1 Introduction and Preliminaries

It is well known that for every continuous function g the Bernstein polynomials Bernstein (1912) converge uniformly to g(x) for all \(x\in [0,1]\). The Bernstein polynomials are defined by

$$\begin{aligned} \left( \mathcal {B}_{r,s }^{*}\;g\right) (x)=\sum _{s =0}^{r}\left( {\begin{array}{c}r\\ s \end{array}}\right) x^{s}(1-x)^{r-s }g\left( \frac{s }{r}\right) , \end{aligned}$$
(1.1)

where \(\left( {\begin{array}{c}r\\ i\end{array}}\right)\) are the binomial coefficients.

The Szász Szász (1950) and Baskakov Baskakov (1957) operators were constructed to approximate the continuous functions defined on the unbounded interval \([0,\infty ).\) The Baskakov operators are define by (see also Al-Abied et al. (2021), Cai and Aslan (2021), Cai et al. (2022), Kajla et al. (2021), Khan et al. (2022), Kilicman et al. (2020), Heshamuddin et al. (2022), Mohiuddine et al. (2020, 2021), Mohiuddine and Özger (2020), Ayman Mursaleen et al. (2022), Rao et al. (2021))

$$\begin{aligned} \left( \mathcal {B}_{r,s }\;g\right) (x)=\sum _{s =0}^{\infty }\left( {\begin{array}{c}r+s -1\\ s \end{array}}\right) \frac{x^{s}}{(1+x)^{r+s }}\;g\left( \frac{s }{r}\right) . \end{aligned}$$

It was Kantorovich (1930) who gave modification of Bernstein operators known as the Bernstein-Kantorovich operators to approximate the functions of wider classes rather than continuous functions. The Bernstein-Kantorovich operators are defined by

$$\begin{aligned} \left( \mathcal {K}_{r,s }\;g\right) (x)=(r+1)\sum _{s =0}^{r}\left( {\begin{array}{c}r\\ s \end{array}}\right) x^{s}(1-x)^{r-s }\int _{\frac{s }{r+1 }}^{\frac{s +1}{r+1 }}g(t)\mathrm {d}t \end{aligned}$$
(1.2)

for functions \(g\in L_{p}[0,1]\) (\(1\le p<\infty\)).

In order to obtain the \(L_{p}\)-approximation, Ditzian and Totik (1987) defined Kantorovich modification of Baskakov operators, known as the Baskakov–Kantorovich operators given by

$$\begin{aligned} \left( \mathcal {K}_{r,s }\;g\right) (x)=r\sum _{s =0}^{\infty }\left( {\begin{array}{c}r+s -1\\ s \end{array}}\right) \frac{x^{s}}{(1+x)^{r+s }}\;\int _{\frac{s }{r}}^{\frac{s +1}{r}}g\left( t\right) \mathrm {d}t. \end{aligned}$$
(1.3)

First, we revisit some fundamental q-calculus notations (Kac and Cheung (2002), Koornwinder (1992)).

For \(q>0\) and any positive integer \(\eta\), q-integer is defined by

$$\begin{aligned}{}[\eta ]=[\eta ]_{q}:=\left\{ \begin{array}{ll} \dfrac{1-q^{\eta }}{1-q}, &{} \qquad q\ne 1, \\ \eta , &{} \qquad q=1, \end{array} \right. \end{aligned}$$

and the q-factorial by

$$\begin{aligned}{}[\eta ]!=[\eta ]_{q}!:=\left\{ \begin{array}{l} [\eta ][\eta -1]...[1],~~~~~~\eta \ge 1; \\ ~1,~~~~~~~~~~~~~~~~~~~~~~~~~\eta =0. \end{array} \right. \end{aligned}$$

For the integers \(0\le k\le \eta\), q-binomial coefficients are defined by

$$\begin{aligned} \left[ \begin{array}{c} \eta \\ k \end{array} \right] =\left[ \begin{array}{c} \eta \\ k \end{array} \right] _{q}:=\dfrac{[\eta ]_{q}!}{[k]_{q}![\eta -k]_{q}!}\text { }(\eta \ge k\ge 0). \end{aligned}$$

For \(q>0\), we write

$$\begin{aligned} \mathbb {N}_{q}:=\left\{ [\eta ],\text { with }\eta \in \mathbb {N}\right\} , \end{aligned}$$

that is,

$$\begin{aligned} \mathbb {N}_{q}=\left\{ 0,1,1+q,1+q+q^{2},....\right\} . \end{aligned}$$

For \(q=1\), \(\mathbb {N}_{q}=\mathbb {N},\) the set of nonnegative integers.

The application of q-calculus emerged as a new area in the field of approximation theory. The first q-analogue of the well-known Bernstein polynomials was introduced by Lupaş (1987) by applying the idea of q-integers. In 1997 Phillips (1997) considered another q-analogue of the classical Bernstein polynomials. Later on, many authors studied the q-generalization of various operators and investigated their approximation properties, e.g. the q-variant of Baskakov operators is defined in Aral and Gupta (2011). Recently, the notion of q-calculus has been applied in sequence spaces and summability (see Mursaleen et al. (2022), Ayman Mursaleen and Serra-Capizzano (2022), Yaying et al. (2021)). In the area of approximation theory, the applicability of q-calculus has developed as a new topic. Lupaş (1987) used the notion of q-integers to introduce the very first q-analogue of the well-known Bernstein polynomials. Phillips (1997) proposed an alternative q-analogue of the classical Bernstein polynomials in 1997. Many authors subsequently studied the q-generalization of certain other operators and their approximation properties. The q-variant of Baskakov operators Aral and Gupta (2011) is defined as follows

$$\begin{aligned} \left( \mathcal {V}_{r,s,q}\;g\right) (x)=\sum _{s=0}^{\infty }B_{r,s,q}(x)\;g\left( \frac{[s]_{q}}{q^{s-1}[r]_{q}}\right) , \end{aligned}$$
(1.4)

where

$$\begin{aligned} B_{r,s,q}(x)=\left[ \begin{array}{c} r+s-1 \\ s \end{array} \right] _{q}\frac{x^{s}}{(1+x)_{q}^{r+s}}q^{\frac{s(s-1)}{2}}, \end{aligned}$$
(1.5)

while the q-Baskakov–Kantorovich operator Gupta and Radu (2009) are defined by

$$\begin{aligned} \left( \mathcal {T}_{r,s,q} \; g\right) (x)= [r]_q \sum _{s =0}^{\infty } q^{s-1}B_{r,s,q}(x) \int _{\frac{q[s]_q}{[r]_q}}^{\frac{[s+1]_q}{[r]_q} }g\left( q^{1-s} t\right) \mathrm {d}_qt. \end{aligned}$$
(1.6)

Lemma 1.1

For the test functions \(e_{j}=t^{j},\;j=0,1,2\), we have

$$\begin{aligned} (1)\quad \left( \mathcal {V}_{r,s,q}\;e_{0}\right) (x)= & {} 1, \\ (2)\quad \left( \mathcal {V}_{r,s,q}\;e_{1}\right) (x)= & {} x, \\ (3)\quad \left( \mathcal {V}_{r,s,q}\;e_{2}\right) (x)= & {} x^{2}+\frac{x}{ [r]_{q}}\left( 1+\frac{x}{q}\right) . \end{aligned}$$

The q-integer \([r]_{q}\), the q-factorial \([r]_{q}!\) and the q-binomial coefficient are defined by (see Kac and Cheung (2002)):

$$\begin{aligned}{}[r]_{q}&:=\left\{ \begin{array}{ll} \frac{1-q^{r}}{1-q}, &{} \hbox {if }q\in \mathbb {R}^{+}\setminus \{1\} \\ r, &{} \hbox {if }q=1, \end{array} \right. \hbox {for }r\in \mathbb {N} \hbox { and }[0]_q=0, \\ [r]_{q}!&:=\left\{ \begin{array}{ll} [r]_{q}[r-1]_{q}\cdots [1]_{q}, &{} {r\ge 1,} \\ 1, &{} {r=0,} \end{array} \right. \\ \left[ \begin{array}{c} r \\ s \end{array} \right] _{q}&:=\frac{[r]_{q}!}{[s ]_{q}![r-s ]_{q}!}, \end{aligned}$$

respectively. The q-analogue of \((1+x)^{r}\) is the polynomial

$$\begin{aligned} (1+x)_{q}^{r}:=\left\{ \begin{array}{ll} (1+x)(1+qx)\cdots (1+q^{r-1}x) &{} \quad r=1,2,3,\cdots \\ 1 &{} \quad n=0. \end{array} \right. \end{aligned}$$

The Gauss binomial formula is given by

$$\begin{aligned} (x+a)_{q}^{r}=\sum \limits _{s=0}^{r}\left[ \begin{array}{c} r \\ s \end{array} \right] _{q}q^{s(s-1)/2}a^{s}x^{r-s}. \end{aligned}$$

The q-derivative \(D_q g\) of a function g is given by

$$\begin{aligned} (D_q g)(x)= \frac{g(x)-g(qx)}{(1-q)x},\; x\ne 0, \end{aligned}$$

and \((D_q g)(0)=g^{\prime }(0),\) provided \(g^{\prime }(0)\) exists. If g is differentiable, then

$$\begin{aligned} \lim _{q \rightarrow 1}D_q g(x)= \lim _{q \rightarrow 1}\frac{g(x)-g(qx)}{(1-q)x}=\frac{d g(x) }{dx}. \end{aligned}$$

For \(r \ge 1\),

$$\begin{aligned} D_q (1+x)_q^r= [r]_q (1+qx)_q^{r-1},\; D_q \left( \frac{1}{ (1+x)_q^r} \right) = -\frac{[r]_q}{(1+x)_q^{r+1 }}, \end{aligned}$$
$$\begin{aligned} D_{q}\bigg(\frac{u(x)}{v(x)}\bigg)=\frac{v(qx)D_{q}u(x)-u(qx)D_{q}v(x)}{ v(x)v(qx)}. \end{aligned}$$

The q-Jackson definite integral is defined by

$$\begin{aligned} \int _{0}^{\infty /A}f(x)d_{q}x=(1-q)\sum _{n=-\infty }^{\infty }f\left( \frac{ q^{n}}{A}\right) \;\frac{q^{n}}{A}\qquad (A\in \mathbb {R}-\{0\}). \end{aligned}$$

2 Construction of Operators

We recall some basics of wavelets Chui (1992); Meyer (1992). The term wavelets refers to set the functions in the form

$$\begin{aligned} \Psi _{\mu ,\nu }(x)=\mu ^{-\frac{1}{2}}\Psi \left( \frac{x-\nu }{\mu } \right) \;\mu >0,\;\nu \in \mathbb {R}, \end{aligned}$$

which are formed by dilations and translations of a single function \(\Psi\) known as the basic wavelet or mother wavelet. In the Franklin-Stro\({\ddot{\text{m}}}\)berg theory the constant \(\mu\) is replaced by \(2^{i}\) and \(\nu\) is replaced by \(2^{i}\;s\) with i and s be the integers. For an arbitrary function \(g\in L_{2}(\mathbb {R}),\) these wavelets play an important role of the orthonormal basis, and the function g is given by:

$$\begin{aligned} g(x)=\sum _{-\infty }^{\infty }\sum _{-\infty }^{\infty }\gamma (i,s )\Psi _{i,s }(x), \end{aligned}$$

where

$$\begin{aligned} \gamma (i,s )=2^{\frac{i}{2}}\Psi _{i,s }(x)\int _{\mathbb {R}}f(x)\Psi (2^{i}x-s )\mathrm {d}x. \end{aligned}$$

Daubechies (1992) constructed an orthonormal basis for \(L_{2}(\mathbb {R} )\) in the form of

$$\begin{aligned} 2^{\frac{i}{2}}\Psi _{k}(x)(2^{i}x-s ) \end{aligned}$$

where k is the positive integer, i, s are the set of integers and the support of \(\Psi _{k}\) is \([0,2k+1]\). For a positive constant \(\xi ,\) if \(\Psi _{k}\) has \(\xi k\) continuous derivatives, then for any \(0\le s \le k,\;k\in \mathbb {N},\)

$$\begin{aligned} \int _{\mathbb {R}}x^{s}\Psi _{k}(x)\mathrm {d}x=0. \end{aligned}$$
(2.1)

Clearly, for \(k=0\), the system reduces to the Haar system. For any \(\Psi \in L_{\infty }(\mathbb {R}),\) we have the following conditions: (i) a finite positive \(\xi\) exits with the property \(\sup \Psi \subset [0,\xi ],\) and (ii) its first k moments vanish: For \(1\le s \le k,\;k\in \mathbb {N},\) we have \(\int _{\mathbb {R}}t^{s}\Psi (t)\mathrm {d}t=0\) and \(\int _{\mathbb {R} }\Psi (t)\mathrm {d}t=1\). Then, by using the Haar basis, the Baskakov type operators are defined by Agratini (1997):

$$\begin{aligned} \left( \mathcal {L}_{r,s }\;g\right) (x)=r\sum _{s =0}^{\infty }\left( {\begin{array}{c}r+s -1\\ s \end{array}}\right) \frac{x^{s}}{(1+x)^{r+s }}\int _{\mathbb {R}}g\left( t\right) \Psi \left( rt-s \right) \mathrm {d}t, \end{aligned}$$
(2.2)

where the operators \(\mathcal {L}_{r,s }\) are extensions of Baskakov–Kantorovich operators and by taking the \(\sup \Psi \subset [0,\xi ],\) the operators \(\mathcal {L}_{r,s }\) can be written in the following form Agratini (1997):

$$\begin{aligned} \left( \mathcal {L}_{r,s} \; g\right) (x)= \sum _{s =0}^{\infty }\left( {\begin{array}{c}r+s-1\\ s\end{array}}\right) \frac{x^s}{(1+x)^{r+s}} \int _{0}^{\xi }g\left( \frac{t+s}{r}\right) \Psi (t) \mathrm {d}t. \end{aligned}$$
(2.3)

In the present section, we construct the q-Baskakov type operators by using Daubechies’ compactly supported wavelets. Suppose \(\int _{ \mathbb {R}}x^{s}\Psi _{k}(x)\mathrm {d}_{q}x=0\) for any \(0\le s \le k,\;k\in \mathbb {N}\) and \(q>0.\)

For all \(\Psi \in L_{\infty }(\mathbb {R}),\) we suppose the following conditions in the sense of wavelets: (i) a finite positive \(\xi\) exits with the property \(\sup \Psi \subset [0,\xi ];\) (ii) and its first k moment vanish: For \(1\le s \le k,\;k\in \mathbb {N}\) we have \(\int _{\mathbb { R}}t^{s}\Psi (t)\mathrm {d}_{q}t=0\) and \(\int _{\mathbb {R}}\Psi (t)\mathrm {d} _{q}t=1\). Then for all \(1\le s \le k,\;k\in \mathbb {N}\) and \(0<q<1,\) we construct the \(q-\)analogue of Baskakov–Kantorovich type wavelets operators by:

$$\begin{aligned} \left( \mathcal {S}_{r,s,q} \; g\right) (x)= [r]_q \sum _{s =0}^{\infty } q^{s-1}B_{r,s,q}(x) \int _{\mathbb {R}}g\left( t\right) \Psi \left( q^{s-1}[r]_qt-[s]_q\right) \mathrm {d}_qt. \end{aligned}$$
(2.4)

In this way our operators \(\mathcal {S}_{r,s,q}(g;x)\) extends the q -Baskakov–Kantorovich operators defined by (1.6). For the choices \(k=0\) and \(\Psi\) Haar basis, we get the exactly q -Baskakov–Kantorovich operators \(\mathcal {T}_{r,s,q}(g;x)\) by (1.6). Moreover, for the choices \(k=0,\;q=1\) and \(\Psi\) Haar basis, we get the Baskakov–Kantorovich operators \(\mathcal {K}_{r,s,q}(g;x)\) by (1.3). By taking the \(\sup \Psi \subset [0,\xi ],\) the operators \(\mathcal {S}_{r,s,q}(g;x)\) can be rewritten in the following form:

$$\begin{aligned} \left( \mathcal {S}_{r,s,q} \; g\right) (x)= \sum _{s =0}^{\infty } B_{r,s,q}(x) \int _{0}^{\xi } g\left( \frac{t+[s]_q}{q^{s-1}[r]_q}\right) \Psi \left( t\right) \mathrm {d}_qt. \end{aligned}$$
(2.5)

It is very clear that for the choice of \(q=1\), we get classical Baskakov–Kantorovich wavelets operators \(\mathcal {L}_{r,s }\) by (2.2) and (2.3).

3 Main Results

Theorem 3.1

Let \(e_{j}=t^{j}\) for all \(0\le j\le k,\;k\in \mathbb {N}\). Then, we have

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}\;e_{j}\right) (x)=\left( \mathcal {V}_{r,s,q}\;e_{j}\right) (x), \end{aligned}$$

where \(x\in [0,\infty )\) and the operators \(\left( \mathcal {V}_{r,s,q}\;g\right) (x)\) are defined as above.

Proof

From the equality (2.5), we can write

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}\;e_{j}\right) (x)= & {} \sum _{s =0}^{\infty }B_{r,s,q}(x)\int _{\mathbb {R}}\left( \frac{t+[s ]_{q}}{q^{s -1}[r]_{q}}\right) ^{j}\Psi \left( t\right) \mathrm {d}_{q}t \\= & {} \sum _{s =0}^{\infty }B_{r,s,q}(x)\frac{1}{\left( q^{s -1}[r]_{q}\right) ^{j}}\bigg(\int _{\mathbb {R}}\sum _{i=0}^{j}\left[ \begin{array}{c} j \\ i \end{array} \right] _{q}t^{i}[s ]_{q}^{j-i}\Psi (t)\mathrm {d}_{q}t\bigg) \end{aligned}$$

From (i) and (ii), we get

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}\;e_{j}\right) (x)= & {} \sum _{s =0}^{\infty }B_{r,s,q}(x)\frac{1}{\left( q^{s-1}[r]_{q}\right) ^{j}}\int _{\mathbb {R}}[s ]_{q}^{j}\Psi (t)\mathrm {d}_{q}t \\= & {} \sum _{s =0}^{\infty }\left[ \begin{array}{c} s +r-1 \\ s \end{array} \right] _{q}\frac{x^{s}}{(1+x)_{q}^{r+s }}q^{\frac{s (s -1)}{2}}\bigg( \frac{[s ]_{q}}{\left( q^{s-1}[r]_{q}\right) }\bigg)^{j} \\= & {} \left( \mathcal {V}_{r,s,q}\;e_{j}\right) (x). \end{aligned}$$

\(\square\)

Theorem 3.2

For all \(g\in C[0,\infty )\cap L_{\infty }[0,\infty )\) and \(0<q<1,\) the following inequalities hold

$$\begin{aligned} (i)\quad ||\mathcal {S}_{r,s,q}\;g||_{\infty }\le & {} \xi ||g||_{\infty }||\Psi ||_{\infty }, \\ (ii)\quad ||\mathcal {S}_{r,s,q}^{\prime }\;g||_{\infty }\le & {} 2[r]_{q}\xi ||g||_{\infty }||\Psi ||_{\infty } \\ (iii)\quad ||\mathcal {S}_{r,s,q}^{\prime \prime }\;g||_{\infty }\le & {} 4[r]_{q}[r+1]_{q}\xi ||g||_{\infty }||\Psi ||_{\infty }. \end{aligned}$$

Proof

If we take into account

$$\begin{aligned} \int _{0}^{\xi } g\left( \frac{t+[s]_q}{q^{s-1}[r]_q}\right) \Psi \left( t\right) \mathrm {d}_qt=\mathcal {I}_g(r,s,q), \end{aligned}$$

then for all \(g \in C[0,\infty )\cap L_\infty [0,\infty )\), the following inequality holds

$$\begin{aligned} \left| \mathcal {I}_g(r,s,q) \right| \le \xi || g||_{\infty } || \Psi ||_{\infty }. \end{aligned}$$

By the equality (2.5), we can write that

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q} \; g\right) (x) \right| \le \sum _{s =0}^{\infty }\left[ \begin{array}{c} s+r-1 \\ s \end{array} \right] _{q} \frac{x^s}{(1+x)_q^{r+s}} q^{\frac{s(s-1)}{2}}\left| \mathcal {I}_g(r,s,q) \right| \le \xi || g||_{\infty } || \Psi ||_{\infty }. \end{aligned}$$

On other hand, to prove the inequalities (ii) and (iii), we use the definition of q-derivative such that

$$\begin{aligned} D_{q}\bigg(\frac{x^{s}}{(1+x)_{q}^{r+s }}\bigg)=\bigg(\frac{(1+qx)[s ]_{q}x^{-1}-q^{s +1}[r+s]_{q}}{(1+qx)}\bigg)\bigg(\frac{x^{s}}{ (1+x)_{q}^{r+s }}\bigg), \end{aligned}$$

where \(D_{q}x^{s}=[s ]_{q}x^{s -1},\) \(D_{q}(1+x)_{q}^{r+s }=q[r+s]_{q}(1+q^{2}x)_{q}^{r+s -1}.\) Applying \([r+s]_{q}=[s ]_{q}+q^{s}[r]_{q}\), then we conclude

$$\begin{aligned} B_{r,s,q}^{\prime }(x)=\frac{[r]_{q}}{x(1+qx)}\left\{\left(1+qx-q^{s +1}x\right)\frac{[s ]_{q}}{[r]_{q}}-q^{2s +1}x\right\}B_{r,s,q}(x), \end{aligned}$$

where \(x>0,\) and \(B_{r,s,q}(x)\) is defined by (1.5).

For all \(s \in \mathbb {N},\) we use the the relation

$$\begin{aligned} \frac{[s ]_{q}}{[r]_{q}}q^{1-s }B_{r,s,q}(x)=xB_{r+1,s -1,q}(x), \end{aligned}$$

then we get

$$\begin{aligned} B_{r,s,q}^{\prime }(x)=\frac{[r]_{q}}{(1+qx)}\left\{\left(1+qx-q^{s +1}x \right)q^{s-1}B_{r+1,s -1,q}(x)-q^{2s +1}B_{r,s,q}(x)\right\}. \end{aligned}$$

Therefore

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)= & {} \frac{[r]_{q}}{(1+qx)} \bigg\{\sum _{s =1}^{\infty }\bigg(1+qx-q^{s +1}x\bigg)q^{s-1}B_{r+1,s -1,q}(x)\mathcal {I}_{g}(r,s,q) \\&-\sum _{s =0}^{\infty }q^{2s +1}B_{r,s,q}(x)\mathcal {I}_{g}(r,s,q) \bigg\} \\= & {} \frac{[r]_{q}}{(1+qx)}\bigg\{\sum _{s =0}^{\infty }\bigg(1+qx-q^{s +2}x \bigg)q^{s}B_{r+1,s,q}(x)\mathcal {I}_{g}(r,s +1,q) \\&-\sum _{s =0}^{\infty }q^{2s +1}B_{r,s,q}(x)\mathcal {I}_{g}(r,s,q) \bigg\}. \end{aligned}$$

Which implies that,

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)\right|= & {} \frac{[r]_{q}}{(1+qx)}\bigg\{\sum _{s =0}^{\infty }\left| \bigg( 1+qx-q^{s +2}x\bigg)q^{s}B_{r+1,s,q}(x)\mathcal {I}_{g}(r,s +1,q)\right| \\+ & {} \left| \sum _{s =0}^{\infty }q^{2s +1}B_{r,s,q}(x)\mathcal {I}_{g}(r,s,q)\right| \bigg\} \\\le & {} \frac{[r]_{q}}{(1+qx)}\bigg\{\sum _{s =0}^{\infty }B_{r+1,s,q}(x)\left| \mathcal {I}_{g}(r,s +1,q)\right| +\left| \left( \mathcal {S}_{r,s,q}\;g\right) (x)\right| \bigg\} \\\le & {} 2[r]_{q}\xi ||g||_{\infty }||\Psi ||_{\infty }. \end{aligned}$$

In a similar way, we conclude that

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}^{\prime \prime }\;g\right) (x)\le & {} -\frac{ [r+1]_{q}}{(1+qx)}\left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x) \\+ & {} \frac{[r]_{q}[r+1]_{q}}{(1+qx)}\left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)\bigg\{\sum _{s =0}^{\infty }B_{r+2,s,q}(x)\mathcal {I} _{g}(r,s +2,q) \\&-\sum _{s =0}^{\infty }B_{r+1,s,q}(x)\mathcal {I}_{g}(r,s +1,q)\bigg\}, \end{aligned}$$

this implies that

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}^{\prime \prime }\;g\right) (x)\right|\le & {} [r+1]_{q}\left| \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)\right| +2[r]_{q}[r+1]_{q}\xi ||g||_{\infty }||\Psi ||_{\infty } \\\le & {} 4[r]_{q}[r+1]_{q}\xi ||g||_{\infty }||\Psi ||_{\infty }. \end{aligned}$$

We get the desired result. \(\square\)

Theorem 3.3

For \(g\in C^{\prime }[0,\infty )\cap L_{\infty }[0,\infty )\) and \(0<q<1,\) we get the inequality

$$\begin{aligned} ||\mathcal {S}_{r,s,q}^{\prime }\;g||_{\infty }\le \xi ||\Psi ||_{\infty } \bigg[\frac{1}{(1+qx)}||g^{\prime }||_{\infty }+\frac{xq^{r}}{ (1+qx)(1+q^{r+s }x)}\left( 1+\frac{x}{q}\right) ||g||_{\infty }\bigg] \end{aligned}$$

Proof

We know the relation

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)= & {} \frac{[r]_{q}}{(1+qx)} \bigg\{\sum _{s =0}^{\infty }\bigg(1+qx-q^{s +2}x\bigg)q^{s}B_{r+1,s,q}(x)\mathcal {I}_{g}(r,s +1,q) \\&-\sum _{s =0}^{\infty }q^{2s +1}B_{r,s,q}(x)\mathcal {I}_{g}(r,s,q) \bigg\}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)= & {} \frac{[r]_{q}}{(1+qx)} \bigg[\sum _{s =0}^{\infty }q^{2s +1}B_{r,s,q}(x)\bigg(\mathcal {I} _{g}(r,s +1,q)-\mathcal {I}_{g}(r,s,q)\bigg) \\&+\sum _{s =0}^{\infty }q^{s}\bigg(\left( 1+qx-q^{s +2}x\right) B_{r+1,s,q}(x)-q^{s +1}B_{r,s,q}(x)\bigg)\mathcal {I}_{g}(r,s +1,q)\bigg] \\\le & {} \frac{[r]_{q}}{(1+qx)}\bigg[\sum _{s =0}^{\infty }B_{r,s,q}(x) \bigg(\mathcal {I}_{g}(r,s +1,q)-\mathcal {I}_{g}(r,s,q)\bigg) \\&+\sum _{s =0}^{\infty }\bigg(B_{r+1,s,q}(x)-B_{r,s,q}(x)\bigg) \mathcal {I}_{g}(r,s +1,q)\bigg]. \end{aligned}$$

Since

$$\begin{aligned} g\left( \frac{t+[s +1]_{q}}{q^{s-1}[r]_{q}}\right) -g\left( \frac{t+[s ]_{q} }{q^{s-1}[r]_{q}}\right) =\frac{q}{[r]_{q}}\;g^{\prime }(\lambda _{r,s,q}), \end{aligned}$$

where \(\frac{t+[s ]_{q}}{q^{s-1}[r]_{q}}<\lambda _{r,s,q}<\frac{t+[s +1]_{q} }{q^{s-1}[r]_{q}}\), we have

$$\begin{aligned} \left| \mathcal {I}_{g}(r,s +1,q)-\mathcal {I}_{g}(r,s,q)\right| \le \frac{q\xi }{[r]_{q}}\;||g^{\prime }||_{\infty }||\Psi ||_{\infty }, \end{aligned}$$

and

$$\begin{aligned} B_{r+1,s,q}(x)-B_{r,s,q}(x)=\frac{q^{r}\frac{[s ]_{q}}{[r]_{q}}-q^{r+s }x}{ (1+q^{r+s }x)}B_{r,s,q}(x). \end{aligned}$$

Taking into account all these relations and applying Cauchy’s inequality, this leads to the following inequality:

$$\begin{aligned}&\left| \left( \mathcal {S}_{r,s,q}^{\prime }\;g\right) (x)\right| \\\le & {} \xi ||\Psi ||_{\infty }\bigg[\frac{1}{(1+qx)}||g^{\prime }||_{\infty } \\&+\frac{[r]_{q}q^{r}}{(1+qx)(1+q^{r+s }x)}||g||_{\infty }\sum _{s =0}^{\infty }B_{r,s,q}(x)\left| \frac{[s ]_{q}}{q^{s-1}[r]_{q}} -x\right| \bigg] \\\le & {} \xi ||\Psi ||_{\infty }\bigg[\frac{1}{(1+qx)}||g^{\prime }||_{\infty } \\&+\frac{[r]_{q}q^{r}}{(1+qx)(1+q^{r+s }x)}||g||_{\infty }\sum _{s =0}^{\infty }B_{r,s,q}(x)\left( \frac{[s ]_{q}}{q^{s-1}[r]_{q}}-x\right) ^{2}\bigg] \\= & {} \xi ||\Psi ||_{\infty }\bigg[\frac{1}{(1+qx)}||g^{\prime }||_{\infty }+ \frac{[r]_{q}q^{r}}{(1+qx)(1+q^{r+s }x)}\mu _{m,2,x}||g||_{\infty }\bigg] \end{aligned}$$

where we take \(\mu _{r,2,q}(x)=\left( \mathcal {V}_{r,s,q}\;\right) \left( \frac{[s ]_{q}}{q^{s-1}[r]_{q}}-x\right) ^{2}=\frac{x}{[r]_{q}}\left( 1+ \frac{x}{q}\right)\), the central moments of q-Baskakove operators of order two. Hence we get our desired result. \(\square\)

Theorem 3.4

For all \(g\in C^{\prime \prime }[0,\infty )\cap C_{B}[0,\infty )\) and \(0<q<1,\) we have

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}\;g\right) (x)-g(x)\right| \le \bigg(\xi ||\Psi ||_{\infty }+1\bigg)K_{2}\bigg(g,\frac{\xi ^{2}}{ 3[r]_{q}^{2}}+\frac{x}{[r]_{q}}\left( 1+\frac{x}{q}\right) \bigg), \end{aligned}$$

where \(K_{2}\) is the Peetre’s \(K-\) functional.

where \(M=\xi ||\Psi ||_{\infty }\), \(\alpha =\frac{\xi ^{2}}{3}\), \(\varphi _{r,s,q}(x)=x\left( 1+\frac{x}{q}\right)\) and \(K_{2}\) is the Peetre’s \(K-\) functional defined by (3.1).

Proof

Let \(f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )\). Then by Taylor series expansion, we write

$$\begin{aligned} f(t)=f(x)+(t-x)f^{\prime }(x)+\int _{x}^{t}(t-u)f^{\prime \prime }(u)\mathrm {d }_{q}u \end{aligned}$$

Thus in the light of Lemma 1.1 and Theorem 3.1, we can write

$$\begin{aligned}&\left| \left( \mathcal {S}_{r,s,q}\;f\right) (x)-f(x)\right| \\= & {} \left| \mathcal {S}_{r,s,q}\bigg(\bigg(\int _{x}^{t}(t-u)f^{\prime \prime }(u)\mathrm {d}_{q}u\bigg),x\bigg)\right| \\= & {} \left| \sum _{s =0}^{\infty }B_{r,s,q}(x)\int _{0}^{\xi }\bigg( \int _{x}^{\frac{t+[s ]_{q}}{q^{s-1}[r]_{q}}}\bigg(\frac{t+[s ]_{q}}{ q^{s-1}[r]_{q}}-u\bigg)|f^{\prime \prime }(u)|\mathrm {d}_{q}u\bigg)\Psi (t)\mathrm {d}_{q}t\right| \\\le & {} \sum _{s =0}^{\infty }B_{r,s,q}(x)\int _{0}^{\xi }\bigg(\left| \int _{x}^{\frac{t+[s ]_{q}}{q^{s-1}[r]_{q}}}\left| \frac{t+[s ]_{q}}{ q^{s-1}[r]_{q}}-u\right| |f^{\prime \prime }(u)|\mathrm {d} _{q}u\right| \bigg)\mathrm {d}_{q}t||\Psi ||_{\infty } \\\le & {} ||\Psi ||_{\infty }||f^{\prime \prime }||_{\infty }\sum _{s =0}^{\infty }B_{r,s,q}(x)\int _{0}^{\xi }\bigg(\frac{t+[s ]_{q}}{ q^{s-1}[r]_{q}}-x\bigg)^{2}\mathrm {d}_{q}t \\= & {} \xi ||\Psi ||_{\infty }||f^{\prime \prime }||_{\infty }\bigg(\frac{\xi ^{2}}{3[r]_{q}^{2}}+\frac{\xi }{[r]_{q}}\mu _{m,1,q}(x)+\mu _{m,2,q}(x) \bigg) \\= & {} \xi ||\Psi ||_{\infty }||f^{\prime \prime }||_{\infty }\bigg(\frac{\xi ^{2}}{3[r]_{q}^{2}}+\frac{x}{[r]_{q}}\left( 1+\frac{x}{q}\right) \bigg). \end{aligned}$$

Now taking infimum over all \(f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )\) and by (i) of Theorem 3.2, we can write

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}\;g\right) (x)-g(x)\right|\le & {} \inf _{f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )}\;\bigg\{||\mathcal {S} _{r,s,q}(g-f)||_{\infty }+||g-f||_{\infty } \\&+\left| \left( \mathcal {S}_{r,s,q}\;f\right) (x)-f(x)\right| \bigg\} \\\le & {} \inf _{f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )}\;\bigg\{\bigg( \xi ||\Psi ||_{\infty }+1\bigg)||g-f||_{\infty }+\xi ||\Psi ||_{\infty }||f^{\prime \prime }||_{\infty } \\&+\bigg(\frac{\xi ^{2}}{3[r]_{q}^{2}}+\frac{x}{[r]_{q}}\left( 1+\frac{x}{q }\right) \bigg)\bigg\} \\\le & {} \left( \xi ||\Psi ||_{\infty }+1\right) \;\;\inf _{f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )}\bigg\{||g-f||_{\infty } \\&+\bigg(\frac{\xi ^{2}}{3[r]_{q}^{2}}+\frac{x}{[r]_{q}}\left( 1+\frac{x}{q }\right) \bigg)||f^{\prime \prime }||_{\infty }\bigg\} \\= & {} \bigg(\xi ||\Psi ||_{\infty }+1\bigg)K_{2}\bigg(g,\frac{\xi ^{2}}{ 3[r]_{q}^{2}}+\frac{x}{[r]_{q}}\left( 1+\frac{x}{q}\right) \bigg), \end{aligned}$$

where \(K_{2}\) is the Peetre’s \(K-\) functional given by

$$\begin{aligned} K_{2}(g,t)=\inf _{f\in C^{2}[0,\infty )\cap C_{B}[0,\infty )}\;\bigg\{ ||g-f||_{\infty }+t||f^{\prime \prime }||_{\infty }\bigg\},\;t>0. \end{aligned}$$
(3.1)

\(\square\)

Corollary 3.5

Let \(g\in C_{B}[0,\infty )\). Then for any \(0<q<1\), the operators \(\left( \mathcal {S}_{r,s,q}\;g\right)\) defined by (2.4) satisfy

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}\;g\right) (x)-g(x)\right| \le \bigg(M+1\bigg)K_{2}\bigg(g,\frac{\varphi _{r,s,q}^{2}(x)}{[r]_{q}}+ \frac{\alpha }{[r]_{q}^{2}}\bigg), \end{aligned}$$

The Peetre’s \(K-\)functional \(K_{2}\) is equivalent to the regular modulus of smoothness, consequently for some positive \(\alpha >0\) and any \(\mu _{0}>0\), one has the relation

$$\begin{aligned} \alpha ^{-1} \omega _2(g,t) \le K_2(g,t^2) \le \alpha \omega _2 (g,t),\; g \in C_B, \; 0<t\le \mu _0, \end{aligned}$$
(3.2)

where

$$\begin{aligned} \omega _2 (g,t)=\sup _{0<h \le t} \left| \Delta _h^2 g \right| _{\infty } \end{aligned}$$

and

$$\begin{aligned} \Delta _h^2 g(x) = {\left\{ \begin{array}{ll} g(x+h)-2g(x)+g(x-h) &{} \quad \text {for }h \le x \\ 0 &{} \quad \text {otherwise }. \end{array}\right. } \end{aligned}$$

Corollary 3.6

Let \(\omega _{2}(g,t)=O(t^{\beta })\) for any \(0<\beta <2\) and \(g\in C_{B}[0,\infty )\). Then

$$\begin{aligned} \left| \left( \mathcal {S}_{r,s,q}\;g\right) (x)-g(x)\right| \le C \bigg(\frac{\varphi _{r,s,q}^{2}(x)}{[r]_{q}}+\frac{\alpha }{[r]_{q}^{2}} \bigg) \end{aligned}$$

Theorem 3.7

Let n be an integer such that \(\xi \le n.\) Then for any \(g\in L_{1}[0,\infty ),\) we have

$$\begin{aligned} \left| \left| \mathcal {S}_{r,s,q}\;g\right| \right| _{1}\le M_{1}\left| \left| g\right| \right| _{1}, \end{aligned}$$

where \(M_{1}=\frac{n[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\).

Proof

For all \(g\in L_{1}[0,\infty ),\) we can write

$$\begin{aligned} \left| \left| \mathcal {S}_{r,s,q}\;g\right| \right| _{1}\le & {} \int _{0}^{\infty /A}\sum _{s =0}^{\infty }\bigg(\int _{0}^{\xi }\left| g\left( \frac{t+[s ]_{q}}{q^{s-1}[r]_{q}}\right) \right| \;\left| \left| \Psi \right| \right| _{\infty }\mathrm {d} _{q}t\bigg)B_{r,s,q}(x)\mathrm {d}_{q}x \\\le & {} \left| \left| \Psi \right| \right| _{\infty }\sum _{s =0}^{\infty }\int _{0}^{\xi }\left| g\left( \frac{t+[s ]_{q}}{ q^{s-1}[r]_{q}}\right) \right| \mathrm {d}_{q}t\;\int _{0}^{\infty /A}B_{r,s,q}(x)\mathrm {d}_{q}x. \end{aligned}$$

Here we note that

$$\begin{aligned} \int _{0}^{\infty /A}\frac{x^{s}}{(1+x)_{q}^{s +r}}\mathrm {d}_{q}x=\frac{1}{ \mathcal {K}(A,s +1)}\mathcal {B}_{q}(s +1,r-1), \end{aligned}$$
$$\begin{aligned} \int _{0}^{\infty /A}B_{r,s,q}(x)\mathrm {d}_{q}x=\frac{1}{[r-1]_{q}}q^{s}, \end{aligned}$$

where from the q-calculus

$$\begin{aligned}&\mathcal {B}_{q}(s,r)=\frac{\Gamma _{q}(s )\Gamma _{q}(r)}{\Gamma _{q}(s +r)} =\mathcal {K}(A,s )\int _{0}^{\infty /A}\frac{x^{s -1}}{(1+x)_{q}^{s +r}} \mathrm {d}_{q}x,\\&\quad \Gamma _{q}(s )=\int _{0}^{1-q}x^{s -1}E_{q}(-qx)\mathrm {d}_{q}x=[s -1]_{q}\Gamma _{q}(s -1),\;s >0,\;\Gamma _{q}(1)=1,\\&\quad \mathcal {K}(x,t)=\frac{1}{x+1}x^{t}\left( 1+\frac{1}{x}\right) _{q}^{t}\left( 1+x\right) _{q}^{1-t}, \end{aligned}$$

in particular, for any positive integer s

$$\begin{aligned} \mathcal {K}(A,s )=q^{\frac{s (s -1)}{2}},\;\mathcal {K}(A,0)=1, \end{aligned}$$

and

$$\begin{aligned} \mathcal {K}(A,t+1)=q^{t}\mathcal {K}(A,t). \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \left| \mathcal {S}_{r,s,q}\;g\right| \right| _{1}= & {} \frac{[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\sum _{s =0}^{\infty }\int _{\frac{[s ]_{q}}{q^{s-1}[r]_{q}}}^{\frac{ \xi +[s ]_{q}}{q^{s-1}[r]_{q}}}\left| g(x)\right| \mathrm {d}_{q}x \\\le & {} \frac{[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\sum _{s =0}^{\infty }\int _{\frac{[s ]_{q}}{ q^{s-1}[r]_{q}}}^{\frac{n+[s ]_{q}}{q^{s-1}[r]_{q}}}\left| g(x)\right| \mathrm {d}_{q}x \\= & {} \frac{[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\sum _{s =0}^{\infty }\sum _{j=0}^{n-1}\int _{\frac{[s +j]_{q}}{q^{s-1}[r]_{q}}}^{\frac{[s +j+1]_{q}}{q^{s -1}[r]_{q}}}\left| g(x)\right| \mathrm {d}_{q}x \\\le & {} \frac{[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\sum _{j=0}^{n-1}\left| \left| g\right| \right| _{1} \\= & {} \frac{n[r]_{q}}{[r-1]_{q}}\left| \left| \Psi \right| \right| _{\infty }\left| \left| g\right| \right| _{1}. \end{aligned}$$

\(\square\)

4 Conclusion

Clearly, for \(k=0\) and the Haar basis \(\Psi ,\) we get exactly the q -Baskakov–Kantorovich operators \(\mathcal {T}_{r,s,q}(g;x)\) by (1.6). For the choices \(k=0,\;q=1\) and the Haar basis \(\Psi ,\) we get exactly the Baskakov–Kantorovich operators \(\mathcal {K}_{r,s,q}(g;x)\) by (1.3). It is very clear that for the choice of \(q=1\), we get classical Baskakov–Kantorovich wavelets operators \(\mathcal {L}_{r,s }\) (2.2) and (2.3) (see Agratini (1997)). Consequently, our operators \(\mathcal {S}_{r,s,q}(g;x)\) are extensions of the q -Baskakov–Kantorovich operators defined by (1.6). Based on these facts, we conclude that our constructed operators are more general.