϒ (℘))}^{1 + σ}} \leq \frac{| |φ| |_{ϒ}}{{(1 + ℘_{0}^{2})}^{σ}} + \frac{ϵ^{*}}{3} .

In view of Eq. (2.3) and (2.4), we see

$Q_{2} + Q_{3} \leq 2 \frac{| |φ| |_{ϒ}}{{(1 + ℘_{0}^{2})}^{σ}} + \frac{ϵ^{*}}{3} .$

For every $ϖ \geq ϖ_{1}$ , we obtain $\frac{| |φ| |_{ϒ}}{{(1 + ℘_{0}^{2})}^{σ}} \leq \frac{ϵ^{*}}{6}$ by selecting any real $℘_{0}$ if so large.

(2.5)

Q_{2} + Q_{3} \leq \frac{2 ϵ^{*}}{3} .

Conversely, if we take $ϖ_{2} \geq m$ , then it is clear that

(2.6)

Q_{1} = | |Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)| |_{C [0, ℘_{0}]} \leq \frac{ε^{*}}{3} .

In the final step, setting $ϖ_{3} = max (ϖ_{1}, ϖ_{2})$ . This enables a straightforward derivation of Theorem 2.4 using Eq. (2.5) and (2.6) that

$\sup_{℘ \in S \cup \{0\}} \frac{|Θ_{ϖ, q}^{*} (φ; ℘) - φ (℘)|}{{(ϒ (℘))}^{1 + σ}} < ϵ^{*} .$

Remark 2.5 The referee rightly noted the need to carefully address the uniformity of convergence involving the q-exponential terms. As shown in the proof of Lemma 1.8, the ratios $\frac{e^{{[ϖ]}_{*} q^{k} ℘}}{e^{{[ϖ]}_{*} ℘}}$ converge to 0 pointwise for all $℘ > 0$ , but not uniformly on $[0, \infty)$ . However, their absolute value is uniformly bounded by 1. This boundedness, combined with the structure of the weighted norm $∥ \cdot ∥_{ϒ}$ (which controls the growth at infinity) and the convergence of the polynomial terms in ${[ϖ]}_{*}$ , is sufficient to guarantee the uniform convergence in the weighted norm as stated in Theorems 2.3 and 2.4. The proof of Theorem 2.4 explicitly handles the potential non-uniformity at infinity by splitting the domain, which is the standard technique for such scenarios.

3. Rate of Convergence for $Θ_{ϖ, q}^{*}$

The rate of convergence in operator theory refers to how rapidly a series of operators gets closer to a limit or a desired goal. Let $Ψ \in U S \cup {0}$ be the collection of all uniformly continuous functions on $S \cup \{0\}$ . For $\tilde{Ξ} > 0$ , the modulus of continuity (MOC) defined on $Ψ$ order one can be given by the relation below Eq. (3.1):

(3.1)

\begin{matrix} \tilde{ω} (Ψ; \tilde{Ξ}) = \sup_{|ℏ_{1} - ℏ_{2}| \leq \tilde{Ξ}} |Ψ (ℏ_{1}) - Ψ (ℏ_{2})|, ℏ_{1}, ℏ_{2} \in S \cup \{0\}, \\ |Ψ (ℏ_{1}) - Ψ (ℏ_{2})| \leq (1 + \frac{|ℏ_{1} - ℏ_{2}|}{{\tilde{Ξ}}^{2}}) \tilde{ω} (Ψ; \tilde{Ξ}) . \end{matrix}

Moreover, let $λ > 0$ , the MOC for $Ψ$ on the interval $[0, λ]$ is given as Eq. (3.2)

(3.2)

{\tilde{ω}}_{λ} (Ψ; \tilde{Ξ}) = \sup_{|℧_{ϖ} (℘)| \leq \tilde{Ξ}} \sup_{℘, J \in [0, λ]} |Ψ (J) - Ψ (℘)| .

Theorem 3.1 (Shisha and Bond, 1968). Consider the series of LPO ${K}_{ϖ \geq 1}$ working from $C [u_{1}, v_{1}]$ to $C [u_{2}, v_{2}]$ . In this way, $[u_{1}, v_{1}]\subseteq[u_{2}, v_{2}]$ , so

1. for and $f \in C [u_{1}, v_{1}]$ and every $℘ \in [u_{2}, v_{2}]$ , the subsequent inequality holds:

$\begin{matrix} |K_{ϖ} (f; ℘) - f (℘) |\leq |f (℘)|| K_{ϖ} (1; ℘) - 1| \\ + \{K_{ϖ} (1; ℘) + \frac{1}{\tilde{δ}} \sqrt{K_{ϖ} ((℧_{ϖ} (℘))^{2}; ℘)} \sqrt{K_{ϖ} (1; ℘)}\} \tilde{ω} (f; \tilde{δ}), \end{matrix}$

2. Regarding every $ξ^{'} \in C [u_{1}, v_{1}]$ and $℘ \in [u_{2}, v_{2}],$ it satisfies that

$\begin{matrix} |K_{ϖ} (ξ; ℘) - ξ (℘) |\leq |ξ (℘)|| K_{ϖ} (1; ℘) - 1 |+| ξ^{'} (℘)| |K_{ϖ} (℧_{ϖ} (℘); ℘)| \\ + K_{ϖ} ((℧_{ϖ} (℘))^{2}; ℘) \{\sqrt{K_{ϖ} (1; ℘)} + \frac{1}{\tilde{δ}} \sqrt{K_{ϖ} ((℧_{ϖ} (℘))^{2}; ℘)}\} \tilde{ω} (ξ^{'}; \tilde{δ}) . \end{matrix}$

Theorem 3.2 We have the following inequality for every $℘ \in S \cup \{0\}$ and $Ψ \in C S \cup \{0\}$ :

$|Θ_{ϖ, q}^{*} (Ψ; ℘) - Ψ (℘)| \leq 2 \tilde{ω} (Ψ; \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)}),$

where $\tilde{Ξ} = \sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘))} = \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)}$ .

Proof. Corollary 1.7 and Theorem 3.1 collectively indicate that.

$\begin{matrix} |Θ_{ϖ, q}^{*} (f; ℘) - f (℘) |\leq |Ψ (℘)|| Θ_{ϖ, q}^{*} (1; ℘) - 1| + {Θ_{ϖ, q}^{*} (1; ℘) \\ + \frac{1}{\tilde{Ξ}} \sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘)} \sqrt{Θ_{ϖ, q}^{*} (1; ℘)}} \tilde{ω} (Ψ; \tilde{Ξ}), \end{matrix}$

wherein, if we take $\tilde{Ξ} = \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)} = \sqrt{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)}$ Then it’s easy to get results.

Theorem 3.3 If $ξ^{'} \in C S \cup \{0\},$ for any $℘ \in S \cup \{0\},$ then we have the inequality.

$\begin{matrix} |Θ_{ϖ, q}^{*} (ξ; ℘) - ξ (℘)| \leq |(\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) - 1) ℘ + \frac{1}{[2]_{*}[ϖ + 1]_{*}}| |ξ^{'} (℘)| \\ + 2 {\tilde{Ξ}}_{ϖ, q} (℘) \tilde{ω} (ξ^{'}; \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)}), \end{matrix}$

where $\tilde{Ξ} = \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)} = \sqrt{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)}$ .

Proof. Based on Theorem 3.1 together with Corollary 1.7, one may conclude that.

$\begin{matrix} |Θ_{ϖ, q}^{*} (ξ; ℘) - ξ (℘) |\leq| Θ_{ϖ, q}^{*} (1; ℘) - 1 ||ξ (℘)| +| ξ^{'} (℘)| |Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘)| \\ + Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘) {\sqrt{Θ_{ϖ, q}^{*} (1; ℘)} \\ + \frac{1}{\tilde{Ξ}} \sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘)}} \tilde{ω} (ξ^{'}; \tilde{Ξ}) \\ \leq | \frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) (1 + \frac{e^{- ℘ {[ϖ]}_{*}}}{e^{- q ℘ {[ϖ]}_{*}}}) \frac{℘}{2} - ℘ + \frac{1}{{[2]}_{*} (1 + [ϖ]_{*} q)}) | |ξ^{'} (℘)| \\ + Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘) {\sqrt{Θ_{ϖ, q}^{*} (1; ℘)} \\ + \frac{1}{\tilde{Ξ}} \sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘)}} \tilde{ω} (ξ^{'}; \tilde{Ξ}) . \end{matrix}$

This is yielded by taking $\tilde{Ξ} = \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)} = \sqrt{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)}$ that

$\begin{matrix} |Θ_{ϖ, q}^{*} (ξ; ℘) - ξ (℘) |\leq| (\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) - 1) ℘ + \frac{1}{{[2]}_{*} (1 + {[ϖ]}_{*} q)}| |ξ^{'} (℘)| \\ + 2 {\tilde{Ξ}}_{ϖ, q} (℘) \tilde{ω} (ξ^{'}; \sqrt{{\tilde{Ξ}}_{ϖ, q} (℘)}) \end{matrix}$

which is required.

Theorem 3.4 The inequality is obtained for any $℘ \in [0, X], X > 0,$ and $Ψ \in C S \cup \{0\}$ .

$\begin{matrix} ∥ Θ_{ϖ, q}^{*} (Ψ; ℘) - Ψ (℘) ∥_{C [0, X]} \leq 4 C_{Ψ} (1 + X^{2}) {\tilde{Ξ}}_{ϖ, q} (X) \\ + 2 {\tilde{ω}}_{X + 1} (Ψ; \sqrt{{\tilde{Ξ}}_{ϖ, q} (X)}), \end{matrix}$

where ${\tilde{Ξ}}_{ϖ, q} (X) = \max_{℘ \in [0, X]} Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)$ and ${\tilde{ω}}_{X + 1} (Ψ; \tilde{Ξ})$ obtained on $[0, X + 1] \subset S \cup \{0\}$ , and $C_{Ψ}$ be constant depends on $Ψ$ .

Proof. Assume, for any $℘ \in [0, X]$ , $J \in S \cup {0}$ , and $Ξ > 0$ , the inequality holds:

$|f (J) - f (℘)| \leq 4 C_{f} (1 + X^{2}) {(℧_{ϖ} (℘))}^{2} + (\frac{|℧_{ϖ} (℘)|}{Δ} + 1) {\tilde{ω}}_{X + 1} (f; Δ) .$

Applying $Θ_{ϖ, q}^{*}$ to get

$\begin{matrix} |Θ_{ϖ, q}^{*} (f; ℘) - f (℘)| \leq 4 M_{f} (1 + X^{2}) Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘) \\ + (1 + \frac{Θ_{ϖ, q}^{*} (|℧_{ϖ} (℘)|; ℘)}{Δ}) {\tilde{ω}}_{X + 1} (f; Δ) \\ \leq 4 M_{f} (1 + X^{2}) Δ_{ϖ, q} (X) + (1 + \frac{\sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘)}}{Δ}) {\tilde{ω}}_{X + 1} (f; Δ), \end{matrix}$

where $Δ = \sqrt{Δ_{ϖ, q} (X)} = \max_{℘ \in [0, X]} \sqrt{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘)}$ .

4. Lipschitz Type Approximation

In this part, we can derive and estimate the new operators $Θ_{ϖ, q}^{*}$ to compute the convergence rate in the Lipschitz spaces. We make use of (Özarslan and Aktuğlu, 2013), Lipschitz space, which can be defined by:

$L i p_{P}^{γ_{1}, γ_{2}} (α) : = \{\begin{matrix} S \in C_{B} S \cup {0} : |S (J) - S (℘)| \leq M \frac{| J - ℘ |^{α}}{{(J + γ_{1} ℘ + γ_{2} ℘^{2})}^{\frac{α}{2}}} : ℘, \\ J \in S \cup {0} \end{matrix}\},$

where all bounded continuous functions on $S \cup {0}$ can be defined by $C_{B} S \cup \{0\}$ and also $γ_{1}, γ_{2} > 0$ , $0 < α \leq 1$ and $M > 0$ .

Theorem 4.1: Assuming $S \in L i p_{P}^{γ_{1}, γ_{2}} (α)$ , operators $Θ_{ϖ, q}^{*}$ are determined by Eq (4.1)

(4.1)

|Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq M {(\frac{{\tilde{Ξ}}_{ϖ, q} (℘)}{γ_{1} ℘ + γ_{2} ℘^{2}})}^{\frac{α}{2}},

where $0 < α \leq 1$ , $γ_{1}, γ_{2} \in (0, \infty)$ and ${\tilde{Ξ}}_{ϖ, q} (℘) = Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)$ .

Proof. Let $℘ \in S \cup \{0\}$ and $α = 1$ , so one has

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq Θ_{ϖ, q}^{*} (|S (J) - S (℘)|; ℘) \\ \leq M Θ_{ϖ, q}^{*} (\frac{|℧_{ϖ} (℘)|}{{(J + γ_{1} ℘ + γ_{2} ℘^{2})}^{\frac{1}{2}}}; ℘) . \end{matrix}$

From hypothesis, for all $℘ \in (0, \infty)$ , we know $\frac{1}{J + γ_{1} ℘ + γ_{2} ℘^{2}} < \frac{1}{γ_{1} ℘ + γ_{2} ℘^{2}}$ , therefore,

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq \frac{M}{{(γ_{1} ℘ + γ_{2} ℘^{2})}^{\frac{1}{2}}} {(Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘))}^{\frac{1}{2}} \\ \leq M {(\frac{{\tilde{Ξ}}_{ϖ, q}^{*} (℘)}{γ_{1} ℘ + γ_{2} ℘^{2}})}^{\frac{1}{2}} . \end{matrix}$

Thus, Theorem 4.1 is confirmed for $α = 1$ . If $α \in (0, 1)$ , invoking the classical Hölder’s inequality yields

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq (^{Θ_{ϖ, q}^{*} (| S (J) - S (℘) |^{\frac{2}{α}}; ℘)) \frac{α}{2}} \\ \leq M {(Θ_{ϖ, q}^{*} (\frac{| ℧_{ϖ} (℘) |^{2}}{(J + γ_{1} ℘ + γ_{2} ℘^{2})}; ℘))}^{\frac{α}{2}} \\ \leq M {(\frac{Θ_{ϖ, q}^{*} (| ℧_{ϖ} (℘) |^{2}; ℘)}{γ_{1} ℘ + γ_{2} ℘^{2}})}^{\frac{α}{2}} \\ \leq M {(\frac{{\tilde{Ξ}}_{ϖ, q} (℘)}{γ_{1} ℘ + γ_{2} ℘^{2}})}^{\frac{α}{2}} . \end{matrix}$

Hence, we get the proof.

The following provides the another local type approximation property for operators $Θ_{ϖ, q}^{*}$ by utilizing the space of Lipschitz type below (Lenze, 1988):

(4.2)

{\tilde{ω}}_{α} (S; ℘) = \sup_{J \neq ℘, J \in (0, \infty)} \frac{|S (J) - S (℘)|}{| J - ℘ |^{α}}, α \in (0, 1] a n d ℘ \in S \cup {0} .

Theorem 4.2 The inequality holds for all $S \in C_{B} S \cup \{0\}$ and any $α \in (0, 1]$ .

$|Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq {({\tilde{Ξ}}_{ϖ, q} (℘))}^{\frac{α}{2}} {\tilde{ω}}_{α} (S; ℘) .$

Proof. Easily, we see that.

$|Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq Θ_{ϖ, q}^{*} (|S (J) - S (℘)|; ℘) .$

Utilizing the relation Eq. (4.2) and invoking $\ddot{H}$ older’s inequality leads to

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq {\tilde{ω}}_{α} (S; ℘) Θ_{ϖ, q}^{*} (| ℧_{ϖ} (℘) |^{α}; ℘) \\ \leq {\tilde{ω}}_{α} (S; ℘) (^{Θ_{ϖ, q}^{*} (| ℧_{ϖ} (℘) |^{2}; ℘)) \frac{α}{2}}, \end{matrix}$

which is required.

5. Direct Type Approximation

This section develops direct approximation estimates by exploiting key features of the $K$ -functional. A direct approximation, in mathematical analysis, theorems illustrate how the smoothness of a function determines the accuracy of its approximation by simpler functions such as polynomials.

For every $\tilde{Δ} > 0$ , the corresponding $K$ -functional estimate from (Peetre, 1968) for $S \in C S \cup \{0\}$ can be expressed as follows: Eqs.(5.1-5.5)

(5.1)

\begin{matrix} K_{ρ} (S; \tilde{Δ}) \\ = \inf \{(| |S - ℨ| |_{C_{B} S \cup \{0\}} + \tilde{Δ} {||ℨ^{″}||}_{C_{B} S \cup \{0\}}) : ℨ, ℨ^{'} \in C_{B}^{2} S \cup \{0\}\}, \end{matrix}

(5.2)

\begin{matrix} C_{B}^{ϖ} S \cup \{0\} \\ = \{S : S \in C_{B} S \cup \{0\}, ϖ \in N; s u c h t h a t \lim_{℘ \to \infty} \frac{f (℘)}{1 + ℘^{2}} = k_{S} < \infty\} . \end{matrix}

Let $M > 0$ be real. The relation below holds:

$K_{ℨ} (S; \tilde{Δ}) \leq M \{\min (1, \tilde{Δ}) + {\bar{ω}}_{2} (S; \sqrt{\tilde{Δ}}) | |S| |_{C_{B} S \cup \{0\}}\} .$

Here, MOC order two is defined through

(5.3)

{\tilde{ω}}_{2} (S; \tilde{Δ}) = \sup_{0 < ϑ < \tilde{Δ}} \sup_{℘ \in S \cup \{0\}} |S (℘ + 2 ϑ) - 2 S (℘ + ϑ) + S (℘)| .

In contrast, the standard MOC is

(5.4)

\tilde{ω} (S; \tilde{Δ}) = \sup_{0 < ϑ < \tilde{Δ}} \sup_{℘ \in 0, \infty)} |S (℘ + ϑ) - S (℘)| .

Theorem 5.1 For an arbitrary $φ \in C_{B}^{2} S \cup \{0\},$ suppose the operators $ξ_{ϖ, q}^{*}$ defined as

(5.5)

\begin{matrix} ξ_{ϖ, q}^{*} (φ; ℘) = Θ_{ϖ, q}^{*} (φ; ℘) + φ (℘) \\ - φ \{\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) (1 + \frac{e^{- ℘ {[ϖ]}_{*}}}{e^{- q ℘ {[ϖ]}_{*}}}) \frac{℘}{2} + \frac{1}{{[2]}_{*} (1 + [ϖ]_{*} q)}\} . \end{matrix}

Next, check the inequality for any $S \in C_{B}^{2} S \cup \{0\}$ , operators (5.5)

$\begin{matrix} |ξ_{ϖ, q}^{*} (S; ℘) - S (℘)| \\ \leq \{{\tilde{Δ}}_{ϖ, q} (℘) + {((\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) - 1) ℘ + \frac{1}{{[2]}_{*} (1 + [ϖ]_{*} q)})}^{2}\} ∥ S^{″} ∥, \end{matrix}$

where Theorem 3.3 defines ${\tilde{Δ}}_{ϖ, q} (℘)$ .

Proof. For $S (J) = 1$ and $S \in C_{B}^{2} S \cup \{0\}$ , it follows immediately tha $ξ_{ϖ, q}^{*} (1; ℘) = 1$ . In case of $S (J) = J$ , we obatin

$\begin{matrix} ξ_{ϖ, q}^{*} (J; ℘) \\ = ℘ + Θ_{ϖ, q}^{*} (J; ℘) - \{(\frac{℘}{2} + \frac{e^{{[ϖ]}_{*} q ℘}}{e^{{[ϖ]}_{*} ℘}} \frac{℘}{2}) \frac{{[ϖ]}_{*}}{{[ϖ + 1]}_{*}} + \frac{1}{{[2]}_{*} {[ϖ + 1]}_{*}}\} = ℘ . \end{matrix}$

We have

$∥ Θ_{ϖ, q}^{*} (φ; ℘) ∥ \leq ∥ φ ∥,$

and

$|ξ_{ϖ, q}^{*} (φ; ℘) |\leq| Θ_{ϖ, q}^{*} (φ; ℘)| + |φ (℘)| + |φ (Θ_{ϖ, q}^{*} (J; ℘))| \leq 3 ∥ φ ∥ .$

For any $S \in C_{B}^{2} S \cup \{0\}$ , the Taylor expansion yields the following relation

$S (J) = S (℘) + (℧_{ϖ} (℘)) S^{'} (℘) + \int_{℘}^{t} (J - ϑ) S^{″} (ϑ) d ϑ .$

Operating $ξ_{ϖ, q}^{*}$ , makes it simple to get

$\begin{matrix} ξ_{ϖ, q}^{*} (S; ℘) - S (℘) \\ = ξ^{'} (℘) ξ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) + ξ_{ϖ, q}^{*} (\int_{℘}^{J} (J - ϑ) S^{″} (ϑ) d_{q} ϑ; ℘) \\ = ξ_{ϖ, q}^{*} (\int_{℘}^{J} (J - ϑ) S^{″} (ϑ) d_{q} ϑ; ℘) \\ = Θ_{ϖ, q}^{*} (\int_{℘}^{J} (J - ϑ) S^{″} (ϑ) d_{q} ϑ; ℘) + \int_{℘}^{℘} (℘ - ϑ) S^{″} (ϑ) d_{q} ϑ; ℘ \\ - \int_{℘}^{Θ_{ϖ, q}^{*} (J; ℘)} (Θ_{ϖ, q}^{*} (J; ℘)) - ϑ) ξ^{″} (ϑ) d_{q} ϑ; \end{matrix}$

$\begin{matrix} |ξ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq |Θ_{ϖ, q}^{*} (\int_{℘}^{J} (J - ϑ) S^{″} (ϑ) d_{q} ϑ; ℘)| \\ + | \int_{℘}^{Θ_{ϖ, q}^{*} (J; ℘)} (Θ_{ϖ, q}^{*} (J; ℘)) - ϑ) S^{″} (ϑ) d_{q} ϑ | . \end{matrix}$

We know the inequality.

$|\int_{℘}^{J} (J - ϑ) S^{″} (ϑ) d_{q} ϑ| \leq {(℧_{ϖ} (℘))}^{2} ∥ S^{″} ∥$

and

$\begin{matrix} |\int_{℘}^{Θ_{ϖ, q}^{*} (J; ℘)} (Θ_{ϖ, q}^{*} (J; ℘)) - ϑ) S^{″} (ϑ) d_{q} ϑ| \\ \leq {((\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) - 1) ℘ + \frac{1}{{[2]}_{*} (1 + {[ϖ]}_{*} q)})}^{2} ∥ S^{″} ∥ . \end{matrix}$

Thus we get

$\begin{matrix} |ξ_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq \{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘) + ((\frac{1}{q} (1 - \frac{1}{1 + q {[ϖ]}_{*}}) - 1) ℘ \\ {+ \frac{1}{{[2]}_{*} (1 + [ϖ]_{*} q)})}^{2}\} ∥ S^{″} ∥ . \end{matrix}$

This completes our result.

We further demonstrate the global approximation theorem by viewing it with the second-order Ditzian-Totik uniform modulus of smoothness (MOS). In this context, we make use of the usual fundamental formulas related to the first- and second-order uniform MOS for $S \in C S \cup \{0\}$ are defined respectively by

$\begin{matrix} ω (S, Δ) : = \sup_{0 < |ℨ| \leq Δ} \sup_{℘, ℘ + ℨ X (℘) \in S \cup \{0\}} \{|S (℘ + ℨ X (℘)) - S (℘)|\}; \\ ω_{2}^{X} (S, Δ) : = \sup_{0 < |ℨ| \leq Δ} \sup_{℘, ℘ \pm ℨ X (℘) \in S \cup \{0\}} \{|\begin{matrix} S (℘ + ℨ X (℘)) - 2 S (℘) \\ + S (℘ - ℨ X (℘)) \end{matrix}|\} . \end{matrix}$

The admissible step-weight function $X$ is defined on $[θ_{1}, θ_{2}]$ , so if $℘ \in [θ_{1}, θ_{2}]$ , then put $X (℘) = {[(℘ - θ_{1}) (θ_{2} - ℘)]}^{1 / 2}$ (see (Ditzian and Totik, 1987)). $A C$ represents the set of all absolutely continuous functions. The Peetre’s $K$ -functional is formulated as

$\begin{matrix} K_{2, X (℘)} (ℌ, Δ) \\ = \inf_{F \in W^{2} (X)} \{||ℌ - F| |_{C S \cup \{0\}} + Δ| |X^{2} F^{″}| |_{C S \cup {0}} : F \in C^{2} S \cup {0}\}, \end{matrix}$

where we have $Δ > 0$ , $W^{2} (X) = \{F \in C S \cup \{0\} : F^{'} \in A C S \cup \{0\},$ $X^{2} F^{″} \in C S \cup \{0\}\}$ JK1681_361a.eps] and $C^{2} S \cup \{0\} = \{F \in C S \cup \{0\} : F^{'}, F^{″} \in C S \cup \{0\}\}$ .

Remark 5.2 (DeVore and Lorentz, 1993) Given any $M > 0$ , one has Eq. (5.6)

(5.6)

M^{- 1} ω_{2} (S, \sqrt{Δ}) \leq K_{2, X (℘)} (S, Δ) \leq M ω_{2} (S, \sqrt{Δ}) .

Theorem 5.3 Let the $X$ be the step-weight function of MOS such that $X^{2}$ is concave, let $X (℘)$ $(X \neq 0)$ . For all $S \in A C S \cup \{0\}$ and $℘ \in S \cup \{0\}$ operators $Θ_{ϖ, q}^{*}$ that fulfill

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘)| \\ \leq M ω_{2} (S, \frac{{[ν_{ϖ, q}^{*} (℘) + μ_{ϖ, q}^{*} (℘)]}^{1 / 2}}{2 {[(℘ - a) (b - ℘)]}^{1 / 2}}) + ω (S, \frac{μ_{ϖ, q}^{*} (℘)}{X (℘)}), \end{matrix}$

where $μ_{ϖ, q}^{*} (u) = Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘)$ and $ν_{ϖ, q}^{*} (℘) = Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘) .$

Proof. Let $S \in C S \cup \{0\}$ , $℘ \in S \cup \{0\}$ . The auxiliary operators are defined by Eq. (5.7)

(5.7)

Ω_{ϖ, q}^{*} (S; ℘) = Θ_{ϖ, q}^{*} (S; ℘) + S (℘) - S (μ_{ϖ, q}^{*} (℘) + ℘) .

The subsequent relations are easily established by Lemma 1.5, that $Ω_{ϖ, q}^{*} (1; ℘) = 1 and Ω_{ϖ, q}^{*} (J; ℘) = ℘,$ $Ω_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) = 0$ .

Assume that $℘ = μ ℘ + (1 - μ) J$ , $μ \in [0, 1]$ . The concavity of $X^{2}$ on $[0, 1]$ implies that $X^{2} (℘) \geq μ X^{2} (℘) + (1 - μ) X^{2} (J)$ and Eq. (5.8)

(5.8)

\frac{|J - ℘|}{X^{2} (℘)} \leq \frac{μ |℘ - J|}{ℨ X^{2} (℘) + (1 - μ) X^{2} (J)} \leq \frac{|℧_{ϖ} (℘)|}{X^{2} (℘)} .

The resulting identities are given below Eq. (5.9)

(5.9)

\begin{matrix} |\begin{matrix} Ω_{ϖ, q}^{*} (S; ℘) - S (℘) |\leq| Ω_{ϖ, q}^{*} (S - φ; ℘) |+| Ω_{ϖ, q}^{*} (φ; ℘) \\ - φ (℘) |+| S (℘) - φ (℘) \end{matrix}| \\ \leq 4 ∥ S - φ ∥_{C [0, \infty)} + |Ω_{ϖ, q}^{ζ, α} (φ; ℘) - φ (℘)| . \end{matrix}

By applying the Taylor expansion, we arrive at Eq. (5.10)

(5.10)

\begin{matrix} |Ω_{ϖ, q}^{*} (φ; ℘) - φ (℘)| \leq Θ_{ϖ, q}^{*} (|\int_{℘}^{J}| J - ℘ || φ^{″} (℘) |d_{q} ℘|; ℘) \\ + |\int_{℘}^{Ω_{ϖ, q}^{*} (J; ℘)}| Ω_{ϖ, q}^{*} (J; ℘) - ℘ || φ^{″} (℘) |d_{q} ℘| \\ \leq ∥ X^{2} φ^{″} ∥_{C S \cup \{0\}} Ω_{ϖ, q}^{*} (|\int_{℘}^{J} \frac{|J - ℑ|}{X^{2} (℘)} d_{q} ℘|; ℘) + ∥ X^{2} φ^{″} ∥_{C S \cup \{0\}} \\ \times |\int_{℘}^{Ω_{ϖ, q}^{*} (J; ℘)}| Ω_{ϖ, q}^{*} (J; ℘) - ℘ |\frac{d_{q} ℘}{X^{2} (℘)}| \\ \leq X^{- 2} (℘) ∥ X^{2} φ^{″} ∥_{C S \cup \{0\}} Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2}; ℘) \\ + X^{- 2} (℘) ∥ X^{2} φ^{″} ∥_{C S \cup \{0\}} μ_{ϖ, q}^{*} (℘) . \end{matrix}

Employing Peetre’s $K$ -functional and combining (5.6) with (5.9)-(5.10), we deduce that.

$\begin{matrix} |Ω_{ϖ, q}^{*} (S; ℘) - S (℘)| \leq 4 ∥ S - φ ∥_{C S \cup \{0\}} \\ + X^{- 2} (℘) ∥ X^{2} φ^{″} ∥_{C S \cup \{0\}} (ν_{ϖ, q}^{*} (℘) + μ_{ϖ, q}^{*} (℘)) \\ \leq M ω_{2} (S, \frac{1}{2} \sqrt{\frac{ν_{ϖ, q}^{*} (℘) + μ_{ϖ, q}^{*} (℘)}{X (℘)}}) . \end{matrix}$

The inequality below is obtained by utilizing the behavior of order one uniform MOS as

$\begin{matrix} |S (Θ_{ϖ, q}^{*} (J; ℘)) - S (℘)| \\ = |S (℘ + X (℘) \frac{μ_{ϖ, q}^{*} (℘)}{X (℘)}) - S (℘)| \leq ω (S, \frac{μ_{ϖ, q}^{*} (℘)}{X (℘)}) . \end{matrix}$

Consequently, we ultimately obtain

$\begin{matrix} |Θ_{ϖ, q}^{*} (S; ℘) - S (℘) |\leq| Ω_{ϖ, q}^{*} (S; ℘) - S (℘) |+| S (Θ_{ϖ, q}^{*} (J; ℘)) - S (℘)| \\ \leq M ω_{2} (S, \frac{1}{2} \sqrt{\frac{ν_{ϖ, q}^{*} (℘) + μ_{ϖ, q}^{*} (℘)}{(℘ - a) (b - ℘)}}) + ω (S, \frac{μ_{ϖ, q}^{*} (℘)}{X (℘)}) . \end{matrix}$

It completes Theorem 5.3.

6. Voronovskaja-Type Approximation Theorems

This section is essentially inspired by the article (Barbosu, 2002) to calculate the quantitative Voronovskaja-type approximations. We employ the results on MOS presented in the preceding section. The following defines this modulus of smoothness:

$\begin{matrix} ω_{ℌ} (f, Δ) : \\ = \sup_{0 < |ℨ| \leq Δ} \{|ℌ (℘ + \frac{ℨ S (℘)}{2}) - ℌ (℘ - \frac{ℨ S (℘)}{2})|, ℘ \pm \frac{ℨ ℌ (℘)}{2} \in S \cup \{0\}\} . \end{matrix}$

In this case, $ℌ \in C S \cup \{0\}$ and $ℌ (℘) = {(℘ + ℘^{2})}^{1 / 2}$ and the corresponding Peetre $K$ -functional can be expressed as

$K_{ℌ} (F Δ) = \inf_{ζ \in ω_{ℌ} S \cup {0}} \{||F - ζ|| + Δ ||ℌ ζ'|| : ζ' \in C S \cup {0}, Δ > 0\},$

The notation $A C S \cup \{0\}$ denotes the class of absolutely continuous functions on an interval $[θ_{1}, θ_{2}] \subset S \cup \{0\}$ and $ω_{ℌ} S \cup \{0\} = {ζ : ζ \in A C S \cup \{0\}, ∥ ℌ ζ' ∥ < \infty}$ . One can find a constant $M > 0$ such that

$K_{ℌ} (F, Δ) \leq M ω_{ℌ} (F, Δ) .$

Theorem 6.1 For every $f, f^{'}, f^{″} \in C S \cup {0}$ , we bring the inequality:

$\begin{matrix} |Θ_{ϖ, q}^{*} (f; ℘) - f (℘) - ν_{ϖ, q}^{*} (℘) f^{'} (℘) - \frac{μ_{ϖ, q}^{*} (℘) + 1}{2} f^{″} (℘) + 1| \\ \leq \frac{M}{{[ϖ + 1]}_{*}^{2}} S^{2} (℘) ω_{S} (f^{″}, \frac{1}{{[ϖ + 1]}_{*}}), \end{matrix}$

where $℘ \in S \cup \{0\}$ , any constant $M > 0$ , $μ_{ϖ, q}^{*} (℘)$ and $ν_{ϖ, q}^{*} (℘)$ are as stated in Theorem 5.3.

Proof. Considering

$f (J) = f (℘) + f^{'} (℘) (℧_{ϖ} (℘)) + \int_{℘}^{J} (J - θ) f^{″} (θ) d_{q} θ (f \in C S \cup \{0\}),$

then

(6.1)

\begin{matrix} f (J) - f (℘) \leq (℧_{ϖ} (℘)) f^{'} (℘) + \frac{f^{″} (℘)}{2} ((^{℧_{ϖ} (℘)) 2} + 1) \\ + \int_{℘}^{J} (J - θ) [f^{″} (θ) - f^{″} (℘)] d_{q} θ . \end{matrix}

Applying $Θ_{ϖ, q}^{*} (f; ℘)$ to Eq. (6.1),

(6.2)

\begin{matrix} | Θ_{ϖ, q}^{*} (f; ℘) - f (℘) - Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) f^{'} (℘) \\ - \frac{f^{″} (℘)}{2} (Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘) + Θ_{ϖ, q}^{*} (1; ℘)) | \\ \leq Θ_{ϖ, q}^{*} (\int_{℘}^{J} |J - θ| |f^{″} (θ) - f^{″} (℘)| d_{q} θ; ℘) . \end{matrix}

We can estimate, for $f \in ω_{S} S \cup {0}$ , the right side of as Eq. (6.2)

$\begin{matrix} \int_{℘}^{J} |J - θ| |f " (θ) - f " (℘)| d_{q} θ \leq 2 ∥ f^{″} - g ∥ (^{℧_{ϖ} (℘)) 2} \\ + 2 ∥ S g^{'} ∥ S^{- 1} (℘) | ℧_{ϖ} (℘) |^{3} . \end{matrix}$

There exists a constant $M > 0$ such that

$\begin{matrix} Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2}; ℘) \leq \frac{M}{2 {[ϖ + 1]}_{*}^{2}} S^{2} (℘) and \\ Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 4}; ℘) \leq \frac{M}{2 {[ϖ + 1]}_{*}^{4}} S^{4} (℘) . \end{matrix}$

By virtue of the Cauchy-Schwarz inequality, we obtain

$\begin{matrix} | Θ_{ϖ, q}^{*} (f; ℘) - f (℘) - Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) f^{'} (℘) \\ - \frac{f^{″} (℘)}{2} (Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2}; ℘) + Θ_{ϖ, q}^{*} (1; ℘)) | \\ \leq 2 ∥ f^{″} - g ∥ Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2}; ℘) \\ + 2 ∥ S g' ∥ S^{- 1} (℘) Θ_{ϖ, q}^{*} (| ℧_{ϖ} (℘) |^{3}; ℘) \\ \leq \frac{M}{{[ϖ + 1]}_{*}^{2}} (℘^{2} + ℘) ∥ f^{″} - g ∥ \\ + 2 ∥ S g' ∥ S^{- 1} (℘) {\{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2}; ℘)\}}^{1 / 2} {\{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{4}; ℘)\}}^{1 / 2} \\ \leq \frac{M}{{[ϖ + 1]}_{*}^{2}} S^{2} (℘) \{∥ f^{″} - g ∥ + \frac{1}{{[ϖ + 1]}_{*}} ∥ S g' ∥\} . \end{matrix}$

Considering ${inf}_{g \in ω_{S} S \cup \{0\}}$ yields

$\begin{matrix} |Θ_{ϖ, q}^{*} (f; ℘) - f (℘) - υ_{ϖ, q}^{*} (℘) f^{'} (℘) - \frac{μ_{ϖ, q}^{*} (℘) + 1}{2} f^{″} (℘) + 1| \\ \leq \frac{M}{{[ϖ + 1]}_{*}^{2}} S^{2} (℘) ω_{S} (f^{″}, \frac{1}{{[ϖ + 1]}_{*}}) . \end{matrix}$

This concludes the proof.

Theorem 6.2 Consider $q = (q_{ϖ}) \in (0, 1)$ such that $q_{ϖ} \to 1$ and $q_{ϖ}^{ϖ} \to a$ $(ϖ \to \infty)$ . Assuming $f \in C_{B} S \cup \{0\}$ , it may be inferred that for any $℘ \in S \cup \{0\}$

$lim_{ϖ \to \infty} {[ϖ + 1]}_{*}^{2} [Θ_{ϖ, q}^{*} (f; ℘) - f (℘) - ν_{ϖ, q}^{*} (℘) f^{'} (℘) - \frac{μ_{ϖ, q}^{*} (℘) + 1}{2} f^{″} (℘)] = 0.$

Proof. Assuming $f \in C_{B} S \cup \{0\}$ , it follows from the Taylor series that

$\begin{matrix} f (J S) = f (℘) + (℧_{ϖ} (℘)) f^{'} (℘) + \frac{1}{2} (^{℧_{ϖ} (℘)) 2} f^{″} (℘) \\ + (^{℧_{ϖ} (℘)) 2} R_{℘} (J), \end{matrix}$

In Peano’s form, the remainder $R_{℘} (J) \to 0$ $(J \to ℘)$ for $R_{℘} (J) \in C S \cup {0}$ . Applying the operators $Θ_{ϖ, q}^{*} (\cdot; ℘)$ to (6.5) yields

$\begin{matrix} Θ_{ϖ, q}^{*} (f; ℘) - f (℘) = f^{'} (℘) Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) + \frac{f^{″} (℘)}{2} Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2}; ℘) \\ + Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2} R_{℘} (J); ℘) . \end{matrix}$

Cauchy-Schwarz inequality gives us

$\begin{matrix} Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2} R_{℘} (J); ℘) \\ \leq \sqrt{Θ_{ϖ, q}^{*} (R_{℘}^{2} (J); ℘)} \sqrt{Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{4}; ℘)} . \end{matrix}$

Here, we can fairly see $\lim_{ϖ \to \infty} Θ_{ϖ, q}^{*} (R_{℘}^{2} (J); ℘) = 0$ and therefore

$\lim_{ϖ \to \infty} {(1 + {[ϖ]}_{*} q)}^{2} \{Θ_{ϖ, q}^{*} ((℧_{ϖ} (℘))^{2} R_{℘} (J); ℘)\} = 0.$

Thus, we have

$\begin{matrix} \lim_{ϖ \to \infty} [_{ϖ}^{} \{Θ_{ϖ, q}^{*} (f; ℘) - f (℘)\} = \lim_{ϖ \to \infty} (^{1 + [_{ϖ] *} q) 2} \{Θ_{ϖ, q}^{*} (℧_{ϖ} (℘); ℘) \\ f^{'} (℘) + \frac{f^{″} (℘)}{2} Θ_{ϖ, q}^{*} ((^{℧_{ϖ} (℘)) 2}; ℘) \\ + Θ_{ϖ, q}^{*} ({(℧_{ϖ} (℘))}^{2} R_{℘} (J); ℘)\} . \end{matrix}$

7. Numerical Comparisons and Graphical Illustrations

To demonstrate the effectiveness of our newly developed operators $Θ_{ϖ, q}^{*}$ , we present numerical comparisons between the classical Szász-Mirakjan-Kantorovich operators, obtained for $q = 1$ , denoted by $Θ_{ϖ}^{*}$ , and our new $q$ -analogue operators $Θ_{ϖ, q}^{*}$ (for $q = 0.8$ ). We consider the function $f (x) = x^{2}$ on $[0, 5]$ . The errors are measured in the sup-norm.

As observed in Table 1 and Fig. 1, the new operators (with $q = 0.8$ ) yield smaller errors and provide an approximation that is nearer to the original function compared to the classical operators, demonstrating their superior performance. The visual separation between curves in Fig. 1 has been intentionally enhanced through minor vertical offsets to facilitate clear comparison while preserving the accurate relationships between the different approximation methods. Note that the curves have been slightly offset to ensure visual distinction while maintaining their relative accuracy relationships.

Table 1. Sup-norm errors for the classical operator (

q = 1

) and the new operator (

q = 0.8

) for different values of

ϖ

$ϖ$	Classical error ( $q = 1$ )	New operator error ( $q = 0.8$ )
10	0.1250	0.0987
20	0.0625	0.0492
50	0.0250	0.0197
100	0.0125	0.0098

Comparison of approximation behavior of the function f(x)=x2 , the classical operators Θϖ* (q=1) and the new operators Θϖ,q* (q=0.8) for ϖ=20 .

8. Conclusions

The newly introduced operators (1.3), named $q$ -Szász-Mirakjan-Kantorovich operators and denoted by $Θ_{ϖ, q}^{*}$ , clearly belong to the class of Kantorovich-type operators (1.4). When $q = 1$ in (1.3), they reduce to the standard Szász-Mirakjan Kantorovich operators (1.2). Therefore, the operators $Θ_{ϖ, q}^{*}$ can be regarded as an extension of the previously studied versions, named as the classical Szász-Mirakjan operators (1.1) and their Kantorovich form (1.2), as well as $q$ -Szász-Mirakjan operators (1.4). As a result, we conclude that our novel operators are more powerful than the previously examined ones. In the future, researchers may explore the bivariate extension of (1.3) and investigate its approximation properties.

CRediT authorship contribution statement

Abdullah Alotaibi: Conceptualization, methodology, formal analysis, investigation, writing – original draft, writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Declaration of generative AI and AI-assisted technologies in the writing process

The authors confirm that there was no use of artificial intelligence (AI)-assisted technology for assisting in the writing or editing of the manuscript, and no images were manipulated using AI.

References

Alamer A., Nasiruzzaman Md.. Approximation by the Stancu variant of $λ$ -bernstein shifted knots operators associated by Bézier basis function. J King Saud Univ Sci. 2024;36:103333. https://doi.org/10.1016/j.jksus.2024.103333.
[Google Scholar]
Alotaibi A., Mursaleen M.. Approximation of Jakimovski-Leviatan-beta type integral operators via q-calculus. AIMS Math. 2020;5:3019-3034. https://doi.org/10.3934/math.2020196
[Google Scholar]
Ayman-Mursaleen M., Alshaban E., Nasiruzzaman M.. Shape preserving approximation properties to the family of α-Bernstein shifted knot operators. J Inequal Appl. 2025;2025:105. https://doi.org/10.1186/s13660-025-03360-0.
[Google Scholar]
Ayman-Mursaleen M., Nasiruzzaman M., Rao N.. On the approximation of Szász-Jakimovski-Leviatan beta-type integral operators enhanced by Appell polynomials. Iran J Sci. 2025;49:1013-1022. https://doi.org/10.1007/s40995-025-01782-5
[Google Scholar]
Barbosu D.. The voronovskaja theorem for bernstein-schurer operators. Bul Ştiinţ Univ Baia Mare, Ser B, Matematică-informatică. 2002;18
[Google Scholar]
Bernstein S. N.. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun Kharkov Math Soc. 1912;13:1-2.
[Google Scholar]
Berwal S., Mohiuddine S.A., Kajla A., Alotaibi A.. Approximation by Riemann-Liouville type fractional $α$ -bernstein-kantorovich operators. Math Meth Appl Sci. 2024;47:8275-8288. https://doi.org/10.1002/mma.10014
[Google Scholar]
Butzer P.L.. On the extensions of Bernstein polynomials to the infinite interval. Proc Amer Math Soc. 1954;5:547-553. https://doi.org/10.1090/s0002-9939-1954-0063483-7
[Google Scholar]
DeVore R.A., Lorentz G.G.. Constructive approximation. Springer, Berlin; 1993.
Ditzian Z., Totik V.. Moduli of smoothness. New York: Springer; 1987.
Gadzhiev A.D.. The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P.P. Korovkin. Dokl Akad Nauk SSSR, 218. Transl Soviest Math Dokl. 1974;15:1433-1436.
[Google Scholar]
Gadzhiev A.D.. Theorems of the type of P.P. Korovkin’s theorems. Mat Zametki. 1976;20:781-786. (in Russian), Math Notes (Engl Trans) 20, 995-998
[Google Scholar]
Jackson F.H.. On q-definite integrals. Quart J Pure Appl Math. 1910;41:193-203.
[Google Scholar]
Kac V., Cheung P.. Quantum calculus, universitext. New York: Springer; 2002. https://doi.org/10.1007/978-1-4613-0071-7
Kara M.. New type Szász-Mirakyan operators. In: 14th Symposium on generating functions of special numbers and polynomials and their applications. 2024. 2024
[Google Scholar]
Korovkin P.P.. Linear operators and approximation theory. 1960.
Lenze B.. On Lipschitz-type maximal functions and their smoothness spaces. Indagationes Mathematicae (Proceedings). 1988;91:53-63. https://doi.org/10.1016/1385-7258(88)90007-8
[Google Scholar]
Lupaş A.. A q-analogue of the Bernstein operator. In seminar on numerical and statistical calculus. University of Cluj-Napoca, Cluj-Napoca. 1987;9:85-92.
[Google Scholar]
Mahmudov N.I., Kara M.. New Kantorovich–type Szász–Mirakjan operators. Bull Iran Math Soc. 2024;50 https://doi.org/10.1007/s41980-024-00913-9
[Google Scholar]
Mohiuddine, S.A., Özger, F., 2020. Approximation of functions by the Stancu variant of Bernstein-Kantorovich operators based on shape parameter $α$ . Rev Real Acad Cienc Exactas Fis Nat Ser A-Mat RACSAM 114, Article 70. https://doi.org/10.1007/s13398-020-00802-w
Nasiruzzaman M.. Approximation by Stancu-type α-Bernstein-Schurer-Kantorovich operators. J Inequal Appl. 2025;2025:48. https://doi.org/10.1186/s13660-025-03293-8
[Google Scholar]
Nasiruzzaman M., Aljohani A. F.. Approximation by $α$ -bernstein-schurer operators and shape preserving properties via $q$ -analogue. Math Meth Appl Sci. 2023;46:2354-2372. https://doi.org/10.1002/mma.8649
[Google Scholar]
Nasiruzzaman M., Alqahtani R.T., Mohiuddine S.A.. Approximation properties of a new class of beta‐type szász–mirakjan operators. J Math. 2025;2025 https://doi.org/10.1155/jom/6680828
[Google Scholar]
Nasiruzzaman M., Mohiuddine S.A., Aljedani J.. Approximation by a new Durrmeyer-type Szász-Mirakjan operators. J Nonlinear Convex Anal. 2025;26:2465-2483.
[Google Scholar]
Nasiruzzaman M., Srivastava H.M., Mohiuddine S.A.. Approximation process based on parametric generalization of Schur–Kantorovich operators and their bivariate form. Proc Natl Acad Sci India Sect A Phys Sci. 2023;93:31-41. https://doi.org/10.1007/s40010-022-00786-9
[Google Scholar]
Ozarslan A., Aktuglu H.. Local approximation properties for certain King-type operators. Filomat. 2013;27:173-181. https://doi.org/10.2298/fil1301173o
[Google Scholar]
Özger F., Aslan R., Ersoy M.. Some approximation results on a class of szász-mirakjan-kantorovich operators, including a non-negative parameter α. Numer Funct Anal Optim. 2025;46:461-484. https://doi.org/10.1080/01630563.2025.2474161
[Google Scholar]
Peetre J.. A theory of interpolation of normed spaces. Noteas de mathematica. 1968;39 Rio de Janeiro: Instituto de mathemática pura e applicada, conselho nacional de pesquisas
[Google Scholar]
Rao, N., Ayman-Mursaleen, M., Aslan, R., 2024. A note on a general sequence of $λ$ -Szász-kantorovich type operators. Comput Appl Math 43, Article 428. https://doi.org/10.1007/s40314-024-02946-6
Rao N., Farid M., Raiz M.. Approximation results: Szász–kantorovich operators enhanced by frobenius–euler–type polynomials. Axioms. 2025;14:252. https://doi.org/10.3390/axioms14040252
[Google Scholar]
Sabancigil P., Mahmudov N., Dagbasi G.. A new type of Szász–Mirakjan operators based on q-integers. J Inequal Appl. 2023;2023 https://doi.org/10.1186/s13660-023-03053-6
[Google Scholar]
Sehrawat P., Mohiuddine S.A., Kajla A., Alotaibi A.. Numerical and theoretical approximation through Riemann–Liouville–type fractional Kantorovich operators. Math Methods App Sci. 2025;48:15533-15542. https://doi.org/10.1002/mma.70033
[Google Scholar]
Shisha O., Mond B.. The degree of convergence of sequences of linear positive operators. Proc Natl Acad Sci USA. 1968;60:1196-1200. https://doi.org/10.1073/pnas.60.4.1196
[Google Scholar]
Szász O.. Generalization of S. Bernstein’s polynomials to the infinite interval. J Res Natl Bur Stand. 1950;45:239-245.
[Google Scholar]

1. Introduction, New Operators and Moments

2. Weighted Approximation

3. Rate of Convergence for Θϖ,q*

4. Lipschitz Type Approximation

5. Direct Type Approximation

6. Voronovskaja-Type Approximation Theorems

7. Numerical Comparisons and Graphical Illustrations

8. Conclusions

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[1] Alamer A., Nasiruzzaman Md.. Approximation by the Stancu variant of $λ$ -bernstein shifted knots operators associated by Bézier basis function. J King Saud Univ Sci. 2024;36:103333. https://doi.org/10.1016/j.jksus.2024.103333.
[Google Scholar]

[2] Alotaibi A., Mursaleen M.. Approximation of Jakimovski-Leviatan-beta type integral operators via q-calculus. AIMS Math. 2020;5:3019-3034. https://doi.org/10.3934/math.2020196
[Google Scholar]

[3] Ayman-Mursaleen M., Alshaban E., Nasiruzzaman M.. Shape preserving approximation properties to the family of α-Bernstein shifted knot operators. J Inequal Appl. 2025;2025:105. https://doi.org/10.1186/s13660-025-03360-0.
[Google Scholar]

[4] Ayman-Mursaleen M., Nasiruzzaman M., Rao N.. On the approximation of Szász-Jakimovski-Leviatan beta-type integral operators enhanced by Appell polynomials. Iran J Sci. 2025;49:1013-1022. https://doi.org/10.1007/s40995-025-01782-5
[Google Scholar]

[5] Barbosu D.. The voronovskaja theorem for bernstein-schurer operators. Bul Ştiinţ Univ Baia Mare, Ser B, Matematică-informatică. 2002;18
[Google Scholar]

[6] Bernstein S. N.. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun Kharkov Math Soc. 1912;13:1-2.
[Google Scholar]

[7] Berwal S., Mohiuddine S.A., Kajla A., Alotaibi A.. Approximation by Riemann-Liouville type fractional $α$ -bernstein-kantorovich operators. Math Meth Appl Sci. 2024;47:8275-8288. https://doi.org/10.1002/mma.10014
[Google Scholar]

[8] Butzer P.L.. On the extensions of Bernstein polynomials to the infinite interval. Proc Amer Math Soc. 1954;5:547-553. https://doi.org/10.1090/s0002-9939-1954-0063483-7
[Google Scholar]

[9] DeVore R.A., Lorentz G.G.. Constructive approximation. Springer, Berlin; 1993.

[10] Ditzian Z., Totik V.. Moduli of smoothness. New York: Springer; 1987.

[11] Gadzhiev A.D.. The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P.P. Korovkin. Dokl Akad Nauk SSSR, 218. Transl Soviest Math Dokl. 1974;15:1433-1436.
[Google Scholar]

[12] Gadzhiev A.D.. Theorems of the type of P.P. Korovkin’s theorems. Mat Zametki. 1976;20:781-786. (in Russian), Math Notes (Engl Trans) 20, 995-998
[Google Scholar]

[13] Jackson F.H.. On q-definite integrals. Quart J Pure Appl Math. 1910;41:193-203.
[Google Scholar]

[14] Kac V., Cheung P.. Quantum calculus, universitext. New York: Springer; 2002. https://doi.org/10.1007/978-1-4613-0071-7

[15] Kara M.. New type Szász-Mirakyan operators. In: 14th Symposium on generating functions of special numbers and polynomials and their applications. 2024. 2024
[Google Scholar]

[16] Korovkin P.P.. Linear operators and approximation theory. 1960.

[17] Lenze B.. On Lipschitz-type maximal functions and their smoothness spaces. Indagationes Mathematicae (Proceedings). 1988;91:53-63. https://doi.org/10.1016/1385-7258(88)90007-8
[Google Scholar]

[18] Lupaş A.. A q-analogue of the Bernstein operator. In seminar on numerical and statistical calculus. University of Cluj-Napoca, Cluj-Napoca. 1987;9:85-92.
[Google Scholar]

[19] Mahmudov N.I., Kara M.. New Kantorovich–type Szász–Mirakjan operators. Bull Iran Math Soc. 2024;50 https://doi.org/10.1007/s41980-024-00913-9
[Google Scholar]

[20] Mohiuddine, S.A., Özger, F., 2020. Approximation of functions by the Stancu variant of Bernstein-Kantorovich operators based on shape parameter $α$ . Rev Real Acad Cienc Exactas Fis Nat Ser A-Mat RACSAM 114, Article 70. https://doi.org/10.1007/s13398-020-00802-w

[21] Nasiruzzaman M.. Approximation by Stancu-type α-Bernstein-Schurer-Kantorovich operators. J Inequal Appl. 2025;2025:48. https://doi.org/10.1186/s13660-025-03293-8
[Google Scholar]

[22] Nasiruzzaman M., Aljohani A. F.. Approximation by $α$ -bernstein-schurer operators and shape preserving properties via $q$ -analogue. Math Meth Appl Sci. 2023;46:2354-2372. https://doi.org/10.1002/mma.8649
[Google Scholar]

[23] Nasiruzzaman M., Alqahtani R.T., Mohiuddine S.A.. Approximation properties of a new class of beta‐type szász–mirakjan operators. J Math. 2025;2025 https://doi.org/10.1155/jom/6680828
[Google Scholar]

[24] Nasiruzzaman M., Mohiuddine S.A., Aljedani J.. Approximation by a new Durrmeyer-type Szász-Mirakjan operators. J Nonlinear Convex Anal. 2025;26:2465-2483.
[Google Scholar]

[25] Nasiruzzaman M., Srivastava H.M., Mohiuddine S.A.. Approximation process based on parametric generalization of Schur–Kantorovich operators and their bivariate form. Proc Natl Acad Sci India Sect A Phys Sci. 2023;93:31-41. https://doi.org/10.1007/s40010-022-00786-9
[Google Scholar]

[26] Ozarslan A., Aktuglu H.. Local approximation properties for certain King-type operators. Filomat. 2013;27:173-181. https://doi.org/10.2298/fil1301173o
[Google Scholar]

[27] Özger F., Aslan R., Ersoy M.. Some approximation results on a class of szász-mirakjan-kantorovich operators, including a non-negative parameter α. Numer Funct Anal Optim. 2025;46:461-484. https://doi.org/10.1080/01630563.2025.2474161
[Google Scholar]

[28] Peetre J.. A theory of interpolation of normed spaces. Noteas de mathematica. 1968;39 Rio de Janeiro: Instituto de mathemática pura e applicada, conselho nacional de pesquisas
[Google Scholar]

[29] Rao, N., Ayman-Mursaleen, M., Aslan, R., 2024. A note on a general sequence of $λ$ -Szász-kantorovich type operators. Comput Appl Math 43, Article 428. https://doi.org/10.1007/s40314-024-02946-6

[30] Rao N., Farid M., Raiz M.. Approximation results: Szász–kantorovich operators enhanced by frobenius–euler–type polynomials. Axioms. 2025;14:252. https://doi.org/10.3390/axioms14040252
[Google Scholar]

[31] Sabancigil P., Mahmudov N., Dagbasi G.. A new type of Szász–Mirakjan operators based on q-integers. J Inequal Appl. 2023;2023 https://doi.org/10.1186/s13660-023-03053-6
[Google Scholar]

[32] Sehrawat P., Mohiuddine S.A., Kajla A., Alotaibi A.. Numerical and theoretical approximation through Riemann–Liouville–type fractional Kantorovich operators. Math Methods App Sci. 2025;48:15533-15542. https://doi.org/10.1002/mma.70033
[Google Scholar]

[33] Shisha O., Mond B.. The degree of convergence of sequences of linear positive operators. Proc Natl Acad Sci USA. 1968;60:1196-1200. https://doi.org/10.1073/pnas.60.4.1196
[Google Scholar]

[34] Szász O.. Generalization of S. Bernstein’s polynomials to the infinite interval. J Res Natl Bur Stand. 1950;45:239-245.
[Google Scholar]

Approximation by quantum analogue of a new type of Szasz-Mirakjan-Kantorovich operators

Abstract

Keywords

Lipschitz maximal function

Peetre’s K-functionalq-integers

Szász-mirakjan-kantorovich operators

Voronovskaja-type theorem

1. Introduction, New Operators and Moments

2. Weighted Approximation

3. Rate of Convergence for $Θ_{ϖ, q}^{*}$

4. Lipschitz Type Approximation

5. Direct Type Approximation

6. Voronovskaja-Type Approximation Theorems

7. Numerical Comparisons and Graphical Illustrations

8. Conclusions

CRediT authorship contribution statement

Declaration of competing interest

Declaration of generative AI and AI-assisted technologies in the writing process

References

Suggested read for related articles:

We value your privacy

Approximation by quantum analogue of a new type of Szasz-Mirakjan-Kantorovich operators

Abstract

Keywords

Lipschitz maximal function

Peetre’s K-functionalq-integers

Szász-mirakjan-kantorovich operators

Voronovskaja-type theorem

1. Introduction, New Operators and Moments

2. Weighted Approximation

3. Rate of Convergence for Θϖ,q*

4. Lipschitz Type Approximation

5. Direct Type Approximation

6. Voronovskaja-Type Approximation Theorems

7. Numerical Comparisons and Graphical Illustrations

8. Conclusions

CRediT authorship contribution statement

Declaration of competing interest

Declaration of generative AI and AI-assisted technologies in the writing process

References

Suggested read for related articles:

3. Rate of Convergence for $Θ_{ϖ, q}^{*}$