Approximation Properties of the Sampling Kantorovich Operators: Regularization, Saturation, Inverse Results and Favard Classes in L p -Spaces

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in L p -setting, 1 ≤ p ≤ +∞ , have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel inﬂuences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order r ∈ N + . Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the L p modulus of smoothness of order r have been established. As a consequence, the qualitative order of approximation is also derived assuming f in suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncov-ered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in L p ( R ) , 1 ≤ p ≤ +∞ , for the sampling Kantorovich operators.


Introduction
In recent years, the sampling Kantorovich operators K χ w , introduced in [4] by Butzer et al., have been deeply studied as an L p -version of the well-known generalized sampling series (see, e.g., [15,16]). In particular, they revealed to be suitable in applications to real world problems based on image reconstruction and enhancement (see [21,22]). From the point of view of Approximation Theory, the above operators have been largely studied in connection to their basic properties, such as convergence and order of approximation, in several functions spaces, such as L p -spaces, or more in general, in Orlicz spaces (see, e.g., [4,46,52]).
It is well-known that, the above mentioned sampling type operators, together with the so-called quasi-projection operators (see, e.g., [33,34,37]) are all approximate versions of the classical Wittaker-Kotel'nikov-Shannon sampling theorem (see, e.g., [32]); the study of extensions of such a classical result has been one of the most studied topic in the last forty years (see, e.g., [2,27,44]).
In order to completely understand what are the real approximation capabilities of a family of approximation operators, it is of fundamental importance the study of the problem of the saturation order, i.e., the best possible order of approximation that can be achieved into a certain space of functions, and to derive, if it is possible, the corresponding saturation class.
More precisely, the problem of establishing the saturation order for the family K χ w , w > 0, consists into determine a class of functions F, a certain subclass E of trivial functions of F, and a positive non-increasing function ϕ(w), w > 0, such that there exists g ∈ F \ E with K χ w g − g = O(ϕ(w)), as w → +∞, and with the property that, for any f ∈ F with: it turns out that f ∈ E, and vice-versa. Here, · denotes any suitable norm on F. In this case, ϕ(w) is said the saturation order of the approximation process K Moreover, using the same procedures exploited for proving the above asymptotic expansions, we also obtain that, under suitable assumptions on χ , the sampling Kantorovich operators are polynomials preserving. Finally, also an asymptotic formula for a family of double-convolution operators has been proved. This result is an auxiliary theorem that is revealed to be crucial in order to establish the saturation order, and for the detection of the Favard classes for K χ w based on bandlimited kernels. The latter fact, i.e., the importance to have at disposal asymptotic type theorems in order to study the saturation phenomenon, is not surprising, and it is known since 1978 by the pioneer work of Nishishirao [45] concerning bounded linear operators. In Sect. 4, the regularization properties of the sampling Kantorovich operators have been investigated. In particular, we proved that the regularity of the kernel function χ influences the regularity of the operator itself. More precisely, we proved that, if χ is continuous then also K χ w f is continuous, for every f ∈ L p (R), 1 ≤ p ≤ +∞. Similar results have been proved for kernels belonging to the Sobolev space W r ,1 (R) and to the Bernstein class B 1 πw (R), obtaining that K χ w f belong respectively to W r , p (R) and B p π (R), for any f ∈ L p (R), 1 ≤ p ≤ +∞. In the bandlimited case (i.e., when χ belongs to B 1 πw (R)), also a closed form for the distributional Fourier transform of the above operators has been stated.
In Sect. 5, direct and inverse theorems of approximation have been proved. In particular, we estimated the order of approximation by the r order modulus of smoothness of L p . Assuming f in a suitable Lipschitz and generalized Lipschitz classes, also the qualitative rate of approximation has been estimated. Further, an inverse theorem of approximation has been proved, allowing to obtain a characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the family K χ w . From this characterization, the case corresponding to the saturation order is not still covered, and it has been treated in the following sections by an ad-hoc strategy.
In Sect. 6, we face the problem of the saturation order for the sampling Kantorovich operators based upon bandlimited kernels. Here, the crucial point of the proof is that the composition K χ w K χ w f can be written as a double-convolution allowing us to use the auxiliary asymptotic theorems given in Sect. 3. Here, we also provide examples of bandlimited kernels for which the saturation theorem (and also the results of the previous sections) holds. The main idea is to consider suitable finite linear combinations of some well-known kernels, such as those mentioned in Sect. 2.
Finally, in Sect. 7 we proved the inverse theorem of approximation corresponding to the saturation order, in fact obtaining a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels. Here, the proof is based on the well-known Helly-Bray and weak* compactness theorem.

Preliminaries and Auxiliary Results
Let L p (R), 1 ≤ p < +∞, be the space of all Lebesgue measurable functions f : R → R, for which the usual norm f p is finite, and, in the case p = +∞, for the sake of simplicity, by L ∞ (R) we refer to the space C(R), i.e., the space of all uniformly continuous and bounded functions on R endowed with the usual sup-norm · ∞ . Note that, also the case of the usual L ∞ (R) (when specified), i.e., the space of measurable functions with finite sup ess, it will be also considered in some result. Moreover, we denote by C r (R), r ∈ N + , and C ∞ (R) the subspaces of C(R) such that the derivatives f (i) , i = 1, . . . , r , and i ∈ N + , exist and belong to C(R), respectively; C r c (R) and C ∞ c (R) are the subsets of C r (R) and C ∞ (R), respectively, of functions with compact support. Finally, we denote by C 0 (R) the space of continuous and bounded functions on R, by BV (R) the space of functions of bounded variation on R, and by AC(R) the space of absolutely continuous functions on R.
According to the notation of the distribution theory, we will denote by D the space of the test functions C ∞ c (R), and by S the Schwartz class. Further, by D and S we define the spaces of the distributions : D → C and of the tempered distributions T : S → C, respectively. We recall that a function f defined on R is said to be tempered, if there exists a positive constant C and a positive integer m such that: For any locally integrable function f , it is possible to define a regular distribution as follows: In view of the above definition, it is usual to identify the regular distribution f by f itself. If the locally integrable function f in (1) is tempered, and we replace the space of test functions D by S, one can obtain the regular tempered distribution T f . As above, T f is usually identified by f itself. Suppose now that ∈ D (or T ∈ S ) and f ∈ C ∞ (R). We can define the multiplication of f with the distribution (or T , see [49] p. 159) as: We can observe that f belongs to D (or f T ∈ S , respectively).
. The biggest open set in which is null is called the null set of (or T ). The complementary of the null set is called the support of the distribution (or T ).
For any ∈ D and n ∈ N + we can define the following distribution: that is called the distributional derivative of order n of . Note that, differently from the ordinary functions, any distribution admits derivatives of any order. For more details about distribution theory, see, e.g., the monographs [49,53]. Further, we recall the definition of Sobolev spaces: where the derivatives f (n) can be intended in the distributional sense. It is well-know that, the definition of the Sobolev spaces can also equivalently formulated (in term of equivalence classes) as follows: Finally, we also introduce the space (see [26] p. 35): Now, we denote by B p σ (R), the so-called Bernstein class (or Bernstein space), for σ ≥ 0 and 1 ≤ p ≤ +∞, containing the functions of L p (R) which can be extended to an entire function f (z) (z = x + iy ∈ C) of exponential type σ , i.e., satisfying: see e.g., [13,14,51]. It is well-known that: with any 1 ≤ p < q. Moreover, we denote by: v ∈ R, the Fourier transform of f ∈ L 1 (R), while for f ∈ L p (R), p ≥ 1, we denote the (distributional) Fourier transform f as the following regular distribution: where ϕ is defined as in (3). Note that, in the case p = 1 the distributional Fourier transform f is, in fact, a usual function and it coincides with the definition in (3), and a similar consideration can be made for the case 1 < p ≤ 2. According to the Paley-Wiener-Schwartz theorem, it is possible to prove that f ∈ B p σ (R), if and only if f is bandlimited (as a function or a distribution), with supp f ⊂ [−σ, σ ]. We recall that, a function f is said to be bandlimited if the support of f (as a function or a distribution) is compact. Now, in order to recall the definition of the families of sampling type operators that will be studied in this paper, we introduce the following notation.
From now on, we will say that a function χ : R → R is a kernel, if it satisfies the following assumptions: (χ 2) the discrete algebraic moment of order 0: (χ 3) the discrete algebraic moment of order 1: (χ 4) there exists β > 2 such that: Now, we also introduce the following notations that will be useful later. We define for χ the so-called continuous algebraic and absolute moments of order ν ∈ N and ν ≥ 0, respectively, by the following integrals: Note that, by condition (χ 4) it turns out that m ν (χ ) ≤ M ν (χ ) < +∞, for every 0 ≤ ν < β − 1 for which m ν (χ ) takes sense.

Lemma 2.2
Let χ : R → R be a given continuous function satisfying (χ 1). Assuming in addition that the function g(u) := −(i u) j χ(u), j ∈ N (u ∈ R and i denotes the complex unit) belongs to L 1 (R), we have that: if and only if where χ ( j) denotes the j-th derivative of χ . It turns out that A Clearly, in case of functions with sufficiently rapid decay, it turns out that (4) of (χ 2) and (χ 3) are equivalent to the condition of Lemma 2.2 for j = 0, A 0 = 1 and j = 1, A χ 1 = −i A 1 , respectively. Condition (7) is known as the Strang-Fix type condition, see [28].
Examples of kernels satisfying the above assumptions (for the details of the proof see [4,24]) are, e.g., the central B-splines of order N (see, e.g., [10,16]), defined by: hence condition (χ 4) is satisfied for every β > 0. Further, we recall the Jackson-type kernels ( [18,41]), defined by: with N ∈ N, α ≥ 1, and c N is a non-zero normalization coefficient, given by: Here, the sinc-function sinc(x) is defined as sin(π x)/π x, if x = 0, and 1 if x = 0. Finally, we also recall the Bochner-Riesz kernels ( [19,50] with η > 0, and where J λ is the Bessel function of order λ ( [20]), and is the usual Euler gamma function. The functions J N and b η are examples of bandlimited (hence belonging to C ∞ (R)) kernels. Moreover, by simple computations involving the inverse Fourier transform (see [24], Sect. 7) we have J Other examples of kernels can be found, e.g., in [12]. Now, we are able to recall the definition of the sampling Kantorovich operators ( [4]). We denote by: the sampling Kantorovich operators of f , where f : R → R is a locally integrable function, such that the above series is convergent for every x ∈ R.
It is well-known that, under the assumptions (χ 1), (χ 2) and (χ 4) on χ , for every f ∈ L p (R), 1 ≤ p ≤ +∞, it turns out that (see Theorem 4.1 and Corollary 5.2 of [4]): and there holds (see Corollary 5.1 and Remark 3.2 (a) of [4]): Note that, the theory of the above operators K χ w (as usually happens for the main families of linear operators) is studied in the L p -setting, only for 1 ≤ p ≤ +∞. The case 0 < p < 1 is not considered since, one of the main tools used in the proofs of the classical approximation results for 1 ≤ p < +∞ (above recalled) is the Jensen inequality, that can not be applied if 0 < p < 1.
In order to study approximation properties for the above sampling Kantorovich series, we also need to recall the definition of the following operators.
In conclusion, we also recall the notion of singular integral (see e.g., [12,40,48]): ( is a kernel of Fejér type with : R → R belonging to L 1 (R); here the symbol " * " refers to the usual convolution product. Approximation results for the singular integrals I w f can be proved, requiring that satisfies: Note that, if χ is a continuous kernel (according to the previous definition), then condition (10) applied to χ is satisfied, by Lemma 2.2 with j = 0 and A 0 = 1. Obviously, it is well-known that, under the assumption (10), the family ( w ) w>0 turns out to be an approximate identity, see [12]. Now, we are able to recall the following lemma that is a direct consequence of the fundamental result of Fourier Analysis, which shows the connections between the (continuous) convolution integrals and the (discrete) convolution sums.

Asymptotic Expansions in L p -Setting
We prove the following asymptotic formula for the sampling Kantorovich operators.
Then, for any f ∈ C r (R): Proof By the Taylor formula with Lagrange remainder until the order r , we have: u, x ∈ R, where θ u,x , is a suitable value between u and x. Hence, replacing (13) in the definition of the operators, and using condition (χ 2), the binomial Newton formula, and (12) one can obtain: On the other hand: Then, we have: We first consider the case p = +∞. By the uniform continuity of f (r ) , for every fixed Thus we can write what follows: We begin estimating S 1 . First of all, we can observe that, if u ∈ [k/w, (k + 1)/w] and |wx − k| ≤ wδ/2, it results: for w > 0 sufficiently large, then, using the convexity of |·| r , r ≥ 1: where M 0 (χ ), M r (χ ) are both finite by Remark 2.1, as a consequence of assumption (χ 4). While for S 2 we have: uniformly with respect to x ∈ R, for w > 0 sufficiently large, by (6) of Remark 2.1. This completes the first part of the proof.
Let now 1 ≤ p < +∞ be fixed. We want to prove that: exploiting the Vitali convergence theorem. By the first part of this theorem, immediately follows that: Let now ε > 0 be fixed, and γ > 0 such that supp f ⊂ [−γ, +γ ]. For any M > γ +1, and using Jensen inequality twice, we can write what follows: Now, using the change of variable t = wx − k: for sufficiently large w > 1, since by condition (χ 4) with β > r p + 1. Therefore, we just proved that in correspondence of ε there exists the interval E ε := [−M, M] such that, for any measurable set F, with F ∩ E ε = ∅, inequality (16) holds. Now, again by the first part of this theorem, it follows that: for sufficiently large w > 0; thus, for any fixed measurable set B ⊂ R, with: |B| < ε 2 − p K − p , we finally get: for every w > 0 sufficiently large. This shows that the integrals (·) |J w (x)| p dx are equi-absolutely continuous. This completes the proof.
Asymptotic expansions can also be proved for functions in Sobolev spaces, assuming that χ has compact support.

Theorem 3.2 Let χ be a kernel with compact support. Then, for every
Further, if χ also satisfies (11) and (12) for r ∈ N + , r > 1, then for every f ∈ W r , p (R), 1 ≤ p ≤ +∞, it turns out that: Proof Let f ∈ W r , p (R), r ∈ N + , r > 1, 1 ≤ p ≤ +∞, be fixed. It is wellknown that f can be written by the Taylor formula (13) with integral remainder [26] p. 37). Hence, proceeding as in the proof of Theorem 3.1, i.e., replacing the above Taylor formula in the definition of the operators K χ w f , and noting that: we immediately obtain: where the above series can be considered only for k ∈ Z such that |wx − k| ≤ T , since supp χ ⊂ [−T , T ], T > 0. Now, for |wx − k| ≤ T , the integrals can be estimated in terms of: Thus, for any w > 0 we have: by Remark 2.1. Now, recalling the generalized Minkowsky type inequality (see [30], p. 148), we have: Now, since f (r ) ∈ L p (R) we know that, for every ε > 0 there exists γ > 0 such that, for every |z| ≤ γ there holds f (r ) (·+z)− f (r ) (·) p < ε, hence for w sufficiently large we finally get: J w p < ε. In the case r = 1, the proof can be deduced analogously.

Remark 3.3
Note that, in case of p = +∞, Theorem 3.2 holds also if we define W r ,∞ (R) as a subspace of the usual L ∞ (R), namely, as the space of all the measurable functions with finite ess sup.
Proceeding as in the proof of the above theorems, the following result can be easily deduced.
Theorem 3.4 Let χ be a kernel, assumed as in Theorem 3.1 with r ∈ N + . Then, for any polynomial p r −1 of degree at most r − 1, it turns out that: Proof Let p r −1 (x) = a r −1 x r −1 + · · · + a 1 x + a 0 , be any polynomial of degree at most r − 1. We first observe that, for every fixed x ∈ R: by condition (χ 4), hence the series is absolutely convergent and the operators K χ w p r −1 are well-defined, for every fixed w > 0. Finally, proceeding as in (14), we immediately get the thesis. Theorem 3.4 shows that the sampling Kantorovich operators are polynomialspreserving, and improves Theorem 3.5 of [23]. Now, the following asymptotic formula for double-convolution type operators can be proved; this will be useful in the next sections. Theorem 3.5 Let χ , ∈ L 1 (R) both satisfying conditions (10) and (χ 4) for β > r +1, with r ∈ N + . If r > 1, we assume in addition that: Then, for any f ∈ C r c (R), and 1 ≤ p ≤ +∞, we have: Proof By the Taylor formula (13), and using (10) for both χ and , we can write what follows: Now, by the change of variable z = w[y − u], the Fubini theorem, the change of variable t = w[x − y], and using (18), respectively, we obtain: On the other hand, we have: thus, by Fubini theorem we easily obtain: w > 0 and x ∈ R. Now, for 1 ≤ p ≤ +∞, using (19) and the generalized Minkowski type inequality, we obtain: Now, since f (r ) ∈ L p (R) we know that, for every ε > 0 there exists δ > 0 such that, for every |u| ≤ δ there holds f (r ) (· + u) − f (r ) (·) p < ε. Then, we can write what follows:
We begin estimating I w,1 . First of all, we can note that, in the domain of integration of I w,1 it results: then: in view of (χ 4). While for I w,2 we have: for w > 0 sufficiently large, since by (χ 4) the integrals |z|>wδ/2 | (z)| |z| dz goes to zero, as w → +∞. Similarly to above, for w > 0 sufficiently large. This completes the proof.
For references about double-convolution type operators, see, e.g., [42,43]. From Theorem 3.5 the following useful corollary which involves the above double-convolution with the sampling Kantorovich operators can be proved.

Corollary 3.6
Let χ be a kernel, and let ∈ L 1 (R) both assumed as in Theorem 3.5.
Then, for any f ∈ C r c (R), r ∈ N + , and g ∈ L p (R), 1 ≤ p ≤ +∞, we have: Further, we also have for g ∈ L p (R): Proof Let 1 ≤ p ≤ +∞ be fixed. Using the Minkowski and the Hölder inequalities: where C > 0 is a suitable positive constant depending only on χ , and f (r ) , that can be determined since (w r I r m ( ) m r − (χ ) , as w → +∞. Now, recalling the convergence result of (8) and using Theorem 3.5 in the case p = +∞ the proof of the first part of the corollary follows immediately. For what concerns the second part of the thesis, using Hölder inequality, we have: where 1/ p + 1/q = 1. Hence the proof follows as above by Theorem 3.5 (applied with 1 ≤ q ≤ +∞) and (8).
Note that, the second part of Corollary 3.6 (i.e., the case with the L 1 -norm) provides a theorem which allows to pass the limit under the integral, for the sequence This will be useful in the proofs of the last two sections of the present paper.

On Regularization Properties of Sampling Kantorovich Operators
We begin this section with the following theorems about the regularity of the sampling Kantorovich operators. Proof Let 1 ≤ p < +∞ and w > 0 be fixed. For every m ∈ N + we consider the sequence: Then, we can write the following inequality: Let now x ∈ R be fixed. Using Jensen inequality twice, we obtain: . Now, passing to the limit for m → +∞ in the above inequality we provide the uniform convergence of the sequence (h w m ) m∈N + to K χ w f on the whole R. Now, observing that any h w m is continuous on R as a finite sum of continuous functions, we finally get that also K χ w f is continuous on R. Further, proceeding as above, one can also prove that: i.e., the operator is bounded on R for every fixed w > 0. For the case p = +∞, i.e., for f ∈ C(R), the proof follows similarly recalling (9) and observing that: and that |k|>m |χ(wx − k)| is the remainder of a uniformly convergent series on R (see Remark 2.1).
For what concerns the second part of the thesis, if χ ∈ C(R), for any fixed w > 0, ε > 0 and x, y ∈ R, we can write: for a fixed sufficiently large m ∈ N + . Moreover, h w m is uniformly continuous on R since it is defined as a finite sum of uniformly continuous functions, thus, if we choose the parameter δ > 0 of the uniform continuity of h w m corresponding to ε/3, and |x − y| < δ, we finally get: The above theorem can be generalized as follows.
Proof Proceeding as in the proof of Theorem 4.1, it turns out that, K χ w f is bounded a.e. on R, then the operator K χ w f ∈ D , i.e., it defines a regular distribution 1 . Let now 1 ≤ p < +∞ be fixed. We can immediately observe that the series K χ w f is absolutely convergent on R by the estimate given in (20). Then, for every ϕ ∈ D, passing the integral under the series, and using the Fubini theorem, we can compute the distributional derivatives of K χ w f as follows: is an ordinary function, and moreover, using Jensen inequality twice, Fubini-Tonelli theorem and the change of variable y = wx − k: In the case p = +∞, the proof follows similarly.
For p = +∞, Theorem 4.2 provide a generalization of the result established in Proposition 4 of [17] in the case r = 1. Clearly, in this case all the derivatives involved must be considered in the usual sense. Further, we can also observe that, for p = +∞, Theorem 4.2 holds also if we consider the classical definition of L ∞ (R) and W r ,∞ (R) (that we recalled in Remark 3.3).
Proof As in Theorem 4.2, K χ w f ∈ C 0 (R), then K χ w f ∈ S . Let now 1 ≤ p < +∞ be fixed. First we can note that the series K χ w f is absolutely convergent on R by the 1 In fact, K χ w f is a regular tempered distribution. estimate given in (20). Then, for every ϕ ∈ S, passing the integral under the series, and by the Fubini theorem, we can write what follows: Recalling that supp χ ⊂ [−πw, +πw] we finally obtain: Let now A := (−∞, −π) ∪ (+π, +∞) be a given open set of R. For every ϕ ∈ S with supp ϕ ⊂ A, by (22) it turns out that: This shows that A is the null set of (K χ w f ) and then supp (K χ w f ) ⊂ [−π, +π ]. This completes the proof. In the case p = +∞, the proof can be done similarly.
At the end of this section, in view of the result proved in Theorem 4.3, the following closed form for the Fourier transform of the sampling Kantorovich operators can be established.
where the above distribution is given by the multiplication of the smooth function e i x/2w sinc x 2πw χ x w and the distribution f .
hence the following relation can be deduced: Now, we claim that I w f ∈ B p πw (R), when f ∈ B p πw (R). Indeed, for every ϕ ∈ S and by the Fubini theorem: where : R → R denotes the characteristic function of the interval [0, 1]. Hence: Now, since f ∈ B p πw (R) we know that A = (−∞, −πw) ∪ (π w, +∞) is the null set of f , and then for every ϕ ∈ S such that supp ϕ ⊂ A it turns out that f [ϕ] = 0. Clearly, in the latter case also supp η u ⊂ A, then we must have f [η u ] = 0 which also implies that (I w f )[ϕ] = 0. Now, recalling the inclusions (2) it is clear that χ ∈ B q πw (R), with q such that 1/ p + 1/q = 1, and using Lemma 2.3 we can finally obtain: Exploiting (24), for every ϕ ∈ S we have: where χ and denote the usual L 1 -Fourier transform. Now, recalling that: we get the thesis.
Note that, Theorem 4.4 extend to all 1 ≤ p ≤ +∞ the result originally proved only for p = 1, in Lemma 3.1 of [25].

Direct and Inverse Results of Approximation in L p -Setting
Now, we recall the definition of Lipschitz classes: where ω 1 ( f , δ) p denotes the usual first-order modulus of smoothness of f ∈ L p (R), 1 ≤ p ≤ +∞. In general, the modulus of smoothness of order r ∈ N + of any given f ∈ L p (R), 1 ≤ p ≤ +∞, is defined by ( [39]): Moreover, the generalized Lipschitz classes are given by: where r is the smallest integer such that r > α, i.e., r = α +1 (where · denotes the integer part of a given number). It is well-known that Lip * (α, L p ) = Lip (α, L p ), for 1 ≤ p ≤ +∞ when α > 0 is not an integer. Note that, the definition of Lip (α, L p ) can be extended for α > 1 setting Lip (α, L p ) := W α 0 (Lip (β, L p )), where α = α 0 + β, with α 0 ∈ N, 0 < β ≤ 1, and: It is known that Lip (r , L p ) = W r , p (R) ⊂ Lip * (r , L p ), if 1 < p ≤ +∞, and Lip (r , L 1 ) = W r −1 (BV ) ∩ L 1 (R) ⊂ Lip * (r , L 1 ) 2 , r ∈ N + , and for any function f in these spaces, there holds: for 1 < p ≤ +∞ and p = 1, respectively, where M > 0 is a suitable positive constant. Now, in order to prove quantitative estimates in L p -setting by means of the modulus of smoothness of order r , we recall the definition of the well-known K-functionals: From a classical result of Johnen [31], we know that: f ∈ L p (R), 1 ≤ p ≤ +∞, for suitable positive constants C 1 and C 2 . Now we are able to prove the following.

Theorem 5.1 Let χ be a given kernel.
In what follows, if the parameter r ∈ N is greater than 1, we also assume that (χ 4) is satisfied for β > r + 1, and conditions (11), (12) hold. Then: (i) for every f ∈ L ∞ (R), there exists C > 0 such that: (ii) if in addition χ satisfies (χ 4) with β > r p+1, 1 ≤ p < +∞, for any f ∈ L p (R), and a suitable C > 0: (iii) if f ∈ Lip * (α, L ∞ ), with 0 < α < r , it turns out that: while for f ∈ Lip(r , L ∞ ) = W r ,∞ (R), we have: , as w → +∞; (iv) under the assumptions required in item (iii) for 1 ≤ p < +∞, if f ∈ Lip * (α, L p ), where 0 < α < r , it turns out that: For what concerns T 1 , using the change of variable y = t − x and Fubini-Tonelli theorem, we have: Further, we can estimate T 2 . By the change of variable y = t − u and Fubini-Tonelli theorem, we have: Finally, we can observe that the items (iii) and (iv) of the statement follows immediately from (i), (ii), (25), and the well-known inequality: This completes the proof.

Remark 5.2
Note that some results concerning quantitative estimates by suitable modulus of smoothness for an abstract class of operators (including those considered in the present paper) are given in [35,36,38]. The main assumptions assumed in the above quoted results are similar (but given in a more abstract form) to those assumed in Theorem 5.1. One of the main difference between the results established in [35,36,38] and Theorem 5.1 is that, there certain local condition on the Fourier transform of the involved kernels and their derivatives are assumed, while here we simply require a sufficiently rapid decay of χ at ±∞. Even if in some special cases the estimates established in Theorem 5.1 can be deduced from the above quoted results, here in the proposed approach the required assumptions are immediate to check, and the proposed proofs are completely different. Further, the families of kernels for which all the above results can be applied are not exactly the same. Indeed, as will be showed in Sect. 6 and can also be deduced, e.g., from Example 4 of [35], in order to provide examples of kernels satisfying all the required assumptions of both theories, suitable finite linear combinations of classical kernels (such as those mentioned in Sect. 2) must be considered. Obviously, such combinations are not, in general, the same. Now, in order to obtain a complete characterization of the spaces Lip * (α, L p ), 1 ≤ p ≤ +∞, also for 0 < α < r , we need of the following lemma.
Proof Let i = 1 and j = 0, 1, be fixed. We have that the series m j (χ , x) is absolutely convergent on R since it is 1-periodic and (χ 4) holds. Now, using the Fubini theorem, noting that χ(· − k), k ∈ Z, defines a regular (tempered) distribution, and that (by (11)) m j (χ , x) is constant on R, we have, for every fixed ϕ ∈ D: differentiability of the operators has been replaced by the (weaker) distributional differentiability, and (ii) we are able to achieve inverse results of approximation also for r > 2. Now, by Theorem 5.1 and Theorem 5.4 it is immediate to deduce a characterization of the generalized Lipschitz classes Lip * (α, L p ), 0 < α < r , r ≥ 1, in term of convergence of sampling Kantorovich operators.

Remark 5.5
Note that, all the results established in Sect. 5 for the case p = +∞, hold also if we consider the usual definition of L ∞ (R) and W r ,∞ (R) in place of those recalled in Sect. 2.
since the limit can be brought under the integral because, using the Holder inequality and (32) (or (33) respectively), we have: where 1/ p +1/q = 1. Now, by Theorem 4.3 we know that K χ w f ∈ B p π (R) ⊂ B p πw (R) for w ≥ 1, and then, in view of what has been established in (24), it turns out that: Further, we also recall that χ satisfies (10) by (χ 2) and Lemma 2.2. Now, using the well-known properties of the convolution, the Fubini theorem and suitable changes of variables: Now, for r = 1, in view of what has been recalled in Sect. 5, by Theorem 5.1, Remark 5.5, Theorem 6.1 and Theorem 7.1, and observing that the inclusion B 1 σ (R) ⊂ W r ,1 (R), r ∈ N + , holds, it follows that: and Lip (1, are the so-called Favard (saturation) classes of the sampling Kantorovich operators based upon bandlimited kernels χ , with m 1 (χ ) = −A χ 1 = 1/2. While, if χ ∈ B 1 πw , w > 0, is a kernel assumed as in Theorem 6.1, with r ∈ N + , r > 1, and it satisfies (χ 4) for suitable values of β, from Theorem 5.1, Remark 5.5, Theorem 6.1 and Theorem 7.1, it follows that, the spaces Lip(r , L p ) = W r , p (R), for 1 < p ≤ +∞, and Lip(r , L 1 ) = W r −1 (BV ) ∩ L 1 (R), for p = 1, are the Favard classes for the sampling Kantorovich operators.

Remark 7.2
Note that, results concerning the saturation classes for the quasi-projection operators have been also established in Theorem 24 of [38] using the well-known Mikhlin's condition (see [29], p. 367). In the present instance, other than the considerations already given in Remark 5.2, we can also stress that the results here proved hold for every 1 ≤ p ≤ +∞ and without requiring the above mentioned Mikhlin's condition, while those proved in [38] hold for 1 < p < +∞ only.