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Introduction
The qualification of approximation for linear positive operators has a significant impact on the approximation theory. Many researchers have studied in this field [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, Bernstein operators and its generalizations have a significant status in Computer-Aided Geometric Design (CAGD) to introduce surfaces and curves and have been investigated in many papers [10, 11, 12]. Some application areas include the numerical solutions of partial differential equations, CAGD, 3D modeling.
In recent years, many articles have centered on the subject of approximating continuous functions with q-Calculus [3, 4, 5, 6, 7, 8, 9]. Initially, Lupas [3] and Philips [4] introduced the q-Bernstein operators generalization and examined approximation qualifications of these operators. Then, Derriennic introduced many qualifications of the q-Durrmeyer operators in [8].
In [13], using discrete linear approximation operators to approximate the nonlinear positive operators were introduced. The operators of max-product type were first used to describe linear operators that used maximum as the name of the sum and provided a Jackson-type error estimate with regard to the continuity modulus [14, 15, 16, 17, 18, 19, 20, 21, 22]. Since the max-product kind of approximation theory is a very rich and useful phenomena of approximating continuous functions, researchers have turned to this new field in recent years. For another approximation theory studies including univariate and bivariate type of operators can be seen via [23, 24, 25, 26, 27, 28, 29, 30, 31, 32].
In [2], the nonlinear Meyer-König and Zeller operators of max-product type were first described from the linear counterpart by using the maximum operator in place of the sum operator, as below
Z_n^{(M)}\left(f \right)(v) = {{\vee _{i = 0}^\infty {s_{n,i}}(v)f\left({{i \over {n + i}}} \right)} \over {\vee _{i = 0}^\infty {s_{n,i}}(v)}},
where
{s_{n,i}}(v) = \left({\matrix{{n + i} \cr i \cr}} \right){\left(v \right)^i}
and
\mathop \vee \nolimits_{i = 0}^\infty {s_{n,i}}(v) = \mathop {\sup}\nolimits_{i \in {\rm{\mathbb N}}} \left\{{{s_{n,i}}(v)} \right\}
.
In [17], Bede et al. proved that
Z_n^{(M)}
operators are a well defined nonlinear operator for all x ∈ [0,1] and
Z_n^{(M)}
operators have approximation conclusions and shape preserving properties. According to the usual Meyer-König and Zeller operators, the max-product kind of these operators also preserve approximation properties over the class of continuous functions. Additionally,
Z_n^{(M)}
operators are continuous for any f > 0 and preserve the quasi-convexity of f on [0,1] and the monocity in [17].
Preliminaries
In this study, we define nonlinear q-Meyer-König and Zeller operators of max-product type and give the approximation qualifications of these operators. Firstly, we indicate some basic definition and general notations. Now, let's consider the operations “∨” (maximum) and “.” (product) over the max-product algebra (ℝ+,∨,·). Assume I ⊂ ℝ is a finite or infinite interval, and set
{\rm{\mathbb A}}: = \left\{{u:I \to {{\rm{\mathbb R}}_ +};u\;continous\;and\;bounded\;on\;I} \right\}.
The max-product type of discrete approximation operators’ standard form is defined as
{L_n}(u)(t) = \mathop \vee \limits_{r = 0}^n {K_n}(t,{t_r})u({t_r}),\;\;\;{L_n}(u)(t) = \mathop \vee \limits_{r = 0}^\infty {K_n}(t,{t_r})u({t_r}),
where n ∈ ℕ, u ∈ 𝔸, Kn(.,tr) ∈ 𝔸 and tr ∈ I, for all r. The pseudo-linearity property is verified by these nonlinear positive operators as below; for u,w : I → ℝ+{L_n}(\zeta.u \vee \gamma.w)(t) = \zeta.{L_n}(u)(t) \vee \gamma.{L_n}(w)(t),
which ζ, γ ∈ ℝ+. Also, the operators of max-product kind are positive homogenous, i.e. ∀λ ≥ 0, Ln(λu) = λLn(u) (for the other details, one can see [17]).
Lemma 1
For n ∈ ℕ, let take Ln : 𝔸 → 𝔸 be a sequence of operators verifying the below circumstances:
For all u,w ∈ 𝔸, Ln (u + w) ≤ Ln (u) + Ln (w),
For all u,w ∈ 𝔸 and t ∈ I, |Ln (u)(t) − Ln (w)(t)| ≤ Ln (|u − w|)(t).
Let's suppose that the sequence Ln provides Ln (e0) = e0for all n ∈ ℕ in addition the conditions given Lemma 1 [18]. Then for all u ∈ 𝔸 and t ∈ I, we get\left| {u(t) - {L_n}\left(u \right)(t)} \right| \le \left[ {{1 \over \delta}{L_n}\left({{\eta _t}} \right)(t) + 1} \right]{\omega _1}\left({u;\delta} \right),where δ> 0, ηt(a) = |a − t| for all a,t ∈ I and ω1 (u;δ) = max {u(t) − u(s)|;t,s ∈ I,|t − s| ≤ δ}.
Let's give some basic description of the q-calculus. For the parameter q > 0 and m ∈ ℕ, one gives the q-integer [m]q as below
{[m]_q} = \left\{{\matrix{{{{1 - {q^m}} \over {1 - q}}} & {{\rm{if}}} & {q \ne 1} \cr m & {{\rm{if}}} & {q = 1} \cr}} \right.,\;\;\;{[0]_q} = 0,
and q-factorial [m]q! as
{[m]_q}! = {[1]_q}{[2]_q} \cdots {[m]_q}\quad {\rm{for}}\quad m \in {\rm{\mathbb N}}\quad {\rm{and}}\quad {[0]_q}! = 1.
For integers 0 ≤ l ≤ m, q-binomial is introduced as:
{\left[ {\matrix{m \cr l \cr}} \right]_q} = {{{{[m]}_q}!} \over {{{[l]}_q}!{{[m - l]}_q}!}}.
Finally, let q-binomial coefficient and 1 ≤ i ≤ m − 1, one get q-Pascal Rules as follows
{\left[ {\matrix{m \cr i \cr}} \right]_q} = {\left[ {\matrix{{m - 1} \cr {i - 1} \cr}} \right]_q} + {q^i}{\left[ {\matrix{{m - 1} \cr i \cr}} \right]_q},{\left[ {\matrix{m \cr i \cr}} \right]_q} = {q^{m - i}}{\left[ {\matrix{{m - 1} \cr {i - 1} \cr}} \right]_q} + {\left[ {\matrix{{m - 1} \cr i \cr}} \right]_q}.
Construction of the operators
In this section, we define nonlinear q-Meyer-König and Zeller operators of max-product type as below:
Z_n^{(M)}\left(f \right)(\varsigma ;q) = {{\mathop \vee \limits_{\zeta = 0}^\infty {t_{n,\zeta}}(\varsigma,q)f\left({{{{{[\zeta ]}_q}} \over {{{[n + \zeta ]}_q}}}} \right)} \over {\mathop \vee \limits_{\zeta = 0}^\infty {t_{n,\zeta}}(\varsigma,q)}},\;\varsigma \in [0,1),
which
{t_{n,\zeta}}(\varsigma,q) = {\left[ {\matrix{{n + \zeta} \cr \zeta \cr}} \right]_q}{\left(\varsigma \right)^\zeta}
. Here, the function f : [0,1] → ℝ+ is continuous.
The operators given in (2) are positive and continuous on the interval [0,1] for a continuous function f : [0,1] → ℝ+. Indeed, f ∈ C+([0,1]) and tn,ζ(ς,q) is positive for all [0,1], we have our operator being positive. For the nonlinearity of
Z_n^{(M)}\left(f \right)(\varsigma ;q)
for any f, h ∈ C+([0,1]), we obtain
Z_n^{(M)}\left({f + h} \right)(\varsigma ;q) \le Z_n^{(M)}\left(f \right)(\varsigma ;q) + Z_n^{(M)}\left(h \right)(\varsigma ;q)
. Also, the pseudo-linearity property is provided by these operators, and these operators are positive homogenous. Also, we handily show that
Z_n^{(M)}\left({f;q} \right)(0) - f\left(0 \right) = Z_n^{(M)}\left({f;q} \right)(1) - f\left(1 \right) = 0
for any n consider that in the indications, proofs and expressions of all approximation conclusions in fact we may assume that 0 < ς< 1. Additionally, we provide an error estimate for the operators
Z_n^{(M)}\left(f \right)(\varsigma ;q)
described by (2) with regard to the modulus of continuity.
For each ζ,γ ∈ {0,1,2,} and
x \in \left[ {{{{{[\gamma ]}_q}} \over {{{[n + \gamma ]}_q}}},{{{{[\gamma + 1]}_q}} \over {{{[n + \gamma + 1]}_q}}}} \right]
, we obtained in the following structure
\matrix{{{P_{\zeta,n,\gamma}}(\varsigma,q) = {{{t_{n,\zeta}}(\varsigma,q)\left| {{{{{[\zeta ]}_q}} \over {{{[n + \zeta ]}_q}}} - \varsigma} \right|} \over {{t_{n,\gamma}}(\varsigma,q)}},} \cr {{p_{\zeta,n,\gamma}}(\varsigma,q) = {{{t_{n,\zeta}}(\varsigma,q)} \over {{t_{n,\gamma}}(\varsigma,q)}}.} \cr}
It follows that if ζ ≥ j + 1, then
{P_{\zeta,n,\gamma}}(\varsigma,q) = {{{t_{n,\zeta}}(\varsigma,q)\left({{{{{[\zeta ]}_q}} \over {{{[n + \zeta ]}_q}}} - \varsigma} \right)} \over {{t_{n,\gamma}}(\varsigma,q)}},
and if ζ ≤ γ, then
{P_{\zeta,n,\gamma}}(\varsigma,q) = {{{t_{n,\zeta}}(\varsigma,q)\left({\varsigma - {{{{[\zeta ]}_q}} \over {{{[n + \zeta ]}_q}}}} \right)} \over {{t_{n,\gamma}}(\varsigma)}}.
Primarily, we demonstrate that 0 ≤ ζ and for fixed n ∈ ℕ, we get
0 \le {t_{n,\zeta + 1}}(\varsigma,q) \le {t_{n,\zeta}}(\varsigma,q)\quad {\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}\quad \varsigma \in \left[ {0,{{{{[\zeta + 1]}_q}} \over {{{[n + \zeta + 1]}_q}}}} \right].
Let's estimate the following inequality
0 \le {\left[ {\matrix{{n + \zeta + 1} \cr {\zeta + 1} \cr}} \right]_q}{\varsigma ^{\zeta + 1}} \le {\left[ {\matrix{{n + \zeta} \cr \zeta \cr}} \right]_q}{\varsigma ^\zeta},
after some simplifications by using q-Pascal rules given in (1), the previous inequality can be reduced to
0 \le \varsigma \le {{{{[\zeta + 1]}_q}} \over {{{[n + \zeta + 1]}_q}}}.
Therefore, if we take ζ = 0,1,⋯,n in the inequality above, we get
\matrix{{{t_{n,1}}(\varsigma,q) \le {t_{n,0}}(\varsigma,q),} \hfill & {\quad {\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}\quad \varsigma \in \left[ {0,{1 \over {{{[n + 1]}_q}}}} \right],} \hfill \cr {{t_{n,2}}(\varsigma,q) \le {t_{n,1}}(\varsigma,q),} \hfill & {\quad {\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}\quad \varsigma \in \left[ {0,{{{{[2]}_q}} \over {{{[n + 2]}_q}}}} \right],} \hfill \cr {{t_{n,3}}(\varsigma,q) \le {t_{n,2}}(\varsigma,q),} \hfill & {\quad {\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}\quad \varsigma \in \left[ {0,{{{{[3]}_q}} \over {{{[n + 3]}_q}}}} \right],} \hfill \cr}
and
{t_{n,\zeta + 1}}(\varsigma,q) \le {t_{n,\zeta}}(\varsigma,q),\quad {{\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}}\quad \varsigma \in \left[ {0,{{{{[\zeta + 1]}_q}} \over {{{[n + \zeta + 1]}_q}}}} \right],
and so on. The result of all these inequalities is
\matrix{{if\quad} \hfill & {\varsigma \in \left[ {0,{1 \over {{{[n + 1]}_q}}}} \right]then\quad {t_{n,\zeta}}(\varsigma,q) \le {t_{n,0}}(\varsigma,q),\;forall\;\zeta = 0,1, \cdots,n;} \hfill \cr {{\rm{if}}\quad} \hfill & {\varsigma \in \left[ {{1 \over {{{[n + 1]}_q}}},{{{{[2]}_q}} \over {{{[n + 2]}_q}}}} \right]{\rm{then}}\quad {t_{n,\zeta}}(\varsigma,q) \le {t_{n,1}}(\varsigma,q),\;{\rm{for}}\,{\rm{all}}\;\zeta = 0,1, \cdots,n;} \hfill \cr {{\rm{if}}\quad} \hfill & {\varsigma \in \left[ {{{{{[2]}_q}} \over {{{[n + 2]}_q}}},{{{{[3]}_q}} \over {{{[n + 3]}_q}}}} \right]{\rm{then}}\quad {t_{n,\zeta}}(\varsigma,q) \le {t_{n,2}}(\varsigma,q),\;{\rm{for}}\,{\rm{all}}\;\zeta = 0,1, \cdots,n;} \hfill \cr}
and
{\rm{if}}\quad \varsigma \in \left[ {{{{{[\gamma ]}_q}} \over {{{[n + \gamma ]}_q}}},{{{{[\gamma + 1]}_q}} \over {{{[n + \gamma + 1]}_q}}}} \right]\;{\rm{then}}\quad {t_{n,\zeta}}(\varsigma,q) \le {t_{n,\gamma}}(\varsigma,q),\;{\rm{for}}\,{\rm{all}}\;\zeta = 0,1, \cdots,n,
which completes the proof.
Approximation degree of
Z_n^{(M)}(f)(x;q)
The Shisha-Mond theorem, which is applicable to nonlinear max-product kind operators and is presented in [13], is used in this section to provide an error estimate for the operators
Z_n^{(M)}(f)(\varsigma ;q)
which is defined in (2), with regard to the modulus of continuity.
Theorem 6
Let's q ∈ (0,1) and the function f is a bounded and continuous on [0,1]. Then, we have\left| {Z_n^{(M)}(f)(\varsigma ;q) - f(\varsigma)} \right| \le 18{\omega _1}\left({f;{{(1 - \varsigma)\sqrt \varsigma} \over {\sqrt {{{[n]}_q}}}}} \right),where n ≥ 4, ς ∈ [0,1] and ω1 ( f ;δ) = sup{|f (ς) − f (ζ)|;ς, ζ∈ [0,1],|ς − ζ| ≤ δ}.
Proof
Since the max-product Meyer-König and Zeller operators based on q-integers supply the conditions in Corollary 2 and we get the following
\left| {Z_n^{(M)}(f)(\varsigma ;q) - f(\varsigma)} \right| \le \left({1 + {1 \over {{\delta _n}}}Z_n^{(M)}({\eta _\varsigma},\varsigma ;q)} \right){\omega _1}\left({f;{\delta _n}} \right),
where ης (t) = |t − ς|. Estimation of the following term is enough for the proof of lemma:
Z_n^{(M)}\left({{\eta _\varsigma},\varsigma ;q} \right) = {{\vee _{\zeta = 0}^\infty {t_{n,\zeta}}(\varsigma,q)\left| {{{{{[\zeta ]}_q}} \over {{{[n + \zeta ]}_q}}} - \varsigma} \right|} \over {\vee _{\zeta = 0}^\infty {t_{n,\gamma}}(\varsigma,q)}}.
Let's assume that
\varsigma \in \left[ {{{{{[\gamma ]}_q}} \over {{{[n + \gamma ]}_q}}},{{{\zeta _n}{{[\gamma + 1]}_q}} \over {{{[n + \gamma + 1]}_q}}}} \right]
, where γ ∈ {0,1,⋯} is fixed and arbitrary. From Lemma 5, we get
Z_n^{(M)}\left({{\eta _\varsigma},\varsigma ;q} \right) = \mathop \vee \limits_{\zeta = 0}^\infty {P_{\zeta,n,\gamma}}(\varsigma,q).
Firstly, for γ = 0 we obtain
Z_n^{(M)}\left({{\eta _\varsigma},\varsigma ;q} \right) \le {{[2](1 - \varsigma)\sqrt \varsigma} \over {\sqrt {{{[n]}_q}}}}
for all
\varsigma \in \left[ {0,{1 \over {{{[n + 1]}_q}}}} \right]
, so we can claim that γ = {1,2,⋯}.
Indeed, for each Pζ,n,γ (ς, q) we determine an upper estimate, for γ = 0,
\varsigma \in \left[ {0,{1 \over {{{[n + 1]}_q}}}} \right]
and ζ ∈ {0,1,⋯,n}. Besides, Lemma 4(i) which indicates that for ζ ≥ 2 one gets Pζ,n,0(ς, q) ≥ Pζ+1,n,0(ς, q) which means that
Z_n^{(M)}\left({{\eta _\varsigma},\varsigma ;q} \right) = ma{x_{\zeta \in \left\{{0,1,2} \right\}}}\left\{{{P_{\zeta,n,0}}(\varsigma,q)} \right\},\varsigma \in \left[ {0,{1 \over {{{[n + 1]}_q}}}} \right]
. For ζ= 0, we have
\matrix{{{P_{\zeta,n,0}}(\varsigma,q)} \hfill & {= \varsigma = \sqrt \varsigma \cdot \sqrt \varsigma \le \sqrt \varsigma \cdot {1 \over {\sqrt {{{[n + 1]}_q}}}} \le \sqrt \varsigma \cdot {1 \over {\sqrt {{{[n]}_q}}}}} \hfill \cr {} \hfill & {\le (1 - \varsigma)\sqrt \varsigma \cdot {1 \over {\sqrt {{{[n]}_q}}}}{1 \over {1 - \varsigma}} \le (1 - \varsigma)\sqrt \varsigma \cdot {1 \over {\sqrt {{{[n]}_q}}}}{{{{[n + 1]}_q}} \over {{{[n]}_q}}}} \hfill \cr {} \hfill & {\le {{[2](1 - \varsigma)\sqrt \varsigma} \over {\sqrt {{{[n]}_q}}}}.} \hfill \cr}
For ζ = 1, we have
{P_{1,n,0}}(\varsigma,q) = n + {11_q}\varsigma \left| {{1 \over {{{[n + 1]}_q}}} - \varsigma} \right| \le \varsigma \le {{[2](1 - \varsigma)\sqrt \varsigma} \over {\sqrt {{{[n]}_q}}}}.
For ζ = 2, we obtain
\matrix{{{P_{2,n,0}}(\varsigma,q)} \hfill & {= n + {{22}_q}{\varsigma ^2}\left| {{2 \over {{{[n + 2]}_q}}} - \varsigma} \right| \le {{{{[n + 1]}_q}{{[n + 2]}_q}} \over 2}{\varsigma ^2}{2 \over {{{[n + 2]}_q}}}} \hfill \cr {} \hfill & {\le {{[n + 1]}_q} \cdot \varsigma \cdot {1 \over {{{[n + 1]}_q}}} \le \varsigma \le {{[2](1 - \varsigma)\sqrt \varsigma} \over {\sqrt {{{[n]}_q}}}}.} \hfill \cr}
Now, let's take γ = 1,2,⋯ is fixed,
\varsigma \in \left[ {{{{{[\gamma ]}_q}} \over {{{[n + \gamma ]}_q}}},{{{{[\gamma + 1]}_q}} \over {{{[n + \gamma + 1]}_q}}}} \right]
and ζ = 0,1,⋯, then we get an upper estimate for each Pζ,n,γ (ς,q). Under these circumstances, the proof will be separated into the following cases:
From
\gamma \ge {1 \over {n + 1}}
and n ≥ 4, we immediately get
\sqrt {{{[n]}_q}x} + 1 + {q^\gamma} \le (1 + \sqrt 5)\sqrt {{{[n]}_q}\varsigma}
and it follows that
{P_{\widetilde \zeta - 1,n,\gamma}}(\varsigma ;q) \le {{(1 + \sqrt 5)\sqrt {{{[n]}_q}\varsigma} (1 - \varsigma)} \over {{{[n]}_q} - 1 - {q^{n + \gamma}} + (1 + {q^{n + \gamma}})\varsigma - \sqrt {{{[n]}_q}\varsigma}}}.
Let
h(\varsigma) = {[n]_q} - 1 - {q^{n + \gamma}} + (1 + {q^{n + \gamma}})\varsigma - \sqrt {{{[n]}_q}\varsigma}
, ς ≥ 0. It is easy to show that h has a global minimum in
{\varsigma _0} = {n \over {4{{(1 + {q^{n + \gamma}})}^2}}}
. It means that
h(\varsigma) \ge h({n \over {4{{(1 + {q^{n + \gamma}})}^2}}}) = {{(3 + 4{q^{n + \gamma}}){{[n]}_q} - 4{{(1 + {q^{n + \gamma}})}^2}} \over {4{{(1 + {q^{n + \gamma}})}^2}}}
. Therefore, we obtain
{P_{\widetilde \zeta - 1,n,\gamma}}(\varsigma ;q) \le {{4(1 + {q^{n + \gamma}})(1 + \sqrt 5)} \over 3}{{(1 - \varsigma)\sqrt \varsigma} \over {\sqrt n}}.Lemma 4 (ii) gives us to
{P_{\widetilde \zeta - 1,n,\gamma}}(\varsigma ;q) \ge {P_{\widetilde \zeta - 2,n,\gamma}}(\varsigma ;q) \ge \cdots \ge {P_{0,n,\gamma}}(\varsigma ;q)
. Hence, we have
{P_{\zeta,n,\gamma}}(x;q) \le {{4(1 + {q^{n + \gamma}})(1 + \sqrt 5)} \over 3}{{(1 - \varsigma)\sqrt \varsigma} \over {\sqrt n}},
for any ζ ≤ γ. Collecting all the estimates obtained above, we have
Z_n^{(M)}\left({{\eta _\varsigma},\varsigma ;q} \right) \le {{4(1 + {q^{n + \gamma}})(1 + \sqrt 5)} \over 3}{{(1 - \varsigma)\sqrt \varsigma} \over {\sqrt n}}
for all ς ∈ [0,1] and choosing
{\delta _n} = {{4(1 + {q^{n + \gamma}})(1 + \sqrt 5)} \over 3}{{(1 - \varsigma)\sqrt \varsigma} \over {\sqrt n}}
in the inequality given in (6), we get the proof of the theorem.
Conclusion
In this paper, nonlinear max-product type q-Meyer-König and Zeller operators have been introduced. Additionally, the modulus of continuity has been used to investigate the degree of approximation and the rate of convergence of the operators. As a result, the max-product type q-Meyer-König and Zeller operators approximated better than the classical linear q-Meyer-König and Zeller operators. In future studies, the shape preservation properties of these operators may be studied, and comparable research may be incorporated into more practical operator frameworks.
Declarations
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper.
Author's contributions
E.A.-Conceptualization, Methodology, Formal Analysis, Writing-Review and Editing. Ö.Ö.G.-Formal Analysis, Validation, Writing-Original Draft. S.K.S.-Formal Analysis, Validation, Resources. All authors read and approved the final submitted version of this manuscript.
Funding
No funding was received to assist with the preparation of this manuscript.
Acknowledgement
Thank you so much to Editor-in-Chief Prof. Dr. Haci Mehmet Baskonus for his guidelines and opinions throughout this process.
Data availability statement
All data that support the findings of this study are included within the article.
Using of AI tools
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.