A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

1 Introduction

The aim of this paper is to give a unitary approach for the simultaneous study of discrete and integral operators. In order to do this, we need to introduce a general setting dealing with a family of linear continuous functionals acting on suitable functional spaces, consisting on functions defined in locally compact Hausdorff topological groups. More precisely, we introduce the following linear integral operators:

where H and G are locally compact Hausdorff topological groups provided with their regular Haar measures μH and μG, respectively, f∈M⁢(G) (the space of measurable functions on G) is such that the above integral defining Tw is well defined, hw:H→hw⁢(H)⊂G are homeomorphisms, (Lhw⁢(t))t∈H is a suitable family of linear operators Lhw⁢(t):M⁢(G)→ℝ, and (χw)w>0 is a family of kernel functionals χw:G→ℝ.

For the above family (1.1), we establish a uniform approximation result for the aliasing error Tw⁢f-f (Theorem 4.2), and also a modular convergence result in the general setting of Orlicz spaces (Theorem 4.7). As pointed out in the paper, this latter result implies convergence in Lp-spaces and in other particular Orlicz spaces as, for example, interpolation spaces, exponential spaces and others.

In order to obtain the general result of Theorem 4.7 we first test the modular convergence for continuous functions with compact support (Theorem 4.3), we establish a modular continuity of family (1.1) (Proposition 4.5), and, using a density result (Proposition 4.6), we are able to prove the main result.

It is worthwhile noting that the operators (1.1) contain various families of discrete and integral operators, very well known in the literature. Among them, by putting G=ℝ, H=ℤ and hw⁢(k)=tk/w, where (tk)k is a suitable sequence of real numbers, there are the generalized sampling series (see [11, 17, 18, 7, 3])

where Ltk/w⁢f=f⁢(tkw), the Kantorovich sampling series (see [8, 52, 23, 53, 24, 54])

where

and the Durrmeyer generalized sampling series (see [5])

where

and ψ is a suitable kernel function.

Note that also perturbated versions of the above operators can be produced, as shown in Section 3.

Moreover, different choices of the topological groups, the corresponding Haar measures and the families hw and Lhw⁢(t) produce classical convolution operators (see [15]) and the Mellin convolution operators (see [1, 2]), as shown in Section 5, together with their multidimensional versions. Since each of the above operators plays an important role in approximation theory and in sampling and signal theory, it seems very useful to have at disposal a unifying approach. Note that the study of the generalized sampling operators has been started by P. L. Butzer and his school at Aachen (see [11, 17, 12, 18, 13, 48, 14, 16]), while the Kantorovich sampling series has been introduced and studied in, e.g., [8, 52, 53]. Both are important from a mathematical point of view and also in signal reconstruction: indeed, the generalized sampling operators represent extensions of the classical sampling theorem, and their Kantorovich version has an important meaning in the applications in signal processing since, by replacing the value f⁢(tkw) with the average of f in a small interval close to tkw (i.e., with the value w⁢∫tk/wtk+1/wf⁢(u)⁢𝑑u), they reduce the “time-jitter” errors. For related references on the above operators, including connections with the classical sampling, see, e.g., [6, 55, 46, 37, 29, 34, 35, 28, 36, 39, 7, 38, 31, 10, 50, 51, 23, 24, 49, 20, 19, 25, 26, 27].

In Section 5, we will show that, for the above particular cases of operators, the assumptions of our approach are all satisfied.

In Section 6, we will present some graphical representations showing the approximation of the considered operators to the function f.

Finally, we point out that the case of Durrmeyer generalized sampling series is important since it represents a case of operators for which continuous functions with compact support are approximated modularly and not with respect to the norm convergence in the Orlicz space, i.e., for which Corollary 4.4 cannot be established and assumption (L3) (given in Section 4) becomes not trivial.

2 Preliminaries

We review some basic concepts and notations which will be used throughout the paper. Let (H,+) be a locally compact Hausdorff topological group, with θH as neutral element. It is well known (see [41, 43]) that there exists a unique (up to a multiplicative constant) left (resp. right) translation invariant regular Haar measure μH (resp. νH). In general, if A is a Borel set of H and if -A={-a∣a∈A}, then μH⁢(-A)=k⁢νH⁢(A) for some nonzero constant k∈ℝ. The measures μH and νH coincide if and only if A is an unimodular group. In this case, one has μH⁢(A)=νH⁢(A)=μH⁢(-A) for every Borel set A⊂H. Compact groups, abelian groups and discrete groups are well-known examples of unimodular groups.

Now, let (G,+) be a locally compact abelian topological group. Let ℬ be a local base of its neutral element θG. Let M⁢(G) denote the set of measurable bounded functions f:M→ℝ. By C⁢(G) (resp. Cc⁢(G)) we denote the subsets of M⁢(G) consisting on uniformly continuous and bounded (resp. continuous with compact support) functions f:G→ℝ. It is understood that C⁢(G) and Cc⁢(G) are equipped with the standard ∥⋅∥∞-norm.

Let φ:ℝ0+→ℝ0+ be a continuous function. It is called a φ-function if the following conditions hold:

1. φ ⁢ ( 0 ) = 0 and φ⁢(x)>0 for all x>0;

2. φ is non-decreasing on ℝ0+;

3. lim x → + ∞ ⁡ φ ⁢ ( x ) = + ∞ .

If a φ-function φ is chosen, the functional

is called modular functional on M⁢(G).

The Orlicz space Lφ⁢(G) is the subset of M⁢(G) consisting of those functions f∈M⁢(G) such that

for some λ>0, equipped with the (strong) norm

The norm ∥⋅∥φ is called Luxemburg norm. Anyway, the most natural notion of convergence in Lφ⁢(G), which is weaker than the norm convergence, is the modular convergence: a sequence (fn)⊂Lφ⁢(ℝ) converges modularly to a function f∈Lφ⁢(ℝ) if there exists a number λ>0 such that

It is not difficult to see that for a sequence to converge strongly in Lφ⁢(G) it is necessary and sufficient that the above limit is true for everyλ>0. However, in some important cases the modular convergence and the Luxemburg norm convergence on Lφ⁢(G) are equivalent: this happens if and only if the so-called Δ2-condition is fulfilled, namely if there exists a number M>0 such that

It can be shown that the Δ2-condition is satisfied if and only if

For references on Orlicz spaces, see, e.g., [40, 42, 44, 45, 7].

Orlicz spaces arise as a natural generalization of Lp spaces, and in fact if 1≤p<∞ and φ⁢(x)=xp, then the Orlicz space generated by IGφ is exactly the Lebesgue space Lp⁢(G). The function φ⁢(x)=xp satisfies the Δ2-condition, hence modular convergence is equivalent to the strong Luxemburg one, and they are in turn equivalent to the standard Lp-norm. However, there are many other examples of Orlicz spaces which play an important role in many different situations like PDEs and functional analysis. We mention the so-called exponential spaces [33], where the φ-function generating the Orlicz space is given by φα⁢(x)=exp⁡(xα)-1 for a certain number α>0. The exponential space Lφα⁢(G) is an example in which the modular convergence is not equivalent to the Luxemburg one, and indeed φα does not satisfy the Δ2-condition. Another interesting example is furnished by the so-called interpolation spaces or Zygmund spaces [47, 9, 30]. The generating φ-function is given by φα,β⁢(x)=xα⁢lnβ⁡(e+x) for fixed α≥1 and β>0. The function φα,β satisfies the Δ2-condition, hence in Lφα,β⁢(G) the modular and the Luxemburg convergence are equivalent.

3 Basic Assumptions and Examples

Let H and G be locally compact Hausdorff topological groups with regular Haar measures μH and μG, respectively. We require that G is abelian (but, a priori, this assumption is not required on H). Assume further that for every w>0 there exists a map hw:H→G which restricts to a homeomorphism from H to hw⁢(H). Along this section and Section 4 we will make assumptions on both the family (Lhw⁢(t))t∈H and on the kernel functions (χw)w>0. In Section 5, we will show several examples of operators for which these assumptions are all satisfied.

For every w>0, let (Lhw⁢(t))t∈H be a family of linear operators Lhw⁢(t):M⁢(G)→ℝ such that the following conditions hold:

1. When restricted to L∞⁢(G), the operators Lhw⁢(t) are uniformly bounded, i.e., if

   L ~ h w ⁢ ( t ) := L h w ⁢ ( t ) ↿ L ∞ ⁢ ( G ) : L ∞ ( G ) → ℝ ,

   then

   ∥ L ~ h w ⁢ ( t ) ∥ := sup ∥ f ∥ ∞ ≤ 1 ⁡ | L h w ⁢ ( t ) ⁢ f | ≤ Υ < ∞

   for some positive constant Υ and for every w>0 and t∈H.

2. If f∈C⁢(G), then the family (Lhw⁢(t))t∈Hpreserves the continuity of f in the sense that for every ε>0 there exist w¯⁢(ε)>0 and a set Bε∈ℬ such that if z-hw⁢(t)∈Bε and w>w¯, then |Lhw⁢(t)⁢f-f⁢(z)|<ε.

Further, let (χw)w>0⊂L1⁢(G) be a family of kernel functionals χw:G→ℝ which is uniformly bounded in L1⁢(G) (i.e., ∥χw∥1<Γ for some constant Γ>0 and for every w>0).

Let us introduce some notation which will be useful from now on. For w>0 and z∈G, if A⊂G is a measurable set, we define

The family (χw)w is assumed to be chosen in such a way that

1. t ↦ χ w ⁢ ( z - h w ⁢ ( t ) ) ∈ L 1 ⁢ ( H ) for every z∈G and w>0;

2. for every w>0 and z∈G,

   ∫ H χ w ⁢ ( z - h w ⁢ ( t ) ) ⁢ 𝑑 μ H ⁢ ( t ) = 1 ;

3. for every w>0,

   M := sup z ∈ G ⁡ ∫ H | χ w ⁢ ( z - h w ⁢ ( t ) ) | ⁢ 𝑑 μ H ⁢ ( t ) < ∞ ;

4. if w>0, z∈G and B∈ℬ, then

   lim w → + ∞ ⁡ ∫ H ∖ B w , z | χ w ⁢ ( z - h w ⁢ ( t ) ) | ⁢ 𝑑 μ H ⁢ ( t ) = 0

   uniformly with respect to z∈G;

5. for every ε>0 and for every compact set K⊂G, there exist a symmetric compact set C⊂G such that, for every t∈Kw, we have

   ∫ G ∖ C Υ w ⁢ ( K ) ⁢ | χ w ⁢ ( z - h w ⁢ ( t ) ) | ⁢ 𝑑 μ G ⁢ ( z ) < ε

   for sufficiently large w>0.

For w>0, we study the operator Tw:M⁢(G)→R defined by

where f∈M⁢(G) is such that the above integral defining Tw is well defined.

There are a lot of known examples of operators which can be expressed in the form (3.1). We describe some (but certainly not all) kinds of operators which are included in the general setting introduced by (3.1).

(1) If G=ℝ, H=ℤ with the counting measure d⁢μH⁢(t), then (3.1) becomes a series, namely

This type of series has been introduced in [54]. According to the form of the operators Lhw⁢(k), we obtain various types of the so-called sampling series.

If, for example, hw⁢(k)=tk/w∈ℝ, and (tk)k is an increasing sequence of real numbers such that

we can define Lhw⁢(k):M⁢(ℝ)→ℝ by

The operator Tw⁢f translates into the generalized sampling series, namely

Instead, we can define Ltk/w⁢f:M⁢(ℝ)→ℝ as

In this case, operators (3.1) give rise to the Kantorovich sampling series

In Tw(2)⁢f the sampling values f⁢(tk/w) are replaced by averages, with the effect of reducing the so-called time-jitter errors and of regularizing the series, allowing to study the convergence in a wider sense (see [8, 52, 21, 22]).

A more general class of sampling series is given by the so-called Durrmeyer generalized sampling series. Let ψ∈L1⁢(ℝ) be such that

and let

We now have the series

Perturbations can be added to the family of operators Lhw⁢(k) to emulate the effect of the so-called time-jitter errors: so, for instance we can take Lhw⁢(k)=f⁢(tk/w+jk⁢(w)), where limw→+∞⁡jk⁢(w)=0 uniformly with respect to k∈ℤ, to obtain a ”time-jitter” generalized sampling series; see, e.g. [14, 53]. We can also produce series analogous to Tw(2)⁢f⁢(x) and Tw(3)⁢f⁢(x) by adding suitable time-jitter perturbations.

(2) If H=G=ℝ, then we retrieve some interesting integral operators together with their Kantorovich versions. For instance, we can take hw⁢(t)=t and Lhw⁢(t)⁢f=Lt⁢f=f⁢(t), and (3.1) translates into a convolution operator, namely

However, other types of integral operators can be generated: for example, if we take hw⁢(t)=t and

we obtain

which is a Kantorovich version of a convolution operator. Note that, whenever f∈Lloc1⁢(ℝ), the factor

converges, as w→+∞, to f⁢(t) for a.a. t∈ℝ (by the Lebesgue–Besicovich theorem). This fact testifies that, asymptotically, the operator Tw(5)⁢f can be compared to the standard convolution operator.

As another example, let us now consider H=G=ℝ+. In this case, the group operation is the product, and the only regular Haar measure on ℝ+ (up to multiplicative constants) is the logarithmic measure d⁢μ⁢(t)=d⁢tt. We can then set hw⁢(t)=t and Lhw⁢(t)⁢f=Lt⁢f=f⁢(t). Under these assumptions, we are left with

which is the Mellin convolution operator.

Moreover, we can modify the operators Lhw⁢(t)⁢f by taking

and obtain a Kantorovich version of the Mellin convolution, namely

Note that, for large values of w>0, the factor 12⁢ln⁡(1+1/w) can be replaced simply by w2.

Finally, the setting of locally compact topological groups allows us to cover also the multidimensional version of the above operators.

4 Approximation Results

In this section, we state and prove the results concerning the convergence of (3.1). Besides Theorem 4.2, which proves the convergence of (3.1) to f uniformly for f∈C⁢(G), Theorem 4.7 proves the convergence of (3.1) to f in a subspace 𝒴 of an Orlicz space Lφ⁢(G), where, from now on, φ is a convex φ-function. We will achieve this result by density, and by a number of preliminary (but important) facts which need to be proved. In the course of the discussion, we will also state two more assumptions on the family of operators (Lhw⁢(t))t∈H, which will in turn both restrict the admissible families (Lhw⁢(t))t∈H and define the subspace 𝒴 in which the main convergence result can be proved.

Now we prove the first result of convergence.

Several examples show that assumptions (L1)–(L2) alone do not guarantee the convergence of Tw⁢f in Orlicz spaces, even in the case when f∈Cc⁢(G). What we need is an assumption which describes the behavior of the family (Lhw⁢(t)) outside the support of a function f∈Cc⁢(G). We state this assumption as follows:

1. Let f∈Cc⁢(G) and let Supp⁡f=K~⊂G. We assume that there exist a compact set K⊃K~ and a constant α>0 such that for every ε>0 there exists w¯>0 such that

   ∫ H ∖ K w φ ⁢ ( α ⁢ | L h w ⁢ ( t ) ⁢ f | ) ⁢ 𝑑 μ H ⁢ ( t ) < ε

   for every w>w¯.

With the help of this property, we can prove the following theorem.

The limit (4.1) is valid when λ⁢M<α. If the family {Lhw⁢(t)} satisfies Lhw⁢(t)⁢f=0 whenever t∉Kw, then assumption (L3) is not necessary, and there is no reason for choosing a particular α in the proof of Theorem 4.3. Hence the operators Tw⁢f converge to f in the stronger Luxemburg norm. We state this result as follows.

We point out that if one wants to extend the convergence in Orlicz spaces to a family larger than Cc⁢(G), then one has technical difficulties to face. Let us show what happens: estimating Iφ⁢(λ⁢Tw⁢f) (λ>0), we obtain

by using again the Jensen inequality and the Fubini–Tonelli theorem. It is now clear that if we want to achieve a desired result of convergence in Orlicz space for a function f∈Lφ⁢(G), the last inequality in (4.3) should be compared to Iφ⁢(λ⁢β⁢f) for some β>0, in order to have a modular continuity of the operators. In general, this is not possible, and in fact one does not even know if the values Iφ⁢(λ⁢Tw⁢f) exist. Hence we need an additional assumption which determines a restriction on both the family {Lhw⁢(t)} and the functions f∈Lφ⁢(G) for which the convergence can be actually obtained. The assumption reads as follows:

1. Let φ be a convex φ-function. There exists a subspace 𝒴⊂Lφ⁢(G) with Cc∞⁢(G)⊂𝒴 such that for every f∈𝒴 and for every λ>0 there exist constants c=c⁢(λ,f),β=β⁢(λ,f)>0 satisfying

   lim sup w → + ∞ ⁡ ∥ χ w ∥ 1 ⁢ ∫ H φ ⁢ ( λ ⁢ | L h w ⁢ ( t ) ⁢ f | ) ⁢ 𝑑 μ H ⁢ ( t ) ≤ c ⁢ I φ ⁢ ( λ ⁢ β ⁢ f ) .

Later, we will discuss some examples for which (L4) is satisfied.

The first consequence of assumption (L4) is the fact that Tw maps 𝒴 into Lφ⁢(G); indeed, we have the following proposition.

We now need a density result (see, e.g., [4]).