A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

Angeloni, Laura; Vinti, Gianluca

doi:https://doi.org/10.1155/2016/3843921

Journal of Function Spaces

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Function Spaces, Approximation Theory, and Their Applications

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Volume 2016 | Article ID 3843921 | https://doi.org/10.1155/2016/3843921

A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

Laura Angeloni¹and Gianluca Vinti¹

Academic Editor: Henryk Hudzik

Received16 May 2016

Accepted20 Jul 2016

Published26 Sept 2016

Abstract

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional concept of -variation in the sense of Tonelli are also presented.

1. Introduction

The aim of the present paper is to give a review on recent results about convergence of integral operators of convolution type with respect to some concepts of multidimensional variation. We will consider the case of classical convolution integral operators of the formwhere and is a family of approximate identities, as well as the case of Mellin integral operators of the form where , , and . The above operators are of convolution type with respect to the homothetic operator and the measure , where is a Borel subset of (which is an invariant measure with respect to the multiplicative operation).

An important tool in order to frame the results of the paper is the setting of the functional spaces we deal with. The BV-spaces, apart from the well-known importance from the mathematical point of view, also play an important role in problems of Image Reconstruction where some of the various approaches make use of integral operators of convolution type (see, e.g., sampling-type operators).

The working space will be the space of functions of bounded multidimensional variation in the sense of Tonelli (defined in Section 2) and, as further extension, in order to deal with a larger class of functions, we will consider the space , where is a -function (see Section 2). We point out that, due to the necessary assumptions on the -function (Assumption ii), the case of BV cannot be obtained as particular case of . This is the reason why the two settings have to be treated independently.

For the above classes of operators we will provide estimates, convergence results, and also a characterization of the absolutely (-absolutely) continuous functions in terms of the respective convergence in variation. Moreover, the rate of approximation has been considered and examples of kernel functions to which the results can be applied are also furnished. Finally, also the nonlinear case for both the convolution and the Mellin-type operators has been considered.

Apart from the well-known importance of the classical convolution integral operators, the Mellin operators are very interesting and widely studied in approximation theory (for the basic theory see [1, 2] while, for results about similar homothetic-type operators, see, e.g., [3–16]), also because of their important applications in several fields. Among them, for example, we recall that Mellin analysis is deeply connected with some problems of Signal Processing, in particular with the so-called Exponential Sampling, which have applications in various problems of engineering and optical physics (see, e.g., [17–20]).

Concerning now the multidimensional concept of -variation introduced in [21], we point out that, due to the lack of an integral representation of the -variation for -absolutely continuous functions as happens for the classical variation, the major results about convergence require a different approach and suitable techniques. In particular, this holds for the convergence of the -modulus of continuity. Again, a similar problem occurs in case of Mellin integral operators on , where the Tonelli integrals are defined via the log-measure. Moreover, in the latter case, in order to prove that the operators are absolutely continuous, in case of regular kernels (this is a crucial point to obtain the characterization of absolute continuity), one has to pass through an equivalent notion of absolute continuity (for the log-absolute continuity, see Section 4) compatible with the setting of equipped with the log-measure.

We finally remark that, of course, all the results of the paper contain, in particular, the one-dimensional case (see [22–27]).

2. Preliminaries and Some Concepts of Variation

We will now recall the multidimensional concept of variation in the sense of Tonelli. Such definition was introduced by Tonelli [28] for functions of two variables and then extended to dimension by Radó [29] and Vinti [30].

Let us introduce some notations. If we are interested in the th coordinate of a vector , we will write so that, for a function , there holds Given an -dimensional interval , by we will denote the -dimensional interval obtained by deleting the th coordinate from ; namely,

Definition 1. A function is said to be of bounded variation if the sections of are a.e. of bounded variation on and their variation is summable; that is, (the usual Jordan one-dimensional variation of the th section of ) is finite a.e. and , for every .

In order to compute the variation of on an interval , the first step is to define the -dimensional integrals (the so-called Tonelli integrals) Let now be the Euclidean norm of the vector ; that is, where we put if for some .

The variation of on an interval is defined aswhere the supremum is taken over all the families of -dimensional intervals which form partitions of .

Finally, the variation of over the whole is defined aswhere the supremum is taken over all the intervals .

Bywe will denote the space of functions of bounded variation on .

We recall that it can be proved that if then exists a.e. in and (see, e.g., [29, 30]).

We point out that, in the multidimensional setting, it is natural to consider functions of bounded variation within the Lebesgue space : indeed this is analogous to the distributional concept of variation given by Cesari [31] and, in equivalent forms, by Krickeberg [32], De Giorgi [33], Giusti [34], and Serrin [35]. We notice that the definition of variation in the sense of Tonelli is equivalent to the distributional one in the class of functions which satisfy some approximate continuity properties (see, e.g., [30]).

We will now recall the concept of absolute continuity in sense of Tonelli.

Definition 2. A function is locally absolutely continuous () if, for every interval and for every , the th section is (uniformly) absolutely continuous for almost every .

Similarly to the one-dimensional case, it is possible to prove that if , then(see, e.g., [29, 30]).

We will denote by the space of all the functions .

In the following we will present also results about convergence for a family of Mellin integral operators. In order to study such kind of operators the most natural way is to consider (the domain of the functions where Mellin operators act), as a group with the multiplicative operation (instead of the additive operation on ) and equipped with the logarithmic Haar-measure ( is a Borel subset of and , ), instead of the usual Lebesgue measure. For this reason, in order to obtain results in BV-spaces for such kind of operators, it seems natural to adapt the definition of the Tonelli variation to this frame: we therefore introduced in [36] a new concept of multidimensional variation in which, in the Tonelli integrals, the Lebesgue measure is replaced by the logarithmic measure .

Definition 3. One will say that is of bounded variation on if the sections are of bounded variation on a.e. and , where denotes the space of the functions such that .

In order to define the multidimensional variation on , for a fixed interval we consider the -dimensional integralswhere denotes the product . The remaining steps for the definition of the variation follow as before.

For the sake of simplicity, we will use the same notations for the variation in both the cases of and : as it is natural, when one works on , it is intended that the measure used is the logarithmic one.

The classical definition of Jordan variation [37] was extended in the literature in several directions: one of the first generalizations was the quadratic variation introduced by Wiener [38], extended to the -variation, [39, 40], and later to the concept of -variation. The -variation was first introduced by Young [41] and then extensively studied by Musielak and Orlicz and their school (see, e.g., [22, 42–49]).

From now on we will assume that is as follows.

Assumption i. is a convex -function, where a -function is a continuous, nondecreasing function on , such that , for , and .

Assumption ii. as

We recall that (see [22]) the -variation of on is defined aswhere the supremum is taken over all the partitions of the interval , and

Definition 4. A function is said to be of bounded -variation () if , for some .

The Musielak-Orlicz -variation was generalized to the multidimensional frame in [50] following the approach of Vitali. However, in order to study approximation problems, the approach of the Tonelli variation seems to be the most natural in this context. For such reason in [21] we introduced a concept of multidimensional -variation inspired by the Tonelli and C. Vinti approach.

Again, the crucial point is to define, for , the Tonelli integrals: in this case we putwhere is the (one-dimensional) Musielak-Orlicz -variation of the jth section of .

Putting now the multidimensional -variation of on an interval is defined as (the supremum is taken over all the partitions of ) and, finally,Bywe will denote the space of functions of bounded -variation over .

Similarly to the classical variation, it is natural to introduce a concept of multidimensional -absolute continuity.

Definition 5. One says that is locally -absolutely continuous () if it is -absolutely continuous in the sense of Tonelli; that is, for every and , the section is (uniformly) -absolutely continuous for almost every .

Here (see [22]) a function is -absolutely continuous if there exists such that the following property holds: for every , there exists for which for all the finite collections of nonoverlapping intervals , , such that

By we will denote the space of all the functions .

As before, in order to obtain results for Mellin integral operators in -spaces in the multidimensional frame, we adapted the previous definition of -variation in the sense of Tonelli to the case of functions defined on equipped with the logarithmic measure . In such concept of multidimensional -variation, introduced in [51], the Tonelli integrals (13) are replaced byAgain, we will use the same notations, as in the case of the variation, for functions defined on both and .

3. Approximation Results for Convolution Integral Operators

In this section we will present results about approximation in variation by means of the convolution integral operators, namely, , for . Here is a family of approximate identities (see, e.g., [52]); that is, is a measurable essentially bounded function such that for an absolute constant and , for every ; for any fixed , , as .

In the following we will say that if and are satisfied.

Of course the operators are well-defined for every and therefore in particular for every function , since

We first recall that the family of operators map into itself. Indeed, the following estimate holds.

Proposition 6 (see [25]). Let If satisfies , then, where is the constant of Assumption .

Remark 7. In the case of nonnegative kernels , Proposition 6 gives the “variation diminishing property” for the operators : indeed in this case , .

In order to obtain the main result about convergence in variation, the following estimate of the error of approximation is essential.

Proposition 8 (see [25]). If then, for every ,

Another important tool is a characterization of the convergence for the modulus of smoothness of , defined aswhere , for every , is the translation operator (see, e.g., [6, 53]).

Theorem 9 (see [25]). Let . Then if and only if

The proof of the sufficient part of this result is a consequence of integral representation (9) of the variation for absolutely continuous functions and of the continuity in of the translation operator. For the necessary part, in [25] it is proved that if the kernel functions are absolutely continuous (as it happens in the most common cases), then also the integral operators belong to . Then, since is a closed subspace of with respect to the convergence in variation [25] and by estimate (23), in case of regular kernels, the convergence of the modulus of smoothness implies that .

By means of Proposition 8 and Theorem 9 it is possible to obtain the main result about convergence for absolutely continuous functions.

Theorem 10 (see [25]). If and , then

Remark 11. We point out that the assumption of absolute continuity of the function is crucial to obtain the main convergence theorem and such result does not hold, in general, if, for example, . For example, in the case , let us consider defined by First of all we point out that as . Let us now consider the Poisson-Cauchy kernel defined asThenand thereforeThis implies that and hence as

In case of regular kernels, by the closure of in , the converse of Theorem 10 is also true. Hence we obtain the following characterization of the space of the absolutely continuous functions, similarly to what happens in the one-dimensional case for the Jordan variation.

Theorem 12 (see [25]). Assume that and . Then if and only if

The previous results were generalized to the case of the multidimensional -variation in [21]. In particular, besides a kind of variation diminishing property, in [21] the following estimate for the error of approximation is obtained.

Proposition 13 (see [21]). Let and let be such that is satisfied. Then, for every and for every ,where is the -modulus of smoothness of .

Using the previous estimate, the main convergence result follows by the singularity assumption on the kernel functions and by the convergence for the -modulus of smoothness.

Theorem 14 (see [54]). Let . Then there exists such that if and only if .

The convergence for the modulus of smoothness, in case of the Tonelli variation, is a direct consequence of the integral representation of the variation for absolutely continuous functions (9). On the contrary, for the -variation there are no results of this kind and, in order to get the convergence in -variation of the -modulus of smoothness, it is necessary to use a different technique. In particular (see [54]), the crucial point is to construct a kind of “step” functions that approximate the function and for which a convergence result can be proved.

Using Theorem 14, it is possible to obtain the main result of convergence in -variation for the convolution integral operators .

Theorem 15 (see [21]). If and , then there exists such that

To give a sketch of the proof, the starting point is the estimate of Proposition 13 for the error of approximation . By Theorem 14, we have that tends to 0, for sufficiently small , while, by Assumption on the kernel functions, in correspondence with such small , converges to 0 for large enough; hence the result follows, taking into account and the fact that .

As before, in case of regular kernels the converse of Theorem 15 is also true.

Theorem 16 (see [21]). Let and let . Then if and only if there exists such that

Remark 17. We point out that it is easy to find examples of kernel functions to which the previous results can be applied. Among them, for example, the Gauss-Weierstrass kernel is defined as or the Picard kernelwhere is the Gamma Euler function (see Figure 1).

(a)

(b)

4. Approximation Results for Mellin Integral Operators

We now turn our attention to Mellin integral operators, defined as , where , . On the kernel functions we assume that is a measurable essentially bounded function such that , for an absolute constant and , for every ; for every fixed , , as ,that is, the assumptions of approximate identities, adapted to the present setting of . If satisfy and , we will write .

For Mellin integral operators it is possible to develop an “approximation theory” similar to the case of the convolution integral operators; however, one of the main differences is the homothetic structure of which leads to the choice of the logarithmic measure and also to the necessity to adapt some definitions. For example, the modulus of smoothness of has to be now defined as, where , for every , is the homothetic operator and is the unit vector of . Such notion of modulus of smoothness is the natural generalization, in the present frame of , of the classical modulus of continuity (see, e.g., [6, 21, 25]).

Of course, due to the presence of the logarithmic measure, we cannot use the integral representation of the Tonelli variation; nevertheless, it is possible to directly prove a result of convergence in variation for the modulus of smoothness in case of AC-functions, using a kind of “separated” variations (, ) which take into account just a single direction instead of all the directions.

Theorem 18 (see [36]). If , then

By means of the previous result and an estimate for the error of approximation analogous to Proposition 13, it is possible to prove the following convergence result.

Theorem 19. Let and . Then

A natural question now is whether, at least in case of AC-kernels, the converse of the previous result holds, as in the case of convolution operators. Actually, due to the form of the operators , such question is now much more delicate and direct approach cannot be used. In order to solve the problem, it is necessary to use another concept of absolute continuity (the log-absolute continuity), equivalent to the classical one, which takes into account the logarithmic measure . We first present the definition in the one-dimensional case.

Definition 20 (see [55]). One says that is log-absolutely continuous on () if for every there exists such that, for every collection of nonoverlapping intervals in such that then By we will denote the space of functions which are of bounded variation on and log-absolutely continuous on , for every .

Now, in the general multidimensional frame, is log-absolutely continuous on if, for every , the th sections of , , are (uniformly) log-absolutely continuous for almost every .

By means of the definition of the log-absolute continuity, in [55] it is proved that Mellin integral operators, as the classical convolution operators, preserve absolute continuity: this, together with the fact that the set of the absolutely continuous functions is a closed subspace of the set of the BV-functions, allows us to obtain the following characterization.

Theorem 21 (see [55]). Let and . Then if and only if .

In [56] approximation properties for Mellin integral operators were studied in the frame of -spaces, using the multidimensional version of the -variation on introduced in [51]. In particular the following theorem is obtained.

Theorem 22 (see [56]). Let and . Then there exists a constant such that

The proof of Theorem 22 is based on the estimate of the error of approximation [56],and on a convergence result for the -modulus of smoothness . This last result was obtained in [51] by means of a direct approach: the function is approximated in -variation by two auxiliary functions, constructed on a grid on which their sections are piecewise constant.

In order to prove the converse of Theorem 22, it is again necessary to use a concept of logarithmic -absolute continuity, which takes into account the homothetic structure of . We report below the definition in the one-dimensional case, while for the multidimensional case it is sufficient to proceed as for the log-absolute continuity.

Definition 23 (see [57]). One says that is log--absolutely continuous on if there exists such that, for every , there exists for whichfor all finite collections of nonoverlapping intervals , , such that

The log--absolute continuity is equivalent to the -absolute continuity and allows obtaining the characterization of in terms of convergence in -variation of Mellin integral operators.

Theorem 24 (see [57]). Let and . Then if and only if there exists such that .

Remark 25. We point out that taking as particular case of Theorem 24 we obtain the characterization of -absolute continuity in the one-dimensional case, namely, for the classical Musielak-Orlicz -variation.

Remark 26. It is not difficult to find examples of kernel functions which fulfill Assumptions () and (). Among them, for example, the moment-type kernels (or average kernels) are defined asIt is easy to see that they fulfill Assumption . Moreover, for every , implies that there exists an index such that ; hence Therefore as ; that is, also is satisfied.

Other families of kernel functions to which the previous results can be applied are the Mellin-Gauss-Weierstrass kernels, defined as or the Mellin-Picard kernels, which are defined asWe point out that these definitions are the natural reformulations, in the present multiplicative setting of , of the classical Gauss-Weierstrass kernels and Picard kernels, respectively (see Remark 17).

5. Further Results

We will now give some hints about further approximation results that were obtained in BV-spaces.

First of all, an interesting problem is to study the nonlinear versions of operators and . We point out that the nonlinear case is much more delicate than the linear one and requires some ad hoc assumptions; on the other side, it not only is interesting from a mathematical point of view, being of course more general than the linear one, but also is important from the point of view of the applications. Indeed, there are several applicative problems that cannot be faced by means of linear processes; an example is furnished by some problems of Signal Processing.

The nonlinear version of the convolution integral operators iswhere is a family of measurable functions of the form for every , Here and with is a Lipschitz kernel for every ; that is, there exists such that

Moreover we assume that () is a measurable function such that , for some and for every , and , for every ; () for any fixed , , as ; () denoted by , , ,uniformly with respect to every (not trivial) bounded interval ; that is, for every there exists such that , for every and for every bounded interval .

Remark 27. We point out that Assumption is due to the nonlinear frame and of course it is obviously satisfied in the linear case (, ). Moreover it is not difficult to provide examples of kernels which fulfill all the previous assumptions. For example, we can consider the kernel functions , , , where are approximate identities,and the definition of is extended in odd way for (see Figure 2).

The problem of the convergence in variation for the nonlinear integral operators was faced in [58, 59]; in particular, the main convergence result reads as follows.

Theorem 28 (see [58]). If and satisfy , , then

Similar approximation results were obtained in [60] for the nonlinear convolution integral operators in the frame of and in [61] for the nonlinear version of the Mellin integral operators in .

We finally point out that, besides the problem of convergence, the rate of approximation has been also studied in all the previously mentioned settings. In order to do it, as it is natural, one has to introduce suitable Lipschitz classes which take into account the variation functional. We point out that, in order to approach the mentioned problem, the assumptions on kernels have to be slightly modified.

For example, let us consider the case of the convolution integral operators in the setting of . In this frame the Lipschitz class is defined as, and Assumption has to be replaced by the following: for any fixed , , as .

Moreover we will say that is an -singular kernel, for , iffor every . Then it is possible to obtain the following result about the order of approximation for the convolution integral operators .

Theorem 29 (see [25]). Let and let be an -singular kernel satisfying and . Moreover assume that there exists such that Thenas .

Similar results in the nonlinear case were obtained in [58], while, for results about the rate of approximation for convolution integral operators with respect to the multidimensional -variation, see [21] and [60] (nonlinear case).

The case of Mellin integral operators was studied in [36] and in [61] (nonlinear case) with respect to the Tonelli variation, while the case of the multidimensional -variation was studied in [56]. We finally refer to [62, 63] for approximation results in the slightly different setting of .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

The authors are Members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was supported by Department of Mathematics and Computer Sciences, University of Perugia. Moreover, the first author of the paper has been partially supported within the GNAMPA-INdAM Project 2016 “Problemi di Regolarità nel Calcolo delle Variazioni e di Approssimazione.”

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Copyright © 2016 Laura Angeloni and Gianluca Vinti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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