On the convergence properties of sampling Durrmeyer-type operators in Orlicz spaces

Lemma 3.2.Under the above assumptions on the kernel φ, we have

• (i)

  $$M_{0}(\varphi ):=\sup _{u\in \mathbb {R} }\sum _{k\in \mathbb {Z}}\left| \varphi (u-k)\right| <+\infty ;$$

• (ii)

  For every $$\gamma >0$$,

  $$$$\begin{equation*} \lim _{w\rightarrow +\infty }\sum _{{\left|wx-k\right|}>\gamma w}{\left| \varphi (wx-k)\right|}=0, \end{equation*}$$$$
  uniformly with respect to $$x\in \mathbb {R}$$.

Remark 3.3.We note the following:

• (a)

  For $$\mu ,\nu >0$$ with $$\mu \le \nu$$, then $$M_\nu (\varphi )<+\infty$$ implies $$M_\mu (\varphi )<+\infty$$, see, for example, [22]. Moreover, if φ has compact support, we immediately have that $$M_\nu (\varphi )<+\infty$$, for every $$\nu \ge 0$$. Finally, in an analogous way, for $$\mu , \nu >0$$ with $$\mu \le \nu$$, $$\tilde{M}_\nu (\psi )<+\infty$$ implies $$\tilde{M}_\mu (\psi )<+\infty$$.

• (b)

  From Lemma 3.2, $$S_w^{\varphi ,\psi }$$ are well defined for every $$f \in L^\infty (\mathbb {R})$$. Indeed,

  $$$$\begin{equation*} {\left|{\left(S_w^{\varphi ,\psi }f\right)}(x)\right|} \le M_{0}(\varphi )\tilde{M_{0}}(\psi )\Vert f \Vert _\infty ,\ x \in \mathbb {R}. \end{equation*}$$$$
  Thus, sampling Durrmeyer operators are bounded linear operators mapping $$L^\infty (\mathbb {R})$$ into itself.

• (c)

  If a function g is bounded and $$g(u)=\mathcal {O}(\left|u\right|^{-\alpha })$$, as $$\left|u\right|\rightarrow +\infty$$, with $$\alpha >\nu +1$$, $$\nu >0$$, then

  $$$$\begin{equation*} \widetilde{M}_\mu (g),\ M_\mu (g)<+\infty ,\text{ for every }0\le \mu \le \nu , \end{equation*}$$$$
  see, for example, [22].

Theorem 4.1.Let $$f\in L^\infty (\mathbb {R})$$. Then,

at any point x of continuity of f. Moreover, if $$f\in C(\mathbb {R})$$, then

Proof.We only prove the second part of the theorem, since the first part can be obtained by similar methods. Let $$\varepsilon >0$$ be fixed. Then, there exists $$\delta >0$$ such that $$\left|f(x)-f(y)\right|<\varepsilon$$ when $$\left|x-y\right|<\delta$$. Let $$x \in \mathbb {R}$$ be fixed. Using (3.1), we have

$$=I_1 + I_2$$. The first term can be further divided into $$=I_{1,1}+I_{1,2}$$. For $$u \in \mathbb {R}$$ such that $$\left|wu-k \right| < \frac{\delta }{2}w$$, if $$\left| wx-k \right| \le \frac{\delta }{2}w$$, we have Thus, Now, by the change of variable $$wu-k=y$$, and recalling that $$\psi \in L^1(\mathbb {R})$$, we have for every $$w\ge 0$$. Thus, Moreover, with since $$\psi \in L^1({\mathbb {R}})$$. Then, there exists a sufficiently large $$\bar{w_1}\in \mathbb {R}$$ such that By similar reasoning, we obtain the following inequality: From property (ii) of Lemma 3.2, there exists a sufficiently large $$\bar{w_2}\in \mathbb {R}$$ such that Setting $$K:=M_{0}(\varphi )\Vert \psi \Vert _1+2\Vert f \Vert _\infty (M_{0}(\varphi )+\Vert \psi \Vert _1)$$ and $$\bar{w}:=\max \left\lbrace \bar{w_1},\bar{w_2}\right\rbrace$$, we have Finally, observing that the above estimate does not depend on $$x\in \mathbb {R}$$, we easily obtain and thus the proof follows by the arbitrariness of $$\varepsilon >0$$.$$\Box$$

Theorem 5.1.Suppose that φ and ψ are such that $$M_1(\varphi )+\tilde{M}_1(\psi )<+\infty$$ and let $$f\in C(\mathbb {R})$$. Then, we have

for every $$w>0$$, where $$C^{\varphi ,\psi }=M_0(\varphi )\left(\tilde{M}_0(\psi )+\tilde{M}_1(\psi )\right)+M_1(\varphi )\tilde{M}_0(\psi )$$.

Proof.Let $$x\in \mathbb {R}$$ be fixed. We have

for every $$w>0$$, where the previous estimate is a consequence of the definition of $$\omega (f,\frac{1}{w})$$ and of (5.1) with $$\lambda = w\left|u-x\right|>0$$ and $$\delta =\frac{1}{w}$$. Now we estimate the following term: So we obtain Now, setting $$C^{\varphi ,\psi }:=M_0(\varphi )\left(\tilde{M}_0(\psi )+\tilde{M}_1(\psi )\right)+M_1(\varphi )\tilde{M}_0(\psi )<+\infty$$, we finally obtain for every $$w>0$$. This completes the proof.$$\Box$$

Theorem 6.1.Let ψ be a kernel such that $$M_0(\psi )<+\infty$$, and $$f\in L^{\eta }(\mathbb {R})$$ be fixed. Then, there exists $$\lambda >0$$ such that

In particular, $$S_w^{\varphi ,\psi }$$ are well defined and belong to $$L^{\eta }(\mathbb {R})$$, for every $$w>0$$.

Proof.Since $$f\in L^{\eta }(\mathbb {R})$$, there exists $$\overline{\lambda }>0$$ such that $$I^{\eta }[\overline{\lambda }f]<+\infty$$. We consider now $$\lambda >0$$ such that

Then, we have $$I^{\eta }[\lambda M_0(\varphi )\tilde{M_0}(\psi )f]<+\infty$$. Applying Jensen inequality twice, the change of variable $$wu-k=t$$, and Fubini–Tonelli theorem, we obtain with the change of variable $$wx-k=y$$.$$\Box$$

Theorem 6.2.Let ψ be a kernel such that $$M_0(\psi )<+\infty$$, and let $$f\in L^{\eta }(\mathbb {R})$$ be fixed. Then, there exists $$\lambda >0$$ such that

Proof.First of all, since $$f\in L^\eta (\mathbb {R})$$, we have that there exist $$\lambda _1, \lambda _2>0$$ such that $$I^\eta [\lambda _1f]<+\infty$$, and

that is, for every fixed $$\varepsilon >0$$, there exists $$\delta >0$$ such that for every $$h\in \mathbb {R}$$ such that $$\left| h\right|\le \delta$$ (see, e.g., [7]). Now, we fix $$\lambda >0$$ such that Thus, by the properties of the convex modular functional $$I^\eta$$, we can write what follows: First, we estimate J₁. Applying Jensen inequality twice similarly to the proof of Theorem 6.1, the change of variable $$wx-k=t$$, and Fubini–Tonelli theorem, we obtain Now, using δ given in (6.1), we can split the above integral as follows: Now, using the inequality in (6.1) with $$h=\frac{t}{w}$$, we have for every $$w>0$$. For what concerns J_{1, 2}, by the convexity of η, we have Now we can observe that, since $$\varphi \in L^1(\mathbb {R})$$, there exists $$\overline{w_1}>0$$ such that for every $$w\ge \overline{w_1}$$. Moreover, noting that for every $$t\in \mathbb {R}$$ and $$w>0$$, we have for every $$w\ge \overline{w_1}$$. Now, we estimate J₂. By the change of variable $$t = u-\frac{k}{w}$$, applying Jensen inequality twice, and Fubini–Tonelli theorem, we have where we have used the change of variable $$y=wt$$. Then, using again δ given in (6.1), we can rewrite the above integral as follows: Thus, similarly to before, using the inequality in (6.1) with $$h=\frac{y}{w}$$, we obtain for every $$w>0$$. Now, for what concerns the last term J_{2, 2}, by the convexity of η, we have Now, we can observe that, since $$\psi \in L^1(\mathbb {R})$$, there exists $$\overline{w_2}>0$$ such that for every $$w\ge \overline{w_2}$$ and, similarly to before, we have for every $$w\ge \overline{w_2}$$. Finally, setting $$\overline{w}:=\max \left\lbrace \overline{w_1},\overline{w_2}\right\rbrace$$ and we have for every $$w\ge \overline{w}$$. Thus, the proof follows by the arbitrariness of ε.$$\Box$$