Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces

Abstract

In this paper, we establish a quantitative estimate for Durrmeyer-sampling type operators in the general framework of Orlicz spaces, using a suitable modulus of smoothness defined by the involved modular functional. As a consequence of the above result, we can deduce quantitative estimates in several instances of Orlicz spaces, such as $L^p$-spaces, Zygmund spaces and the exponential spaces. By using a direct approach, we also provide a further estimate in the particular case of $L^p$-spaces, with $1\le p <+\infty$, that turns out to be sharper than the previous general one. Moreover, we deduce the qualitative order of convergence, when functions belonging to suitable Lipschitz classes are considered.

1 Introduction

The theory of the generalized sampling type operators has been introduced since the eighties by P. L. Butzer and his school in Aachen, in order to provide an approximate version of the celebrated Whittaker–Kotel’nikov–Shannon sampling theorem. Butzer’s work is in fact the first rigorous approach for solving the many disadvantages of the classical sampling formula. It represents the formulation of a mathematical model for an approximate reconstruction of signals and images. The above operators have been firstly introduced and studied within their natural frame of continuous functions, and then applied also to the case of not necessarily continuous signals (see, e.g., [3, 8,9,10,11,12, 26, 33]). In the following years, the study of other suitable versions of the generalized sampling series in more and more general functional spaces has generated a great interest in the field of approximation theory, from both the theoretical and the applications point of view.

An important contribution in this direction has been given more recently by Bardaro et al. [2]. In this work, they introduced the so-called Kantorovich-sampling type operators, defined by

$$\begin{aligned} (S_w^{\varphi }f)(x):=\sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w \int _{\frac{k}{w}}^{\frac{k+1}{w}}f\left( u\right) du,\,\,\,w>0,\,x\in {\mathbb {R}}, \end{aligned}$$

(1)

that turn out to be well-defined and bounded in $L^p({\mathbb {R}})$, under suitable assumptions on the kernel function. The approximation properties of Kantorovich-sampling type operators have been widely studied in the general framework of Orlicz spaces, in which the $L^p$-spaces and several other well-known cases of functional spaces are included. In particular, the study of the approximation order together with saturation results and inverse theorems of approximation have been established in [19, 20, 22]. Moreover, many and various applications to the real-world problems have been studied, especially in image reconstruction and enhancement (see [13, 18]).

A more general version of sampling type operators has been given by Bardaro and Mantellini in [5], via the so-called Durrmeyer-sampling type operators, defined by

$$\begin{aligned} \left( S_w^{\varphi ,\psi }f\right) (x):= \sum _{k\in {\mathbb {Z}}} \varphi (wx-k) w\int _{{\mathbb {R}}}\psi (wu-k)f(u)du,\,\,\,w>0,\,x\in {\mathbb {R}}, \end{aligned}$$

(2)

where $\varphi$ and $\psi$ are two kernel functions satisfying the usual moment conditions (see also [4, 14, 15, 23,24,25]). Here, the Durrmeyer modification of $S_w^{\varphi }f$ has been obtained by replacing the integral averages of the function in a neighbourhood of the node $\frac{k}{w}$ by a convolution integral between f and an additional kernel $\psi$. It is easy to see that the kernel $\psi$ generates a Fejér-type approximate identity by the formula $\psi _w(\cdot ):=w\psi (w\,\cdot )$, $w>0$. This leads to read (2) as semi-discrete operators of the following form

$$\begin{aligned} \left( S_w^{\varphi ,\psi }f\right) (x):= \sum _{k\in {\mathbb {Z}}} \varphi (wx-k) \left( \psi _w*f\right) \left( \frac{k}{w}\right) ,\,\,\,w>0,\,x\in {\mathbb {R}}, \end{aligned}$$

in which we have a double convolution, first continuous and then discrete. Even for this type of sampling operators, a unifying theory for the convergence in Orlicz spaces has been achieved both in one and in multidimensional frame (see, e.g., [16, 17, 34]). Moreover, we have already observed in [16] that (2) extends (1), by taking as $\psi$ the characteristic function of the interval [0, 1] and it also extends the generalized sampling type operators by taking $\psi$ as the Dirac delta distribution. Furthermore, the study of the approximation order for (2) has been already approached in [16], in the classical case of uniformly continuous and bounded functions, i.e., $C({\mathbb {R}})$ endowed with the uniform norm $\Vert \cdot \Vert _\infty$, by using the modulus of continuity of the function being approximated (see also [1]).

However, the study of quantitative estimates in the general setting of Orlicz spaces in terms of the related modulus of smoothness for (2), is still an open problem and it represents the chief purpose of the present paper.

Therefore, in Sect. 2 we firstly recall the notion of the modulus of smoothness in $L^\varphi ({\mathbb {R}})$, which is defined via the modular functional of the space, together with other notions and preliminaries useful for the paper. In Sect. 3, we present the main quantitative result (Theorem 3.1), and we deduce a qualitative result (Corollary 3.1) introducing the Lipschitz classes in Orlicz spaces. We remark that working in the general setting of Orlicz spaces leads to a unifying theory that covers several functional spaces, such as $L^p$-spaces, Zygmund spaces, exponential spaces, and many others; in particular, in Sect. 4, we face the above problem also in the $L^p$-spaces, by using a direct approach, providing a quantitative estimate that turns out to be sharper than the general one, thanks to the well-known properties of the usual $L^p$-modulus of smoothness. Finally, the qualitative version of the above estimate (Corollary 4.1) is deduced, considering functions belonging to suitable Lipschitz classes.

2 Preliminaries and notations

We begin by recalling some basic features of Orlicz spaces, that have been introduced by W. Orlicz as an extension of Lebesgue spaces.

Let $\varphi : {\mathbb {R}}^+_0 \rightarrow {\mathbb {R}}^+_0$ be a convex $\varphi$-function, i.e., $\varphi$ satisfies the following assumptions:

$\Phi _1$):

$\varphi$ is convex in ${\mathbb {R}}^+_0$;

$\Phi _2$):

$\varphi (0) = 0$ and $\varphi (u) > 0,$ for every $u >0$.

Denoted by $M({\mathbb {R}})$ the space of all (Lebesgue) measurable function $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$, the corresponding convex modular functional on $M({\mathbb {R}})$ is defined by

$$\begin{aligned} I^{\varphi }[f] := \int _{{\mathbb {R}}}\varphi (|f(x)|)dx, ~~f \in M({\mathbb {R}}), \end{aligned}$$

(see, e.g., [6, 29]). Therefore, the Orlicz space generated by $\varphi$ is given by

$$\begin{aligned} L^{\varphi }({\mathbb {R}}) = \{f \in M({\mathbb {R}}): I^{\varphi }[\lambda f] < + \infty , \text{ for } \text{ some }~\lambda >0\}. \end{aligned}$$

It is immediate to see that if we consider $\varphi (u)=u^p$, $1\le p <+\infty$, $I^\varphi [f]=\Vert f\Vert _p^p$, then $L^{\varphi }({\mathbb {R}})=L^{p}({\mathbb {R}})$.

We point out that the convexity of the $\varphi$-function $\varphi$ is not strictly necessary for the definition of Orlicz spaces. Indeed, the assumption of convexity can be weakened by requiring that $\varphi$ is continuous and non-decreasing in ${\mathbb {R}}^+_0$. However, in order to obtain estimates or other approximation results, we need to assume that the involved $\varphi$-function is convex.

The most natural notion of convergence in Orlicz spaces is called modular convergence (introduced in [30]). We will say that a net of functions $(f_w)_{w >0} \subset L^{\varphi }({\mathbb {R}})$  is modularly convergent to a function $f \in L^{\varphi }({\mathbb {R}})$ if

$$\begin{aligned} \lim _{w \rightarrow +\infty }I^{\varphi }[\lambda (f_w -f)] = 0, \end{aligned}$$

for some $\lambda >0.$ This notion induces a topology in $L^{\varphi }({\mathbb {R}})$, called modular topology.

Now, we recall the definition of the modulus of smoothness in Orlicz spaces $L^\varphi ({\mathbb {R}})$, with respect to the modular $I^\varphi$.

Let $f\in L^\varphi ({\mathbb {R}})$, we define

$$\begin{aligned} \omega (f,\delta )_\varphi :=\displaystyle \sup _{|t|\le \delta }I^\varphi [f(\cdot +t)-f(\cdot )], \end{aligned}$$

with $\delta >0$. It is well-known (see [6]) that if $f\in L^\varphi ({\mathbb {R}})$, then there exists $\lambda >0$ such that

$$\begin{aligned} \lim _{\delta \rightarrow 0^+} \omega (\lambda f,\delta )_\varphi =0. \end{aligned}$$

For other general references concerning Orlicz spaces, see also [27, 28, 31, 32].

Now, in order to recall the definition of the family of operators that will be studied in the next sections, we give the following definitions.

We introduce a pair of functions $\varphi$ and $\psi$ belonging to $L^1({\mathbb {R}})$, that we will call respectively discrete and continuous kernel, such that $\varphi$ is bounded in a neighbourhood of the origin, and satisfies

$$\begin{aligned} \sum _{k\in {\mathbb {Z}}} \varphi (u-k)=1,\text { for every} u\in {\mathbb {R}}, \end{aligned}$$

(3)

while

$$\begin{aligned} \int _{{\mathbb {R}}} \psi (u)du=1. \end{aligned}$$

(4)

Thus, for $w>0$, we can recall the definition of the Durrmeyer-sampling type operators based on $\varphi$ and $\psi$ (see [4, 5]), as

$$\begin{aligned} \left( S_w^{\varphi ,\psi }f\right) (x):= \sum _{k\in {\mathbb {Z}}} \varphi (wx-k) w\int _{{\mathbb {R}}}\psi (wu-k)f(u)du,\,\,\,\, x\in {\mathbb {R}}, \end{aligned}$$

(5)

for any given function $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ such that the above series is convergent, for every $x\in {\mathbb {R}}$.

Concerning the kernel $\varphi$ and $\psi$ involved in the definition of $S_w^{\varphi ,\psi }$, we need to assume suitable conditions concerning the so-called discrete and continuous absolute moments.

We recall that for any $r>0$, the discrete absolute moment of order r for a function $\xi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is defined by:

$$\begin{aligned} M_{r}(\xi ):=\sup _{u\in {\mathbb {R}} }\sum _{k\in {\mathbb {Z}}}\left| \xi (u-k)\right| \left| u-k\right| ^r, \end{aligned}$$

while, the continuous absolute moment of order r is defined by

$$\begin{aligned} {\widetilde{M}}_{r}(\xi ):=\int _{{\mathbb {R}} } \left| \xi (u)\right| \left| u\right| ^r\,du. \end{aligned}$$

In particular, if we assume that $M_0(\varphi )<+\infty$ for the discrete kernel $\varphi$, it follows that the Durrmeyer-sampling type operators are well-defined in $L^\infty ({\mathbb {R}})$. Moreover, we can state the following [16].

Theorem 2.1

Let $M_0(\varphi )<+\infty$.

• i) If $f\in L^\infty ({\mathbb {R}})$, we have

  $$\begin{aligned}\left| \left( S_w^{\varphi ,\psi }f\right) (x)\right| \le M_{0}(\varphi )\Vert \psi \Vert _1\Vert f \Vert _\infty ,\ x\in {\mathbb {R}},\,w>0,\end{aligned}$$

  i.e., the operators $S_w^{\varphi ,\psi }$ are linear and continuous in $L^\infty ({\mathbb {R}})$.

• ii) Let $\eta$ be a convex $\varphi$-function. If $f\in L^{\eta }({\mathbb {R}})$ and $M_0(\psi )<+\infty$, then there exists $\lambda >0$ such that

  $$\begin{aligned}I^{\eta }[\lambda S_w^{\varphi ,\psi }f ]\le \displaystyle \frac{M_0(\psi )\Vert \varphi \Vert _1}{M_0(\varphi )\displaystyle \Vert \psi \Vert _1} I^{\eta }[\lambda M_0(\varphi )\Vert \psi \Vert _1f],\,\,w>0,\end{aligned}$$

  i.e., the operators $S_w^{\varphi ,\psi }$ are linear and continuous in $L^\eta ({\mathbb {R}})$.

3 The main result

In order to provide a quantitative estimate for the Durrmeyer-sampling type operators by using the modulus of smoothness in Orlicz spaces, we introduce the following condition.

For any $0<\alpha <1$, we will say that a function $\xi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ satisfies the integral decay condition (${\mathcal {D}}_\alpha$), if

for suitable positive constants M and $\mu$ depending on $\alpha$ and $\xi$.

Now, we can state the main result of this section. In order to do this, in all the rest of the paper we always consider a pair of discrete and continuous kernels, respectively, such that

$$\begin{aligned}M_0(\varphi )+M_0(\psi )<+\infty .\end{aligned}$$

Theorem 3.1

Let $\eta$ be a convex $\varphi$-function and let $f\in L^\eta ({\mathbb {R}})$. Moreover, let $0<\alpha <1$ such that both the kernels $\varphi$ and $\psi$ satisfy (${\mathcal {D}}_\alpha$).

Then, there exist two positive constants ${\overline{M}}$ and ${\overline{\mu }}$ depending on $\alpha$, $\varphi$ and $\psi$, such that

$$\begin{aligned} \begin{aligned} I^\eta \left[ \lambda \left( S_w^{\varphi ,\psi }f-f\right) \right]&\le \left( \frac{M_0(\psi )\Vert \varphi \Vert _1+M_0(\varphi )\Vert \psi \Vert _1}{2M_0(\varphi )\Vert \psi \Vert _1}\right) \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,\frac{1}{w^\alpha }\right) _\eta \\&\quad +{\overline{M}}\left( \frac{M_0(\varphi )+M_0(\psi )}{2M_0(\varphi )\Vert \psi \Vert _1}\right) I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] w^{-{\overline{\mu }}}, \end{aligned} \end{aligned}$$

for $\lambda >0$ and for every sufficiently large $w>0$. In particular, if $\lambda >0$ is sufficiently small, the above inequality implies the modular convergence of the Durrmeyer-sampling type operators $S_w^{\varphi ,\psi }f$ to f.

Proof

Let $\lambda >0$ be fixed. Since $\eta$ is a convex $\varphi$-function, we can write what follows:

$$\begin{aligned} I^\eta \left[ \lambda \left( S_w^{\varphi ,\psi }f-f\right) \right]&=\int _{{\mathbb {R}}} \eta \left( \lambda \left| \left( S_w^{\varphi ,\psi }f\right) (x)-f(x)\right| \right) dx \\&=\int _{{\mathbb {R}}}\eta \left( \lambda \left| \left( S_w^{\varphi ,\psi }f\right) (x)-\sum _{k\in {\mathbb {Z}}} \varphi (wx-k)w\right. \right. \\&\times \left. \left. \int _{{\mathbb {R}}}\psi (wu-k)f\left( u+x-\frac{k}{w}\right) du\right. \right. \\&+\sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\int _{{\mathbb {R}}}\psi (wu-k) \\&\left. \left. \times f\left( u+x-\frac{k}{w}\right) du-f(x)\right| \right) dx \\&\le \frac{1}{2}\left\{ \int _{{\mathbb {R}}}\eta \left( 2\lambda \left| \left( S_w^{\varphi ,\psi }f\right) (x) -\sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\right. \right. \right. \\&\times \left. \left. \left. \int _{{\mathbb {R}}}\psi (wu-k)f\left( u+x-\frac{k}{w}\right) du\right| \right) dx\right. \\&\left. +\int _{{\mathbb {R}}}\eta \left( 2\lambda \left| \sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\int _{{\mathbb {R}}}\psi (wu-k) \right. \right. \right. \\&\times \left. \left. \left. f\left( u+x-\frac{k}{w}\right) du-f(x)\right| \right) dx\right\} \\&=:\frac{1}{2}\left\{ I_1+I_2\right\} . \end{aligned}$$

First, we estimate $I_1$. Applying Jensen inequality twice, we have

$$\begin{aligned} \begin{aligned} I_1&\le \int _{{\mathbb {R}}}\eta \left( 2\lambda \sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\int _{{\mathbb {R}}}|\psi (wu-k)|\left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) dx \\&\le \frac{1}{M_0(\varphi )}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|\\&\quad \times \left[ \eta \left( 2\lambda M_0(\varphi )w\int _{{\mathbb {R}}}|\psi (wu-k)|\left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) \right] dx \\&\le \frac{1}{M_0(\varphi )\Vert \psi \Vert _1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|\\&\quad \times \left[ \int _{{\mathbb {R}}}w|\psi (wu-k)|\eta \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| \right) du\right] dx. \end{aligned} \end{aligned}$$

Now, considering the change of variable $y=x-\frac{k}{w}$ and Fubini–Tonelli theorem, we obtain

$$\begin{aligned} \begin{aligned} I_1&\le \frac{1}{M_0(\varphi )\Vert \psi \Vert _1}\int _{{\mathbb {R}}}w|\varphi (wy)|\left\{ \int _{{\mathbb {R}}} \left[ \sum _{k\in {\mathbb {Z}}}|\psi (wu-k)|\right] \right. \\ {}&\quad \times \eta \left. \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f\left( u+y\right) -f(u)\right| \right) du\right\} dy \\ {}&\le \frac{M_0(\psi )}{M_0(\varphi )\Vert \psi \Vert _1}\int _{{\mathbb {R}}}w|\varphi (wy)|\left[ \int _{{\mathbb {R}}}\eta \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f\left( u+y\right) -f(u)\right| \right) du\right] dy. \end{aligned} \end{aligned}$$

Let $0<\alpha <1$ the parameter of condition (${\mathcal {D}}_\alpha$). Thus, we can split $I_1$ as follows

$$\begin{aligned} \begin{aligned} I_1&\le \frac{M_0(\psi )w}{M_0(\varphi )\Vert \psi \Vert _1}\displaystyle \left\{ \int _{|y|\le \frac{1}{w^\alpha }}+\int _{|y|>\frac{1}{w^\alpha }} \right\} |\varphi (wy)|\\&\quad \times \left[ \int _{{\mathbb {R}}}\eta \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f\left( u+y\right) -f(u)\right| \right) du\right] dy \\&=:I_{1,1}+I_{1,2}. \end{aligned} \end{aligned}$$

Let us now focus on $I_{1,1}$. By the definition of the modulus of smoothness of $L^\eta ({\mathbb {R}})$, we have

$$\begin{aligned} \begin{aligned} I_{1,1}&\le \frac{M_0(\psi )w}{M_0(\varphi )\Vert \psi \Vert _1}\int _{|y|\le \frac{1}{w^\alpha }}|\varphi (wy)|\cdot \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,|y|\right) _\eta dy \\&\le \frac{M_0(\psi )\Vert \varphi \Vert _1}{M_0(\varphi )\Vert \psi \Vert _1}\cdot \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,\frac{1}{w^\alpha }\right) _\eta . \end{aligned} \end{aligned}$$

For what concerns $I_{1,2}$, by the convexity of $I^\eta$, we get

$$\begin{aligned} \begin{aligned} I_{1,2}&\le \frac{M_0(\psi )w}{M_0(\varphi )\Vert \psi \Vert _1}\int _{|y|>\frac{1}{w^\alpha }}|\varphi (wy)|\cdot \frac{1}{2}\left\{ I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] \right. \\&\quad \left. +I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f(\cdot +y)\right] \right\} dy \\&=\frac{M_0(\psi )w}{M_0(\varphi )\Vert \psi \Vert _1}\int _{|y|>\frac{1}{w^\alpha }}|\varphi (wy)|I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] dy, \end{aligned} \end{aligned}$$

where the above equality easily follows observing that

$$\begin{aligned} I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] =I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f(\cdot +y)\right] \end{aligned}$$

for every $y\in {\mathbb {R}}$. Moreover, since $\varphi$ satisfies (${\mathcal {D}}_\alpha$), there exist two positive constants $M_\varphi$ and $\mu _\varphi$ such that

$$\begin{aligned} w\int _{|y|>\frac{1}{w^\alpha }}|\varphi (wy)|dy\le M_\varphi w^{-\mu _\varphi }, \end{aligned}$$

for w sufficiently large. Thus, for $I_{1,2}$, we finally obtain

$$\begin{aligned} I_{1,2}\le \frac{M_0(\psi )}{M_0(\varphi )\Vert \psi \Vert _1}I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] M_\varphi w^{-\mu _\varphi }, \end{aligned}$$

for w sufficiently large.

Now, we estimate $I_2$. Using (3), (4) and the change of variable $t=u-\frac{k}{w}$, we have

$$\begin{aligned} \begin{aligned} I_2&=\int _{{\mathbb {R}}}\eta \left( 2\lambda \left| \sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\int _{{\mathbb {R}}}\psi (wu-k)f\left( u+x-\frac{k}{w}\right) du-f(x)\right| \right) dx \\&=\int _{{\mathbb {R}}}\eta \left( 2\lambda \left| \sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\int _{{\mathbb {R}}}\psi (wu-k)\left[ f\left( u+x-\frac{k}{w}\right) -f(x)\right] du\right| \right) dx \\&=\int _{{\mathbb {R}}}\eta \left( 2\lambda \left| \sum _{k\in {\mathbb {Z}}}\varphi (wx-k)w\int _{{\mathbb {R}}}\psi (wt)\left[ f\left( t+x\right) -f(x)\right] dt\right| \right) dx \\&\le \int _{{\mathbb {R}}}\eta \left( 2\lambda \sum _{k\in {\mathbb {Z}}}\left| \varphi (wx-k)\right| w\int _{{\mathbb {R}}}\left| \psi (wt)\right| \left| f\left( t+x\right) -f(x)\right| dt\right) dx. \end{aligned} \end{aligned}$$

Thus, applying Jensen inequality twice, we obtain

$$\begin{aligned} \begin{aligned} I_2&\le \frac{1}{M_0(\varphi )}\int _{{\mathbb {R}}}\left[ \sum _{k\in {\mathbb {Z}}}\left| \varphi (wx-k)\right| \right] \eta \left( 2\lambda M_0(\varphi )w \int _{{\mathbb {R}}}\left| \psi (wt)\right| \left| f\left( t+x\right) -f(x)\right| dt\right) dx \\&\le \frac{1}{M_0(\varphi )}\int _{{\mathbb {R}}}M_0(\varphi )\,\eta \left( 2\lambda M_0(\varphi )w\int _{{\mathbb {R}}}\left| \psi (wt)\right| \left| f\left( t+x\right) -f(x)\right| dt\right) dx \\&\le \frac{w}{\Vert \psi \Vert _1}\int _{{\mathbb {R}}}\left| \psi (wt)\right| \left[ \int _{{\mathbb {R}}}\eta \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f(t+x)-f(x)\right| \right) dx\right] dt \\&=\frac{w}{\Vert \psi \Vert _1}\left\{ \int _{|t|\le \frac{1}{w^\alpha }}+\int _{|t|>\frac{1}{w^\alpha }}\right\} \left| \psi (wt)\right| I^\eta \left[ 2\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f(t+\cdot )-f(\cdot )\right| \right] dt \\&:=I_{2,1}+I_{2,2}, \end{aligned} \end{aligned}$$

where $0<\alpha <1$ is again the parameter of condition (${\mathcal {D}}_\alpha$).

As concerns $I_{2,1}$, we have:

$$\begin{aligned} \begin{aligned} I_{2,1}&\le \frac{w}{\Vert \psi \Vert _1}\int _{|t|\le \frac{1}{w^\alpha }}\left| \psi (wt)\right| \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,|t|\right) _\eta dt \\&\le \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,\frac{1}{w^\alpha }\right) _\eta \cdot \frac{w}{\Vert \psi \Vert _1}\int _{|t|\le \frac{1}{w^\alpha }}\left| \psi (wt)\right| dt \\&\le \omega \left( 2\lambda M_0(\varphi )\Vert \psi \Vert _1f,\frac{1}{w^\alpha }\right) _\eta . \end{aligned} \end{aligned}$$

Now, similarly to above, we estimate $I_{2,2}$. Hence

$$\begin{aligned} \begin{aligned} I_{2,2}&\le \frac{w}{\Vert \psi \Vert _1}\int _{|t|>\frac{1}{w^\alpha }}|\psi (wt)|\cdot \frac{1}{2}\left[ \int _{{\mathbb {R}}}\eta \left( 4\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f(t+x)\right| \right) dx\right. \\&\quad \left. +\int _{{\mathbb {R}}}\eta \left( 4\lambda M_0(\varphi )\Vert \psi \Vert _1\left| f(x)\right| \right) dx\right] dt \\&=\frac{w}{\Vert \psi \Vert _1}\int _{|t|>\frac{1}{w^\alpha }}|\psi (wt)|I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] dt, \end{aligned} \end{aligned}$$

and since also $\psi$ satisfies (${\mathcal {D}}_\alpha$), there exist two positive constants $M_\psi$ and $\mu _\psi$ such that

$$\begin{aligned} w\int _{|t|>\frac{1}{w^\alpha }}|\psi (wt)|dt\le M_\psi w^{-\mu _\psi }, \end{aligned}$$

for w sufficiently large. Thus, for $I_{2,2}$, we have

$$\begin{aligned} I_{2,2}\le \frac{1}{\Vert \psi \Vert _1}I^\eta \left[ 4\lambda M_0(\varphi )\Vert \psi \Vert _1f\right] M_\psi w^{-\mu _\psi }, \end{aligned}$$

for w sufficiently large. In conclusion, setting

$$\begin{aligned} {{\overline{M}}}:=\max \left\{ M_\varphi ,M_\psi \right\} \text{ and } {{ \overline{\mu }}}:=\min \left\{ \mu _\varphi ,\mu _\psi \right\} >0, \end{aligned}$$

we achieve the thesis. $\square$

Remark 3.1

In order to find examples of kernel functions satisfying the previous decay condition (${\mathcal {D}}_\alpha$), we may show two useful sufficient conditions.

Let $\xi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ and $0<\alpha <1$.

• a) If $\xi (u)={\mathcal {O}}(|u|^{-\theta })$, as $|u|\rightarrow +\infty$, $\theta >1$, then it is easy to see that

  $$\begin{aligned} w\int _{|u|>1/w^\alpha }|\xi (wu)|du\le \frac{C}{\theta -1}w^{-(1-\alpha )(\theta -1)}, \end{aligned}$$

  that is condition (${\mathcal {D}}_\alpha$) is satisfied with $M:=\displaystyle \frac{C}{\theta -1}$ and $\mu :=(1-\alpha )(\theta -1)$.

• b) If the continuous absolute moment of order $\theta >0$ is finite (i.e., ${\widetilde{M}}_\theta (\xi )<+\infty$), then we have

  $$\begin{aligned} w\int _{|u|>1/w^\alpha }|\xi (wu)|du\le {\widetilde{M}}_\theta (\xi ) w^{-\theta (1-\alpha )}, \end{aligned}$$

  that is condition (${\mathcal {D}}_\alpha$) is satisfied with $M:=\displaystyle {\widetilde{M}}_\theta (\xi )$ and $\mu :=\theta (1-\alpha )$.

From the above quantitative estimate we can directly deduce the qualitative order of approximation, assuming f in suitable Lipschitz classes. To this aim, we recall the definition of Lipschitz classes in Orlicz spaces $L^\eta ({\mathbb {R}})$.

We denote by $Lip_\eta (\nu )$, with $0<\nu \le 1$, the space of all functions $f\in L^\eta ({\mathbb {R}})$ such that there exists $\lambda >0$ for which

$$\begin{aligned} I^\eta [\lambda \left( f(\cdot )-f(\cdot +t)\right) ]=\int _{{\mathbb {R}}}\eta \left( \lambda |f(x)-f(x+t)|\right) dx={\mathcal {O}}\left( |t|^\nu \right) , \end{aligned}$$

as $|t|\rightarrow 0$.

As a direct consequence of Theorem 3.1, we obtain the following.

Corollary 3.1

Let $f\in Lip_\eta (\nu )$, with $0<\nu \le 1$. Under the assumptions of Theorem 3.1 with $0<\alpha <1$, there exist $K>0$ and $\lambda >0$ such that

$$\begin{aligned} I^\eta \left[ \lambda \left( S_w^{\varphi ,\psi }f-f\right) \right] \le Kw^{-\rho }, \end{aligned}$$

for sufficiently large $w>0$, where $\rho :=\min \left\{ \alpha \nu ,\mu \right\}$ and $\mu >0$ is the parameter arising from Theorem 3.1.

In addition to the $L^p$-spaces, that we are going to examine in depth in the next section, other remarkable functional spaces generated by suitable convex $\varphi$-functions are included in Orlicz spaces. For instance, $\varphi _{\alpha ,\beta }(u):=u^\alpha \log ^\beta (u+e)$ (with $\alpha \ge 1$, $\beta >0$ and $u\ge 0$) and $\varphi _\gamma (u):=e^{u^{\gamma }}-1$ (with $\gamma >0$ and $u\ge 0$) generate the $L^\alpha \log ^\beta L$-spaces (or Zygmund spaces) and the exponential spaces, respectively. In particular, the first ones are widely used in the theory of partial differential equations, while the second ones for obtaining embedding theorems. The convex modular functionals corresponding to $\varphi _{\alpha ,\beta }$ and $\varphi _\gamma$ are

$$\begin{aligned} I^{\varphi _{\alpha ,\beta }}[f]:=\int _{\mathbb {R}}\left| f(x)\right| ^\alpha \log ^\beta (e+\left| f(x)\right| )dx,\qquad f\in {\mathcal {M}}({\mathbb {R}}), \end{aligned}$$

and

$$\begin{aligned} I^{\varphi _\gamma }[f]:=\int _{\mathbb {R}}\left( e^{|f(x)|^\gamma }-1\right) dx,\qquad f\in {\mathcal {M}}({\mathbb {R}}), \end{aligned}$$

respectively.

4 The L$^p$-case

In this section, we want to obtain a quantitative estimate for functions belonging to $L^p({\mathbb {R}})$, $1\le p <+\infty$. These spaces represent a particular case of Orlicz spaces, as it is shown in Sect. 2. We study the present case by the following direct approach and using the well-known modulus of smoothness in $L^p$-space since, in that case, we are able to establish a direct quantitative estimate that is sharper than that one obtained in the general case of Orlicz spaces considered in Theorem 3.1. This is due to a certain property of the $L^p$-modulus of smoothness that, in general, is not valid for $\omega (f,\delta )_\varphi$.

To this aim, we recall the definition of the $L^p$-first order modulus of smoothness of $f\in L^p({\mathbb {R}})$, given by

$$\begin{aligned} \omega (f,\delta )_p:=\sup _{|h|\le \delta }\left( \int _{{\mathbb {R}}}|f(t+h)-f(t)|^p dt\right) ^{\frac{1}{p}}, \end{aligned}$$

with $\delta >0$ and $1\le p <+\infty$.

Theorem 4.1

Let $\varphi$ and $\psi$ be such that

$$\begin{aligned} {\widetilde{M}}_p(\varphi )+{\widetilde{M}}_p(\psi )<+\infty , \end{aligned}$$

(6)

for some $p\in [1,+\infty [$. Then, for every $f\in L^p({\mathbb {R}})$, we obtain the following quantitative estimate

$$\begin{aligned}{} & {} \Vert S_w^{\varphi ,\psi }f-f\Vert _p\\{} & {} \quad \le M_0(\varphi )\left( 2\Vert \psi \Vert _1\right) ^{\frac{p-1}{p}} \\{} & {} \qquad \times \left[ \left( \displaystyle M_0(\psi )\cdot \frac{\Vert \varphi \Vert _1+{\widetilde{M}}_p(\varphi )}{M_0(\varphi )}\right) ^\frac{1}{p}+\left( \Vert \psi \Vert _1+{\widetilde{M}}_p(\psi )\right) ^\frac{1}{p}\right] \omega \left( f,\frac{1}{w}\right) _p, \end{aligned}$$

for every sufficiently large $w>0$.

Proof

Similarly to the first part of Theorem 3.1, and using the Minkowsky inequality, we have

$$\begin{aligned} \begin{aligned} \Vert S_w^{\varphi ,\psi }f-f\Vert _p&\le \left[ \int _{{\mathbb {R}}}\left( \sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\int _{{\mathbb {R}}}|\psi (wu-k)|\right. \right. \\&\quad \times \left. \left. \left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) ^pdx\right] ^{\frac{1}{p}} \\&\quad +\left[ \int _{{\mathbb {R}}}\left( \sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\int _{{\mathbb {R}}}|\psi (wu-k)|\right. \right. \\&\quad \times \left. \left. \left| f\left( u+x-\frac{k}{w}\right) -f(x)\right| du\right) ^pdx\right] ^{\frac{1}{p}} \\&:=I_1+I_2. \end{aligned} \end{aligned}$$

Now we focus on $I_1$. Applying Jensen inequality twice

$$\begin{aligned} \begin{aligned} I_1^p&=\int _{{\mathbb {R}}}\left( \sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\int _{{\mathbb {R}}}|\psi (wu-k)|\left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) ^pdx \\ {}&\le M_0(\varphi )^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|\left( w\int _{{\mathbb {R}}}|\psi (wu-k)|\left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) ^pdx \\ {}&= M_0(\varphi )^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w \left( \Vert \psi \Vert _1\int _{{\mathbb {R}}}\frac{|\psi (wu-k)|}{\Vert \psi \Vert _1}\right. \\ {}&\quad \times \left. \left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| du\right) ^pdx \\ {}&\le M_0(\varphi )^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w \left[ \Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}|\psi (wu-k)|\right. \\ {}&\quad \times \left. \left| f\left( u+x-\frac{k}{w}\right) -f(u)\right| ^pdu\right] dx. \end{aligned} \end{aligned}$$

Now, putting $y=x-\frac{k}{w}$ and using Fubini–Tonelli theorem, we have

$$\begin{aligned} \begin{aligned} I_1^p&=M_0(\varphi )^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wy)|w\left[ \Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}|\psi (wu-k)|\left| f\left( u+y\right) -f(u)\right| ^pdu\right] dy \\&=M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}w|\varphi (wy)|\left[ \int _{{\mathbb {R}}}\left[ \sum _{k\in {\mathbb {Z}}}|\psi (wu-k)|\right] \left| f\left( u+y\right) -f(u)\right| ^pdu\right] dy \\&\le M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\psi )\int _{{\mathbb {R}}}w|\varphi (wy)|\,\omega \left( f,|y|\right) ^p_pdy \\&\le M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\psi )\,\omega \left( f,\frac{1}{w}\right) ^p_p\int _{{\mathbb {R}}}w|\varphi (wy)|\left( 1+w|y|\right) ^pdy, \end{aligned} \end{aligned}$$

where in the last step we have used the well-know inequality¹:

$$\begin{aligned} \omega (f,\lambda \delta )_p\le (1+\lambda )\,\omega (f,\delta )_p,\qquad \lambda ,\delta >0. \end{aligned}$$

(7)

Exploiting again the convexity of $|\cdot |^p$ with $p\ge 1$, we obtain

$$\begin{aligned} \begin{aligned} I_1^p&\le M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\psi )\,\omega \left( f,\frac{1}{w}\right) ^p_p\cdot 2^{p-1}\int _{{\mathbb {R}}}w|\varphi (wy)|\left( 1+(w|y|)^p\right) dy \\&= M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\psi )\,\omega \left( f,\frac{1}{w}\right) ^p_p\cdot 2^{p-1}\\&\quad \times \left\{ \int _{{\mathbb {R}}}w|\varphi (wy)|dy+\int _{{\mathbb {R}}}w|\varphi (wy)|(w|y|)^pdy\right\} \\&= M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\psi )\,\omega \left( f,\frac{1}{w}\right) ^p_p\cdot 2^{p-1}\left( \Vert \varphi \Vert _1+{\widetilde{M}}_p(\varphi )\right) <+\infty , \end{aligned} \end{aligned}$$

for every $w>0$, where $\Vert \varphi \Vert _1$ and ${\widetilde{M}}_p(\varphi )$ are both finite in view of the hypothesis $\varphi \in L^1({\mathbb {R}})$ and (6) respectively.

Now, we analyze the second term $I_2$. Applying the change of variable $y=u-\frac{k}{w}$, Jensen inequality twice and Fubini-Tonelli theorem, we obtain

$$\begin{aligned} \begin{aligned} I_2^p&=\int _{{\mathbb {R}}}\left( \sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\int _{{\mathbb {R}}}|\psi (wy)|\left| f\left( x+y\right) -f(x)\right| dy\right) ^pdx \\&\le M_0(\varphi )^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|\left( w\int _{{\mathbb {R}}}|\psi (wy)|\left| f\left( x+y\right) -f(x)\right| dy\right) ^pdx \\&\le M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}\sum _{k\in {\mathbb {Z}}}|\varphi (wx-k)|w\left( \int _{{\mathbb {R}}}|\psi (wy)|\left| f\left( x+y\right) -f(x)\right| ^pdy\right) dx \\&\le M_0(\varphi )^{p-1}\Vert \psi \Vert _1^{p-1}M_0(\varphi )\int _{{\mathbb {R}}}w\left( \int _{{\mathbb {R}}}|\psi (wy)|\left| f\left( x+y\right) -f(x)\right| ^pdy\right) dx \\&=M_0(\varphi )^p\Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}w|\psi (wy)|\left( \int _{{\mathbb {R}}}\left| f\left( x+y\right) -f(x)\right| ^pdx\right) dy \\&\le M_0(\varphi )^p\Vert \psi \Vert _1^{p-1}\int _{{\mathbb {R}}}w|\psi (wy)|\,\omega \left( f,|y|\right) _p^pdy. \end{aligned} \end{aligned}$$

Now, using again the well-known inequality (7), the convexity of $|\cdot |^p$ and arguing as before, we get

$$\begin{aligned} \begin{aligned} I_2^p&\le M_0(\varphi )^p\Vert \psi \Vert _1^{p-1}\cdot 2^{p-1}\omega \left( f,\frac{1}{w}\right) _p^p\left( \Vert \psi \Vert _1+{\widetilde{M}}_p(\psi )\right) <+\infty , \end{aligned} \end{aligned}$$

for every $w>0$, where $\Vert \psi \Vert _1+{\widetilde{M}}_p(\psi )<+\infty$ by the assumptions.

Hence, the theorem is completely proved. $\square$

Remark 4.1

Note that in Theorem 4.1, we do not use condition (${\mathcal {D}}_\alpha$) but the stronger (although more classical) assumption (6) since we can obtain, thanks to (7), a sharper result.

As made in the general frame of Orlicz spaces, we can directly deduce a qualitative estimate on the order of approximation also in this particular case of $L^p$-spaces. Here, we recall the definition of Lipschitz class of Zygmund-type, which is defined as follows:

$$\begin{aligned} Lip(\nu ,p):=\left\{ f\in L^p({\mathbb {R}}):\Vert f(\cdot +t)-f(\cdot )\Vert _p={\mathcal {O}}(|t|^\nu ),\,\text { as }|t|\rightarrow 0\right\} , \end{aligned}$$

(8)

with $0<\nu \le 1$ and $1\le p <+\infty$ (see, e.g., [7]).

Now, we can establish the following result as a consequence of Theorem 4.1.

Corollary 4.1

Let $\varphi$ and $\psi$ be such that

$$\begin{aligned} {\widetilde{M}}_p(\varphi )+{\widetilde{M}}_p(\psi )<+\infty , \end{aligned}$$

for some $p\in [1,+\infty [$. Then, for every $f\in Lip(\nu ,p)$, with $0<\nu \le 1$, we have

$$\begin{aligned} \Vert S_w^{\varphi ,\psi }f-f\Vert _p\le & {} M_0(\varphi )\left( 2\Vert \psi \Vert _1\right) ^{\frac{p-1}{p}}\\{} & {} \times \left[ \left( \displaystyle M_0(\psi )\cdot \frac{\Vert \varphi \Vert _1+{\widetilde{M}}_p(\varphi )}{M_0(\varphi )}\right) ^\frac{1}{p}+\left( \Vert \psi \Vert _1+{\widetilde{M}}_p(\psi )\right) ^\frac{1}{p}\right] C w^{-\nu }, \end{aligned}$$

for every sufficiently large $w>0$, where the constant $C>0$ arising from definition (8).

For examples of kernel functions satisfying all the required assumptions, see, e.g., [21].