1 Introduction

The study of the linear methods of approximation, which are given by sequences of linear positive operators, became a strongly ingrained part of Approximation Theory. Due to their special properties, over time, these approximation processes have been proved very useful in approximating various signals. Our paper will bring into light two sequences of integral operators known in the literature as Picard \((P_n,\ n\ge 1)\), respectively Gauss \((W_n,\ n\ge 1)\) operators. Their classical forms are described by the following formulas

$$\begin{aligned} (P_n f)(x)= & {} \displaystyle \frac{n}{2}\int _\mathbb {R}f(x+t)e^{-n|t|}dt,\quad x\in \mathbb {R},\end{aligned}$$
(1)
$$\begin{aligned} (W_n f)(x)= & {} \sqrt{\displaystyle \frac{n}{\pi }}\int _\mathbb {R}f(x+t)e^{-nt^2}dt,\quad x\in \mathbb {R}, \end{aligned}$$
(2)

where the function f is selected such that the integrals are finite.

These operators have been investigated in several works. We mention the monograph [2] and the references therein. By using probabilistic schemes, Gauss-Weierstrass operators are reconstructed in [1, Section 5.2.9]. For each \(n\in \mathbb {N}\), both operators are linear and positive. Moreover,

$$\begin{aligned} (P_n e_0)(x)=(W_n e_0)(x)=1,\quad x\in \mathbb {R}, \end{aligned}$$
(3)

where \(e_0\) represents the constant function on \(\mathbb {R}\) of constant value 1.

Throughout the paper \(e_j\) stands for monomial of j-degree, \(e_j(t)=t^j\), \(t\in \mathbb {R}\).

We amend the classical operators defined by (1) and (2), such that they will be able to reproduce not only \(e_0\) but also a certain exponential function. The proposed generalizations of the above operators are defined as follows:

$$\begin{aligned} (P_n^* f)(x)=\displaystyle \frac{\sqrt{n}}{2}\int _\mathbb {R}f(\alpha _n(x)+t)e^{-\sqrt{n}|t|}dt,\quad n\ge n_a,\quad x\in \mathbb {R}, \end{aligned}$$
(4)

and

$$\begin{aligned} (W_n^* f)(x)=\sqrt{\displaystyle \frac{n}{\pi }}\int _\mathbb {R}f(\beta _n(x)+t)e^{-nt^2}dt,\quad n\in \mathbb {N},\quad x\in \mathbb {R}, \end{aligned}$$
(5)

where

$$\begin{aligned} \alpha _n(x)= & {} x-\displaystyle \frac{1}{2a}\log \left( \displaystyle \frac{n}{n-4a^2}\right) ,\quad n\ge n_a,\end{aligned}$$
(6)
$$\begin{aligned} \beta _n(x)= & {} x-\displaystyle \frac{a}{2n},\quad n\ge 1, \end{aligned}$$
(7)

and \(a>0\). In the above \(n_a=[4a^2]+1\), \([\cdot ]\) indicating the integer part function or the so-called floor function. The domains of the sequences \(P^*=(P_n^*)_{n\ge n_a}\), \(W^*=(W_n^*)_{n\ge 1}\) are denoted by \(\mathcal {F}(P^*)\) and \(\mathcal {F}(W^*)\), respectively.

Also, we introduce the function \(\varphi _a\) given by formula

$$\begin{aligned} \varphi _a(x)=e^{2ax},\quad x\in \mathbb {R}. \end{aligned}$$
(8)

For a tending to zero, the original versions of the operators are reobtained.

Relating to operators defined by (4) and (5) we study their approximation properties in polynomial weighted spaces including Voronovskaja-type formulas. The final section is devoted to bringing to light properties of these operators that spring from the notion of generalized convexity.

2 Preliminary results

At first we calculate all the moments of both classes of operators.

Lemma 1

Let \(P_n^*\), \(n\ge n_a\), be the operators given at (4) and (6). For each integer p, \(p\ge 0\), we have

$$\begin{aligned} (P_n^* e_p)(x)=\sum _{s=0}^{[p/2]}\displaystyle \frac{(2s)!}{n^s}\left( {\begin{array}{c}p\\ 2s\end{array}}\right) \alpha _n^{p-2s}(x),\quad x\in \mathbb {R}. \end{aligned}$$
(9)

Proof

Setting \(I_k=\int _\mathbb {R}t^k e^{-\sqrt{n}|t|}dt\), for k odd we deduce \(I_k=0\). For k even, \(k=2s\), we obtain

$$\begin{aligned} I_{2s}=2\displaystyle \frac{(2s)!}{\left( \sqrt{n}\right) ^{2s+1}},\quad 0\le 2s\le p. \end{aligned}$$
(10)

Further,

$$\begin{aligned} (P_n^* e_p)(x)&=\displaystyle \frac{\sqrt{n}}{2}\int _\mathbb {R}\sum _{k=0}^p \left( {\begin{array}{c}p\\ k\end{array}}\right) \alpha _n^{p-k}(x)t^k e^{-\sqrt{n}|t|}dt\\&=\sqrt{n}\sum _{s=0}^{[p/2]}\left( {\begin{array}{c}p\\ 2s\end{array}}\right) \alpha _n^{p-2s}(x) \displaystyle \frac{(2s)!}{\left( \sqrt{n}\right) ^{2s+1}}, \end{aligned}$$

and thus we arrive at relation (9). \(\square \)

As particular cases we obtain

$$\begin{aligned} P_n^* e_0=e_0,\quad P_n^* e_1=\alpha _n,\quad P_n^* e_2=\alpha _n^2 +\displaystyle \frac{2}{n}. \end{aligned}$$
(11)

Lemma 2

Let \(W_n^*\), \(n\ge 1\), be the operators given at (5) and (7). The moments of these operators have the following values

$$\begin{aligned}&(W_n^* e_0)(x)=1,\quad (W_n^* e_1)(x)=\beta _n(x),\\&(W_n^* e_p)(x)=\beta _n^p(x)+\displaystyle \sum _{s=1}^{[p/2]}\displaystyle \frac{(2s-1)!!}{(2n)^s}\left( {\begin{array}{c}p\\ 2s\end{array}}\right) \beta _n^{p-2s}(x),\quad p\ge 2,\nonumber \end{aligned}$$
(12)

where \(x\in \mathbb {R}\).

Proof

For \(p=0\) and \(p=1\) identities are established immediately. Let \(p\ge 2\) be fixed. Setting \(J_k=\int _\mathbb {R}t^k e^{-nt^2}dt\), for k odd we get \(J_k=0\). For k even, \(k=2s\), we have

$$\begin{aligned} J_{2s}=\displaystyle \frac{(2s-1)!!}{(2n)^s}J_0 \quad \text{ and }\quad J_0=\sqrt{\displaystyle \frac{\pi }{n}}, \end{aligned}$$
(13)

where \(s\in \mathbb {N}\), \(1\le 2s\le p\).

Further we can write

$$\begin{aligned} (W_n^* e_p)(x)=\sqrt{\displaystyle \frac{n}{\pi }}\left( \beta _n^p(x)J_0 +\sum _{s=1}^{[p/2]}\left( {\begin{array}{c}p\\ 2s\end{array}}\right) \beta _n^{p-2s}(x)J_{2s}\right) \end{aligned}$$

which leads us to the desired relation. \(\square \)

As particular case we obtain

$$\begin{aligned} W_n^* e_2=\beta _n^2+\displaystyle \frac{1}{2n}. \end{aligned}$$
(14)

Denoting by \(\mu _{r}(L_n;\cdot )\) the central moment of r order of the operator \(L_n\), this means \(\mu _{r}(L_n,x)=L_n((\cdot -x)^r;x)\), \(r=0,1,2,\ldots \), we can enunciate

Lemma 3

Let \(P_n^*\) and \(W_n^*\) be the operators defined by (4) and (5), respectively.

  1. (i)

    \(\mu _{0}(P_n^*;x)=1\), \(\mu _{1}(P_n^*;x)=\alpha _n(x)-x\), \(\mu _{2}(P_n^*;x)=(\alpha _n(x)-x)^2+\displaystyle \frac{2}{n}\), \(n\ge n_a\),

  2. (ii)

    \(\mu _{0}(W_n^*;x)=1\), \(\mu _{1}(W_n^*;x)=\beta _n(x)-x\), \(\mu _{2}(W_n^*;x)=(\beta _n(x)-x)^2+\displaystyle \frac{1}{2n}\), \(n\ge 1\),

where \(\alpha _n\) and \(\beta _n\) are defined by (6) and (7), respectively.

Proof

All the above identities are implied by relations (11), (12) and (14). \(\square \)

Lemma 4

Let \(P_n^*\) and \(W_n^*\) be the operators defined by (4) and (5), respectively. The following relations take place:

  1. (i)

    \(\mu _{6}(P_n^*;x)=(\alpha _n(x)-x)^6+\displaystyle \frac{30}{n}(\alpha _n(x)-x)^4 +\displaystyle \frac{360}{n^2}(\alpha _n(x)-x)^2+\displaystyle \frac{720}{n^3}\),

  2. (ii)

    \(\mu _{6}(W_n^*;x)=(\beta _n(x)-x)^6+\displaystyle \frac{15}{2n}(\beta _n(x)-x)^4 +\displaystyle \frac{45}{4n^2}(\beta _n(x)-x)^2+\displaystyle \frac{15}{8n^3}\),

  3. (iii)

    \(\lim \limits _{n\rightarrow \infty }\displaystyle \frac{\mu _{6}(P_n^*;x)}{\mu _{2}(P_n^*;x)}=0\), \(\lim \limits _{n\rightarrow \infty }\displaystyle \frac{\mu _{6}(W_n^*;x)}{\mu _{2}(W_n^*;x)}=0\).

Proof

  1. (i)
    $$\begin{aligned} \mu _{6}(P_n^*;x)= & {} \displaystyle \frac{\sqrt{n}}{2}\displaystyle \int _\mathbb {R}((\alpha _n(x)-x)+t)^6 e^{-\sqrt{n}|t|}dt\\= & {} (\alpha _n(x)-x)^6(P_n^* e_0)(x)+\displaystyle \frac{\sqrt{n}}{2}(15(\alpha _n(x)-x)^4I_2\\&+15(\alpha _n(x)-x)^2 I_4+I_6), \end{aligned}$$

    where \(I_{2s}\), \(s\in \mathbb {N}\), are indicated at (10).

  2. (ii)

    \(\mu _{6}(W_n^*;x)\) is computed in the same manner taking into account the relation (13).

  3. (ii)

    For the sake of simplicity, we denote \(\alpha _n(x)-x=a_n\), where

    $$\begin{aligned} a_n=-\displaystyle \frac{1}{2a}\log \left( \displaystyle \frac{n}{n-4a^2}\right) ,\quad n\ge n_a. \end{aligned}$$

    We get

    $$\begin{aligned} \displaystyle \frac{\mu _{6}(P_n^*;x)}{\mu _{2}(P_n^*;x)} =\displaystyle \frac{a_n^6+30n^{-1}a_n^4+360n^{-2}a_n^2+720n^{-3}}{a_n^2+2n^{-1}},\quad n\ge n_a. \end{aligned}$$

    Since \(\lim \nolimits _{n\rightarrow \infty }a_n=0\) and \(\lim \nolimits _{n\rightarrow \infty }na_n^2=0\), the shown identity occurs. Similarly we proceed to second limit.

\(\square \)

3 Weighted approximation

For proceed further, we need a result due to Gadzhiev [3]. The author considered a continuous and strictly increasing function \(\varphi \) defined on \(\mathbb {R}\) and \(\rho (x)=1+\varphi ^2(x)\) such that \(\lim \nolimits _{x\rightarrow \pm \infty }\rho (x)=\infty \).

Set

$$\begin{aligned} B_\rho (\mathbb {R})=\{f:\mathbb {R}\rightarrow \mathbb {R}:\ |f(x)|\le M_f \rho (x)\}, \end{aligned}$$

where \(M_f\) is a constant depending on f,

$$\begin{aligned} C_\rho (\mathbb {R})= & {} B_\rho (\mathbb {R})\cap C(\mathbb {R}),\\ C_\rho ^*(\mathbb {R})= & {} \left\{ f\in C_\rho (\mathbb {R}):\ \lim \limits _{|x|\rightarrow \infty }\displaystyle \frac{f(x)}{\rho (x)} \text{ exists } \text{ and } \text{ it } \text{ is } \text{ finite }\right\} . \end{aligned}$$

If the space \(B_\rho (\mathbb {R})\) is endowed with the norm \(\Vert \cdot \Vert _\rho \) defined by

$$\begin{aligned} \Vert f\Vert _\rho =\sup _{x\in \mathbb {R}}\displaystyle \frac{|f(x)|}{\rho (x)}, \end{aligned}$$
(15)

then the same norm is considered in the other two spaces defined above.

Theorem 1

[3, Theorem 2] Let \((A_n)_{n\ge 1}\) be a sequence of linear positive operators mapping \(C_\rho (\mathbb {R})\) into \(B_\rho (\mathbb {R})\). If

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert A_n \varphi ^\nu -\varphi ^\nu \Vert _\rho =0,\quad \nu =0,1,2, \end{aligned}$$
(16)

then, for any \(f\in C_\rho ^*(\mathbb {R})\) we have

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert A_n f-f\Vert _\rho =0. \end{aligned}$$
(17)

Our aim is to study the approximation property of \(P_n^*\) and \(W_n^*\) operators on some weighted spaces. We consider a weight commonly used in defining spaces of function with polynomial growth. We choose

$$\begin{aligned} \varphi (x)=x \quad \text{ and }\quad \rho (x)=1+x^2,\ x\in \mathbb {R}. \end{aligned}$$
(18)

This choice meets the conditions specified formerly.

Theorem 2

Let \(P_n^*\), \(n\ge n_a\), be the operators defined by (4) and (6). For each \(f\in C_\rho ^*(\mathbb {R})\) the following relation

$$\begin{aligned} \lim \limits _{n\rightarrow \mathbb {R}}\Vert P_n^* f-f\Vert _\rho =0 \end{aligned}$$
(19)

holds, where \(\rho \) is stated at (18).

Proof

Based on (15), for linear positive operators \(P_n^*\) defined on \(C_\rho (\mathbb {R})\), we have

$$\begin{aligned} |(P_n ^* f)(x)|\le \Vert f\Vert _\rho (P_n^* \rho )(x),\ x\in \mathbb {R}. \end{aligned}$$

Lemma 1 guarantees that our operators map \(C_\rho (\mathbb {R})\) into \(C_\rho (\mathbb {R})\subset B_\rho (\mathbb {R})\).

We check the three conditions of relation (16).

Since \(P_n^* e_0=e_0\), for \(\nu =0\) the condition is fulfilled.

For \(\nu =1\), on the basis of (11), we have

$$\begin{aligned} \Vert P_n^* e_1-e_1\Vert _\rho&=\sup _{x\in \mathbb {R}}\displaystyle \frac{|(P_n^* e_1)(x)-x|}{1+x^2}\\&=\sup _{x\in \mathbb {R}}\displaystyle \frac{\left| \frac{1}{2a}\log \frac{n}{n-4a^2}\right| }{1+x^2} \le \displaystyle \frac{1}{2a}\log \displaystyle \frac{n}{n-4a^2}. \end{aligned}$$

Consequently, \(\lim \nolimits _{n\rightarrow \infty }\Vert P_n^* e_1-e_1\Vert _\rho =0\).

Finally, for \(\nu =2\), on the basis of (11), we get

$$\begin{aligned} \Vert P_n^* e_2-e_2\Vert _\rho&=\sup _{x\in \mathbb {R}}\displaystyle \frac{\left| \left( x-\frac{1}{2a}\log \frac{n}{n-4a^2}\right) ^2+\frac{2}{n}-x^2\right| }{1+x^2}\\&=\sup _{x\in \mathbb {R}}\displaystyle \frac{\left| -\frac{x}{a}\log \frac{n}{n-4a^2}+\frac{1}{4a^2}\log ^2 \frac{n}{n-4a^2} +\frac{2}{n}\right| }{1+x^2}\\&\le \displaystyle \frac{1}{a}\log \displaystyle \frac{n}{n-4a^2}+\displaystyle \frac{1}{4a^2}\log ^2 \displaystyle \frac{n}{n-4a^2}+\displaystyle \frac{2}{n}. \end{aligned}$$

Again, \(\lim \nolimits _{n\rightarrow \infty }\Vert P_n^* e_2-e_2\Vert _\rho =0\).

In view of Theorem 1, relation (19) follows. \(\square \)

Following the same route and using relations (12) and (14) we can formulate

Theorem 3

Let \(W_n^*\), \(n\ge 1\), be the operators defined by (5) and (7). For each \(f\in C_\rho ^*(\mathbb {R})\) the following relation

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert W_n^* f-f\Vert _\rho =0 \end{aligned}$$

holds, where \(\rho \) is stated at (18).

4 Quantitative Voronovskaja formulas

In this section we establish the asymptotic behavior for our operators.

In order to measure the rate of convergence on \(C_\rho ^*(\mathbb {R})\) we use a weighted modulus of smoothness. Following [4] we consider

$$\begin{aligned} \Omega (f,\delta )=\sup _{\begin{array}{c} x\in \mathbb {R}\\ |h|\le \delta \end{array}} \displaystyle \frac{|f(x+h)-f(x)|}{(1+h^2)(1+x^2)},\quad f\in C_\rho ^*(\mathbb {R}). \end{aligned}$$
(20)

Among its properties we recall the following: \(\lim \nolimits _{\delta \rightarrow 0^+}\Omega (f,\delta )=0\), \(\Omega (f,\cdot )\) is an increasing function and for each \(\lambda >0\)

$$\begin{aligned} \Omega (f,\lambda \delta )\le 2(1+\lambda )(1+\delta ^2)\Omega (f,\delta ). \end{aligned}$$
(21)

Lemma 5

For each \(f\in C_\rho ^*(\mathbb {R})\) let \(\Omega (f,\cdot )\) be defined by (20). For any \((t,x)\in \mathbb {R}\times \mathbb {R}\) and any \(\delta >0\) the following relation

$$\begin{aligned} |f(t)-f(x)|\le 4\left( 1+\displaystyle \frac{(t-x)^4}{\delta ^4}\right) (1+\delta ^2)^2(1+x^2)\Omega (f,\delta ). \end{aligned}$$
(22)

holds.

Proof

Let \(\delta >0\) be arbitrary fixed and \((x,t)\in \mathbb {R}\times \mathbb {R}\). Set \(t-x=h\).

$$\begin{aligned} \displaystyle \frac{|f(t)-f(x)|}{(1+(t-x)^2)(1+x^2)}&\le \sup _{\begin{array}{c} x\in \mathbb {R}\\ |h|=|t-x| \end{array}}\displaystyle \frac{|f(x+h)-f(x)|}{(1+h^2)(1+x^2)}\\&\le \sup _{\begin{array}{c} x\in \mathbb {R}\\ |\widetilde{h}|\le |t-x| \end{array}} \displaystyle \frac{|f(x+\widetilde{h})-f(x)|}{(1+\widetilde{h}^2)(1+x^2)} =\Omega (f,|t-x|)\\&\le 2\left( 1+\displaystyle \frac{|t-x|}{\delta }\right) (1+\delta ^2)\Omega (f,\delta ). \end{aligned}$$

In the last increase we used (21) with \(\lambda :=|t-x|/\delta \). We got

$$\begin{aligned} |f(t)-f(x)|\le 2\left( 1+\displaystyle \frac{|t-x|}{\delta }\right) (1+(t-x)^2)(1+x^2)(1+\delta ^2)\Omega (f,\delta ). \end{aligned}$$

If we prove

$$\begin{aligned} \left( 1+\displaystyle \frac{|t-x|}{\delta }\right) (1+(t-x)^2) \le 2\left( 1+\displaystyle \frac{(t-x)^4}{\delta ^4}\right) (1+\delta ^2), \end{aligned}$$
(23)

i.e., \((1+y)(1+(t-x)^2)\le 2(1+y^4)(1+\delta ^2)\), where \(y=|t-x|/\delta \), then (22) is true. We justify (23) on two cases.

For \(y\le 1\), \((1+y)(1+(t-x)^2)\le 2(1+\delta ^2)\) and (23) is evident.

For \(1<y\), \((1+y)(1+(t-x)^2)\le 2y(y^2+\delta ^2y^2)=2y^3(1+\delta ^2)\) and again (23) is true. The proof is completed. \(\square \)

Theorem 4

Let \(P_n^*\), \(n\ge n_a\), be given by (4) and (6). Let \(f\in C_\rho ^*(\mathbb {R})\) such that f is twice differentiable and \(f',f''\) belong to \(C_\rho ^*(\mathbb {R})\). For any \(x\in \mathbb {R}\) we have

  1. (i)
    $$\begin{aligned}&|n((P_n^* f)(x)-f(x))+2af'(x)-f''(x)| \le |A_n(x)||f'(x)|\\&\quad +|B_n(x)||f''(x)|+16n(1+x^2)\mu _{2}(P_n^*;x) \Omega \left( f'';\root 4 \of {\displaystyle \frac{\mu _{6}(P_n^*;x)}{\mu _{2}(P_n^*;x)}}\right) , \end{aligned}$$

    where

    $$\begin{aligned} A_n(x)=n\mu _{1}(P_n^*;x)+2a \quad \text{ and }\quad B_n(x)=\displaystyle \frac{n}{2}\mu _{2}(P_n^*;x)-1. \end{aligned}$$
  2. (ii)

    \(\lim \limits _{n\rightarrow \infty }n((P_n^* f)(x)-f(x))=-2af'(x)+f''(x)\).

Proof

  1. (i)

    Let x be arbitrarily fixed and \(t\in \mathbb {R}\). By Taylor’s formula with Lagrange form of the remainder, we have

    $$\begin{aligned} f(t)=f(x)+(t-x)f'(x)+\displaystyle \frac{(t-x)^2}{2}f''(x)+\displaystyle \frac{(t-x)^2}{2}h(\xi _{t,x}), \end{aligned}$$
    (24)

    where \(\xi _{t,x}\) is a certain real number between t and x. In the above

    $$\begin{aligned} h(\xi _{t,x})=f''(\xi _{t,x})-f''(x) \end{aligned}$$
    (25)

    is a continuous function. If \(t\rightarrow x\), then \(\xi _{t,x}\rightarrow x\) and h vanishes at x. Applying the operator \(P_n^*\) to both sides of identity (24), knowing that \(P_n^* e_0=e_0\), we obtain

    $$\begin{aligned} (P_n^* f)(x)-f(x)=\mu _{1}(P_n^*;x)f'(x)+\mu _{2}(P_n^*;x)\displaystyle \frac{f''(x)}{2} +\displaystyle \frac{1}{2}P_n^*((\cdot -x)^2h;x). \end{aligned}$$

    This identity can be rewritten in the following way

    $$\begin{aligned}&|n((P_n^* f)(x)-f(x))+2af'(x)-f''(x)|\nonumber \\&\quad \le |A_n(x)||f'(x)|+|B_n(x)||f''(x)|+\displaystyle \frac{n}{2}P_n^* ((\cdot -x)^2 |h|,x). \end{aligned}$$
    (26)

    By using both (24) and (22) applied for \(f''\), we get

    $$\begin{aligned} |h(\xi _{t,x})|=|f''(\xi _{t,x})-f''(x)| \le 4\left( 1+\displaystyle \frac{(t-x)^4}{\delta ^4}\right) (1+\delta ^2)(1+x^2)\Omega (f'';\delta ) \end{aligned}$$

    and, in the factor \((1+\delta ^2)^2\) considering \(\delta \le 1\), we can write

    $$\begin{aligned} nP_n^*((\cdot -x)^2|h|;x) \le 16n(1+x^2)\mu _{2}(P_n^*;x)\left( 1+\displaystyle \frac{\mu _{6}(P_n^*;x)}{\delta ^4\mu _{2}(P_n^*;x)}\right) \Omega (f'';\delta ). \end{aligned}$$

    Further, a rank \(N_1\ge n_a\) exists such that for any \(n\ge N_1\) we can choose \(\delta ^4=\mu _{6}(P_n^*;x)/\mu _{2}(P_n^*;x)\le 1\). This choice is allowed because of Lemma 4(iii). Returning at (26) the required inequality is proved.

  2. (ii)

    Easily obtain

    $$\begin{aligned} \lim \limits _{n\rightarrow \infty }A_n(x)=0,\quad \lim \limits _{n\rightarrow \infty }B_n(x)=0,\quad \lim \limits _{n\rightarrow \infty }n\mu _{2}(P_n^*;x)=2 \end{aligned}$$

    and taking into account Lemma 4(iii), the statement follows.

\(\square \)

Theorem 5

Let \(W_n^*\), \(n\ge 1\), be given by (5) and (7). Let \(f\in C_\rho ^*(\mathbb {R})\) such that f is twice differentiable and \(f',f''\) belong to \(C_\rho ^*(\mathbb {R})\). For any \(x\in \mathbb {R}\) we have

  1. (i)
    $$\begin{aligned}&\left| n((W_n^* f)(x)-f(x))+\displaystyle \frac{a}{2}f'(x)-\displaystyle \frac{1}{4}f''(x)\right| \le |C_n(x)||f'(x)|\\&\quad +|D_n(x)||f''(x)|+16n(1+x^2)\mu _{2}(W_n^*;x) \Omega \left( f'';\root 4 \of {\displaystyle \frac{\mu _{6}(W_n^*;x)}{\mu _{2}(W_n^*;x)}}\right) \end{aligned}$$

    where

    $$\begin{aligned} C_n(x)=n\mu _{1}(W_n^*;x)+\displaystyle \frac{a}{2} \quad \text{ and }\quad D_n(x)=\displaystyle \frac{n}{2}\mu _{2}(W_n^*;x)-\displaystyle \frac{1}{4}. \end{aligned}$$
  2. (ii)

    \(\lim \limits _{n\rightarrow \infty }n((P_n^* f)(x)-f(x))=-\displaystyle \frac{a}{2}f'(x)+\displaystyle \frac{1}{4}f''(x)\).

For achieving the proof we appeal, inter alia, at relation (24), the central moments \(\mu _{k}(W_n^*;\cdot )\), \(k\in \{1,2\}\), Lemmas 4 and 6. Actually, the technique proceed with arguments identical with those used in the proof of Theorem 4, consequently we omit it.

5 A property implied by generalized convexity

Lemma 6

The operators \(P_n^*\), \(n\ge n_a\), and \(W_n^*\), \(n\ge 1\), reproduce the function \(\varphi _a\) defined by (8).

Proof

We have

$$\begin{aligned} (P_n^* \varphi _a)(x)&=e^{2a\alpha _n(x)}\displaystyle \frac{\sqrt{n}}{2}\int _\mathbb {R}e^{2at-\sqrt{n}|t|}dt\\&=e^{2a\alpha _n(x)}\displaystyle \frac{n}{n-4a^2}=\varphi _a(x). \end{aligned}$$

Similarly, relation \((W_n^* \varphi _a)(x)=\varphi _a(x)\) is deduced by direct calculation.  \(\square \)

This way, we infer that besides the function \(e_0\), function \(\varphi _a\) is also a fixed point for all operators \(P_n^*\), \(n\ge n_a\), and \(W_n^*\), \(n\ge 1\). Further, we use the couple \((e_0,\varphi _a)\).

On the basis of [5, Definition 2] and taken in view Ziegler’s remark [5, page 426] we present the following

Definition

A function f defined on \(\mathbb {R}\) is said to be convex with respect to \((e_0,\varphi _a)\), provided

$$\begin{aligned} \left| \begin{array}{ccc} 1 &{} 1 &{} 1\\ \varphi _a(x_1) &{} \varphi _a(x_2) &{} \varphi _a(x_3)\\ f(x_1) &{} f(x_2) &{} f(x_3) \end{array}\right| \ge 0,\quad -\infty<x_1<x_2<x_3<\infty . \end{aligned}$$
(27)

The set of functions satisfying (27) is denoted by \(\mathcal {C}(e_0,\varphi _a)\).

Theorem 6

Let the operators \(P_n^*\), \(n\ge n_a\), \(W_n^*\), \(n\ge 1\), be given. For every function \(f\in C(\mathbb {R})\cap \mathcal {C}(e_0,\varphi _a)\), we have

$$\begin{aligned} (P_n^* f)(x)\ge f(x) \quad \text{ and }\quad (W_n^* f)(x)\ge f(x),\ x\in \mathbb {R}. \end{aligned}$$

Proof

Since our operators reproduce the functions \(e_0\) and \(\varphi _a\), we can apply Theorem 2 of the paper [5]. We are considering the fact that this result of Ziegler also works for functions defined on unbounded intervals. We emphasize that the condition to be \(e_0\) and \(\varphi _a\) fixed points for our operators are indispensable [5, Theorem 3]. \(\square \)