Multidimensional sampling-Kantorovich operators in BV-spaces

1 Introduction

In recent years, the study of the approximation properties of Kantorovich-type operators has been a challenging topic, and a wide literature has been devoted to the subject (see, e.g., [1–8]). As it is well known, the pioneering idea goes back to Kantorovich [9], who introduced the operators ( B K n f ) ( x ) ≔ ( n + 1 ) ∫ k n + 1 k + 1 n + 1 f ( u ) d u p k , n ( x ) , x ∈ [ 0 , 1 ] , n ∈ N , a modified version of the classical Bernstein polynomials ( B n f ) ( x ) ≔ ∑ k = 0 n f k n p k , n ( x ) , p k , n ( x ) ≔ n k x k ( 1 − x ) n − k , replacing the values of the function over the points k n by means of an integral mean. Such operators, linked to the Bernstein polynomials by the relation ( B n f ) ′ ( x ) = ( B K n − 1 f ′ ) ( x ) in case of an absolutely continuous function f , allowed to obtain, in the L p -spaces, the analog of the classical Weierstrass approximation theorem in C ( [ 0 , 1 ] ) : indeed Lorentz [10] proved that ‖ B K n f − f ‖ L p ( [ 0 , 1 ] ) → 0 , as n → + ∞ .

The idea to replace the values of the function by means of an integral mean was applied to a wide range of operators, including the following generalized sampling operators:

where f : R → R is a bounded function and χ is a kernel (see Section 2). Such operators, introduced with the aim to provide a generalized version of the classical sampling theorem, have been widely studied in last 40 years, together with other related operators (see, e.g., [11–21]), also in view of their deep and natural connections to applicative problems of signal and image reconstruction. Their Kantorovich version was introduced in [22], where the authors obtain convergence results in the general setting of Orlicz spaces, and later on, the approximation properties of such operators were deeply investigated in several function spaces, as well as in multidimensional settings [23–26]. We point out that the multidimensional version of such operators

introduced in [24], proved to be very useful in order to solve some applicative problems of digital image reconstruction and processing (see, e.g., [27,28]). In this direction, among the several function spaces, the setting of the spaces of functions of bounded variation is of particular interest. Indeed, in [29], the problem of the estimate in variation for the operators { K w f } w > 0 is studied and an applicative interpretation of the variation diminishing-type estimates is given. Besides estimates in variation, it is natural to face the problem of the convergence in variation, that is the natural notion of convergence in BV-spaces. Results about convergence in variation have been obtained in the one-dimensional case (see [30] for { B K n } n ∈ N and [31] for { K w } w > 0 ), but the multidimensional case, much more delicate but nevertheless crucial for applications to digital images, is still an open problem.

In this article, we address this issue and obtain a result of convergence in variation (Theorem 3) in the general case of averaged-type kernels (see Section 2) for multidimensional sampling-Kantorovich operators { K w f } w > 0 .

We will work in the frame of BV-spaces in the sense of Tonelli. As it is well known, several generalizations of the Jordan variation to the multidimensional case have been proposed in the literature, including the distributional variation, the Vitali variation, the Cesari generalized variation, the Ascoli-Arzelà variation, and others. We refer to the monograph by Appell et al. [32] for an exhaustive presentation of the different notions of variation, also in the multidimensional frame. We choose to work with the variation introduced by Tonelli [33] for functions of two variables and later extended to the case of N -variables by Radó [34] and Vinti [35], since this concept seems to be very suitable in order to obtain approximation results for families of integral and discrete operators (see, e.g., [25,36–41]). Moreover, the natural geometrical aspects connected to the definition and construction of the Tonelli variation allow us to discuss the previously mentioned applicative issues about digital images [18,29].

In order to reach our goal, we will use an indirect approach. In particular, starting from a natural relation between the sampling-Kantorovich operators and the generalized sampling series applied to a singular integral (see (2) of Section 2), we will prove the main theorem (Theorem 3) using a convergence result for the singular integrals { I ψ w f } w > 0 (Theorem 2), together with an estimate in variation and a convergence result for the generalized sampling operators [18,19]. As it is natural, we have to work within a suitable subspace of B V ( R N ) (see Section 2).

The article is organized as follows: after a preliminary section where the main notations and definitions are presented (Section 2), the main results are proved in Section 3 and examples of kernels to which the results can be applied are presented in Section 4.

2 Preliminaries

We now recall the definition of the space B V ( R N ) in the sense of Tonelli [33–35].

For x = ( x 1 , … , x N ) ∈ R N , f : R N → R and I = ∏ i = 1 N [ a i , b i ] ⊂ R N , we will use the following notations:

1. x j ′ = ( x 1 , … , x j − 1 , x j + 1 , … , x N ) ∈ R N − 1 , x = ( x j ′ , x j ) and f ( x ) = f ( x j ′ , x j ) , j = 1 , … , N ;

2. I j ′ = [ a j ′ , b j ′ ] = ∏ i ≠ j [ a i , b i ] and I = [ a , b ] = [ a j ′ , b j ′ ] × [ a j , b j ] , j = 1 , … , N ;

3. α x = ( α x 1 , … , α x N ) , for α ∈ R , and, for α ≠ 0 , x α = x 1 α , … , x N α .

By V [ f ] ≔ sup [ a , b ] ⊂ R V [ a , b ] [ f ] , we denote the Jordan variation of f over R , where V [ a , b ] [ f ] = sup ∑ i = 1 n ∣ f ( x i ) − f ( x i − 1 ) ∣ , the supremum being taken over all the possible partitions a = x 0 < x 1 < … < x n = b of [ a , b ] , is the Jordan variation of f on [ a , b ] .

For I = ∏ i = 1 N [ a i , b i ] ⊂ R N and j = 1 , … , N , we define the so-called Tonelli integrals, namely the ( N − 1 ) -dimensional integrals, as follows:

where V [ a j , b j ] [ f ( x j ′ , ⋅ ) ] is the one-dimensional Jordan variation of the jth section of f , and their Euclidean norm Φ ( f , I ) ≔ ∑ j = 1 N Φ j 2 ( f , I ) 1 2 . As usual, Φ ( f , I ) = + ∞ if Φ j ( f , I ) = + ∞ for some j = 1 , … , N .

The variation of f on I ⊂ R N is defined as follows:

where the supremum is taken over all the finite families of N -dimensional intervals { J 1 , … , J m } , which form partitions of I . Moreover,

where the supremum is taken over all the intervals I ⊂ R N , is the Tonelli variation of f on R N .

We will also use the notation

so that V j [ f ] : R N − 1 → R , j = 1 , … , N , where f ( x j ′ , ⋅ ) are the jth sections of f .

It is obvious that, for every f ∈ B V ( R N ) , ∇ f exists a.e. in R N and ∂ f ∂ x j ∈ L 1 ( R N ) , for every j = 1 , … , N .

We now recall the definition of local absolute continuity on R N .

We will denote by A C ( R N ) ≔ B V ( R N ) ∩ A C loc ( R N ) the space of the absolutely continuous functions on R N . We recall that, for every f ∈ A C ( R N ) , V [ f ] = ∫ R N ∣ ∇ f ( x ) ∣ d x [34,35].

We will now introduce the family of sampling-Kantorovich operators, namely

where f : R N ⟶ R is a bounded function and χ : R N → R is a kernel.

The operators { K w f } w > 0 have been introduced in [24] as the Kantorovich version of the multivariate generalized sampling series

In [29], the following estimate in variation was obtained for the sampling-Kantorovich operators { K ¯ w m f } w > 0 , with kernels χ ¯ m of averaged type (see definition (1) below), proving that such operators map B V ( R N ) into itself.

In this article, we will go a step further and prove a result of convergence in variation for { K ¯ w m f } w > 0 . Obviously, the setting will be the same, namely we will consider sampling-Kantorovich operators with kernels χ ¯ m of averaged type, denoted by

We will similarly denote by S ¯ w m f the multivariate generalized sampling series of f with averaged kernel χ ¯ m .

Saying that the kernels are of averaged-type means that

where

for some m ∈ N , and χ i : R ⟶ R is a one-dimensional kernel for every i = 1 , … , N , i.e., it satisfies the following conditions:

( χ 1 ) χ i ∈ L 1 ( R ) is such that ∑ k ∈ Z χ i ( u − k ) = 1 for every u ∈ R ;

( χ 2 ) A χ i ≔ sup u ∈ R ∑ k ∈ Z ∣ χ i ( u − k ) ∣ < + ∞ , where the convergence of the series is uniform on the compact sets of R .

One can immediately verify that χ ¯ i , m is a kernel itself and that

for every i = 1 , … , N .

Moreover, χ ¯ m turns out to be a multidimensional kernel, namely it satisfies the multidimensional versions of ( χ 1 ) and ( χ 2 ) , that is,

( χ 1 N ) χ ¯ m ∈ L 1 ( R N ) is such that ∑ k ∈ Z N χ ¯ m ( u − k ) = 1 for every u ∈ R N ;

( χ 2 N ) A χ ¯ m ≔ sup u ∈ R N ∑ k ∈ Z N ∣ χ ¯ m ( u − k ) ∣ < + ∞ , where the convergence of the series is uniform on the compact sets of R N .

We refer to Section 4 for examples of kernels that fulfill all the above assumptions.

It is immediate to see that, if χ is a multidimensional kernel, then both the family of operators { K w f } w > 0 and { S w f } w > 0 are well-defined, for instance, for every bounded function, and therefore, for every f ∈ B V ( R N ) : indeed, if ∣ f ( t ) ∣ ≤ C for some C > 0 , by ( χ 2 ) , ∣ ( K w f ) ( t ) ∣ ≤ C ∑ k ∈ Z N ∣ χ ( w t − k ) ∣ ≤ C A χ for every t ∈ R N .

In order to prove the main result about the convergence in variation for { K ¯ w m f } w > 0 , we will use an indirect approach. In particular, it is well known that, in the one-dimensional case, it is possible to write the sampling-Kantorovich operators as the generalized sampling series of a singular integral (see, e.g., [31]). Here, we will make use of the analogous identity in the multidimensional case. In particular, let us denote by

t ∈ R N , w > 0 , the singular integral of f with kernel { φ w } w > 0 . The family φ w is, as usual, an approximate identity, namely it satisfies the following assumptions:

1. φ w ∈ L 1 ( R N ) , ‖ φ w ‖ L 1 ( R N ) ≤ A , for some constant A > 0 and ∫ R N φ w ( u ) d u = 1 , for every w > 0 ;

2. for every fixed δ > 0 , lim w → + ∞ ∫ ∣ u ∣ > δ ∣ φ w ( u ) ∣ d u = 0 .

Let us now consider ψ w ( t ) ≔ w N χ [ − 1 , 0 ] N ( w t ) , where χ [ − 1 , 0 ] N ( t ) = 1 , t ∈ [ − 1 , 0 ] N , 0 , otherwise , is the characteristic function of [ − 1 , 0 ] N . Then, it is immediate to see that { ψ w } w > 0 is an approximate identity (with A = 1 ) and

for every w > 0 , t ∈ R N .

Using such relation, we will prove the main convergence result by means of an estimate in variation [19], a convergence result for the generalized sampling operators [18] and for the singular integrals { I ψ w f } w > 0 (Theorem 2 of Section 3). In order to establish it, we have to introduce a subspace of L 1 ( R N ) (see [16]) and, of course, some notations.

We recall that an admissible partition over the i -th axis is a partition Σ i ≔ ( x i , j i ) j i ∈ Z such that

A sequence Σ = ( x j ) 𝚓 ∈ Z N ⊂ R N , x j = ( x 1 , j 1 , … , x N , j N ) , and j = ( j 1 , … , j N ) ∈ Z N , is said to be an admissible sequence if it is the cartesian product of admissible partitions Σ i = ( x i , j i ) j i ∈ Z . For a fixed admissible sequence Σ , the l p ( Σ ) -norm of f : R N → R is defined as follows:

where Q j = ∏ i = 1 N [ x i , j i − 1 , x i , j i [ and Δ j ≔ ∏ i = 1 N ( x i , j i − x i , j i − 1 ) denotes the volume of Q j .

The sampling grid, that is, the cartesian product of k i w k i ∈ Z , i = 1 , … , N , is indeed an admissible sequence: for its importance, it will be denoted by Σ w N . Similarly, by Σ w N − 1 , we will denote the cartesian product of k i w k i ∈ Z , i ≠ j , that is, the sampling grid on R N − 1 , excluding the jth coordinate.

Then, the subspace Λ p ( R N ) , p ≥ 1 , is defined as follows:

We recall that Λ p ( R N ) is a proper linear subspace of L p ( R N ) that contains, among others, all the measurable functions with compact support: this one and other properties of Λ p ( R N ) can be found in [16] and [42].

We will also use the following notation introduced in [19]: by B V Λ ( R N ) , we denote the space of functions f ∈ M ( R N ) such that the jth sections f ( x j ′ , ⋅ ) are of bounded variation on R for a.e. in x j ′ ∈ R N − 1 and V j [ f ] ∈ Λ 1 ( R N − 1 ) , for every j = 1 , … , N .

Of course, B V Λ ( R N ) is a subspace of B V ( R N ) and, for example, it contains all the functions of bounded variation with compact support.

Finally, R loc will denote the space of all the locally Riemann integrable functions f : R N → R .

3 Convergence in B V ( R N ) by means of sampling-Kantorovich operators

In this section, we will prove the main result, that is, the convergence in variation by means of sampling-Kantorovich operators. In order to do this, we will provide an estimate (Proposition 1) and a convergence result (Theorem 2) for the singular integrals, which will be an intermediate step to reach the main result.

Results about convergence in L p by means of singular integrals are well known (see, e.g., [43]). We will now study the operators { I ψ w } w in the subspace Λ p ( R N ) . First, we will state an estimate in B V Λ ( R N ) for the singular integrals

We will now prove a result of convergence in Λ p ( R N ) for the singular integrals I ψ w g , where ψ w ( t ) = w N χ [ − 1 , 0 ] N ( w t ) , t ∈ R N , w > 0 , which will be used in the proof of the main convergence result.