Order of approximation for nonlinear sampling

COMMENTATIONES MATHEMATICAE Vol. 53, No. 2 (2013), 271-292

Danilo Costarelli, Gianluca Vinti

Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces To Professor Julian Musielak, a Master, a Mentor and a sincere friend, with high esteem and friendship

Abstract. In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively. The general setting of Orlicz spaces allow us to deduce directly the results concerning the rate of convergence in Lp -spaces, 1 ≤ p < ∞, very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fejér and B-spline kernels are also considered. 2010 Mathematics Subject Classification: 41A25, 41A30, 46E30, 47A58, 47B38, 94A12. Key words and phrases: Nonlinear sampling Kantorovich operators, Orlicz spaces, order of approximation, Lipschitz classes, irregular sampling.

1. Introduction. In this paper, we study the rate of convergence for the nonlinear sampling Kantorovich operators introduced in [57] and we generalize the results obtained in [31] to the nonlinear setting. Here, we consider both the cases of uniform approximation, for uniformly continuous and bounded functions belonging to Lipschitz classes and the modular approximation for functions belonging to Orlicz spaces. In the latter case, we introduce Lipschitz class of the Zygmund-type which take into account of the modular functional involved (see e.g. [31]). The theory of nonlinear integral operators in connection with approximation problems has been started in [42, 44, 45, 46] by the work of the polish mathematician J. Musielak, a direct student of W. Orlicz. Successively, in [10] the above theory

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Order of approximation for nonlinear sampling Kantorovich operators

has been carried out in collaboration with C. Bardaro and G. Vinti in the general context of modular spaces, such as in Orlicz spaces (see e.g. [6, 7, 12, 8, 9, 15, 14, 47, 48, 16, 55, 41]). The sampling operators of the Kantorovich-type were first introduced in [3] by Bardaro, Butzer, Stens and Vinti, in the linear case and in one dimensional setting. Later on, various extensions of the theory have been produced; for instance, in [29, 30] the multivariate theory of the sampling Kantorovich operators has been studied in the linear and nonlinear setting, with applications to Image Processing. Moreover, extensions to a more general context are given in [58, 5]. Nonlinear sampling Kantorovich operators are important both from the mathematical and the applications point of view; as a matter of fact there are signals that need to be reconstructed by using a nonlinear process. The nonlinear sampling Kantorovich operators here considered (introduced in Section 3 ), are of the form ! Z tk+1 /w X w f (u)du , x ∈ R, (I) χ wx − tk , (Sw f )(x) := ∆k tk /w k∈Z

where f : R → R is a locally integrable function such that the series is convergent for every x ∈ R, χ : R2 → R represents a kernel function satisfying certain properties, and (tk )k∈Z is a suitable strictly increasing sequence of real numbers with ∆k := tk+1 − tk > 0, k ∈ Z. Examples of kernels χ satisfying the assumptions of the theory of the present paper, can be constructed using, e.g., central B-splines of order n ∈ N+ , or the Fejér’s kernel (see e.g. [22, 10, 3, 57]). The choice of (tk )k∈Z allows us to sample signals by an irregular sampling scheme. If tk = k, k ∈ Z, we obtain the uniform spaced sampling. The main results of the paper show that, for f belonging to a Lipschitz class and under certain assumptions, kSw f − f k∞ = O(w−ε ),

as

w → +∞,

for suitable ε > 0 (see Theorem 4.1 of Section 4 ) and I ϕ [µ(Sw f − f )] = O(w−θ ),

as

w → +∞,

for suitable constants µ, θ > 0, where I ϕ denotes the modular of the Orlicz space under consideration (see Theorem 5.2 of Section 5 ). The importance of the above operators (I) lies in theory of Signal Processing, where they can be used to reconstruct and approximate signals; in a multivariate context the above operators are suitable to reconstruct and approximate images (see [29, 30, 28]). The operators (I), represent the natural extension of the nonlinear generalized sampling operators to the Lp -setting, 1 ≤ p < +∞. These latter operators, were first introduced by the German mathematician P.L. Butzer and his school at Aachen in the linear form and subsequently, they were extended to the nonlinear setting in [14]. For references on sampling series see e.g. [19, 23, 52, 20, 24, 25, 21, 26, 27, 11, 13, 14, 54, 55, 17, 1, 33, 4]. The theory of the generalized sampling operators

D. Costarelli, G. Vinti

273

represents an approximate version of the classical Sampling Theory, based on the classical Whittaker-Kotelnikov-Shannon sampling theorem (see e.g. [53, 37, 32, 34, 18, 35, 36, 2]). One of the advantages, with respect to the Butzer’s operators, given by the series of the form (I) is that, instead of the sampling values f (k/w), one has an average of f on a small interval containing k/w (here instead of k we have tk ). In practice, this situation very often occurs in Signal Processing when one cannot match exactly the node tk : this represents the so-called time-jitter error. Therefore, the sampling Kantorovich operators reduces time-jitter errors calculating the information in a neighborhood of a point rather than exactly at that point. The integrals in (I) and in particular the presence of the mean values, become our operators suitable for the study in Lp -spaces, or more generally in Orlicz spaces Lϕ (R). We point out that, the setting of Orlicz spaces is important since allows the reconstruction of not necessarily continuous signals; this situation very often occurs especially in the applications to Image Processing, where digital images are represented by discontinuous functions. Important examples of Orlicz spaces here considered, are given by the Zygmund spaces, also known as interpolation spaces, very important e.g. in the theory of partial differential equations, and the exponential spaces, very used for embedding theorems between Sobolev spaces, see e.g. [43, 10, 3]. Finally, in Section 6 we furnish applications to concrete operators constructed by means of special kernels, as e.g. Fejér or B-spline kernels. 2. Preliminaries. In this section, we introduce some notations that will be very useful along the paper. In what follows, we denote by C(R) the set of all uniformly continuous and bounded functions f : R → R endowed with the usual sup-norm k · k∞ . In order to study the rate of convergence of a family of nonlinear discrete operators in C(R), we introduce the Lipschitz class of the Zygmund-type with which we will work. We define Lip∞ (ν), 0 < ν ≤ 1, by Lip∞ (ν) := {f ∈ C(R) : kf (·) − f (· + t)k∞ = O(|t|ν ), as t → 0} ,

where for any two functions f , g : R → R, f (t) = O(g(t)) as t → 0, means that there exist constants C, γ > 0 such that |f (t)| ≤ C |g(t)| for every |t| ≤ γ ( see e.g. [10, 56]). We now recall the definition and some basic properties of Orlicz spaces. + We will say that a function ϕ : R+ 0 → R0 is a ϕ-function, if it satisfies the following conditions: 1. ϕ is a continuous function with ϕ(0) = 0; 2. ϕ is a non-decreasing function and ϕ(u) > 0 for every u > 0; 3.

lim ϕ(u) = +∞.

u→+∞

Let now introduce the functional I ϕ associated to the ϕ-function ϕ and defined by Z ϕ I [f ] := ϕ(|f (x)|) dx, R

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for every f ∈ M (R), i.e. for every (Lebesgue) measurable function f : R → R. As it is well-known, I ϕ is a modular functional (see e.g. [43, 50, 10]). It is easy to prove that if ϕ is convex than I ϕ is convex. The Orlicz space generated by ϕ is defined by Lϕ (R) := {f ∈ M (R) : I ϕ [λf ] < +∞, for some λ > 0} . A natural notion of convergence in Orlicz spaces, called modular convergence, can be introduced (see [49]). Namely, we will say that a net of functions (fw )w>0 ⊂ Lϕ (R) is modularly convergent to f ∈ Lϕ (R), if there exists λ > 0 such that Z I ϕ [λ(fw − f )] = ϕ(λ|fw (x) − f (x)|) dx → 0, as w → +∞. R

Moreover, we recall for the sake of completeness that in Lϕ (R) can be also given a strong notion of convergence, i.e., the Luxemburg-norm convergence, see e.g. [43, 10]. The modular convergence induces a topology in Lϕ (R) called modular topology. Now, we introduce the Lipschitz class used in Orlicz spaces in order to study the rate of convergence for the nonlinear sampling Kantorovich operators we will deal with. By Lipϕ (ν), 0 < ν ≤ 1, we define the set of all functions f ∈ M (R) such that, there exists λ > 0 with Z I ϕ [λ(f (·) − f (· + t))] = ϕ (λ |f (x) − f (x + t)|) dx = O(|t|ν ), R

as t → 0, see [31]. For further results concerning Orlicz spaces, see [39, 43, 38, 40, 50, 51]. 3. Nonlinear sampling Kantorovich operators. In this section, we first recall the definition of the nonlinear sampling Kantorovich operators introduced in [57]. Let π = (tk )k∈Z be a sequence of real numbers such that −∞ < tk < tk+1 < +∞ for every k ∈ Z, limk→±∞ tk = ±∞ and such that δ ≤ ∆k := tk+1 − tk ≤ ∆, for every k ∈ Z and for some δ, ∆ > 0. In what follows, a function χ : R × R → R will be called a kernel if it satisfies the following conditions: (χ1) k 7→ χ(wx − tk , u) ∈ `1 (Z), for every (x, u) ∈ R2 and w > 0; (χ2) χ(x, 0) = 0, for every x ∈ R; (χ3) χ is a (L, ψ)-Lipschitz kernel, i.e., there exist a measurable function L : R → R+ 0 and a ϕ-function ψ, such that |χ(x, u) − χ(x, v)| ≤ L(x)ψ(|u − v|), for every x, u, v ∈ R;

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(χ4) there exists θ0 > 0 such that 1 X Tw (x) := sup χ(wx − tk , u) − 1 = O(w−θ0 ) u6=0 u k∈Z

as w → +∞, uniformly with respect to x ∈ R.

Moreover, we will assume that the function L of condition (χ3) satisfies the following properties: (L1) L ∈ L1 (R) and is bounded in a neighborhood of the origin; (L2) there exists β0 > 0 such that mβ0 ,π (L) := sup x∈R

X

k∈Z

L(wx − tk )|wx − tk |β0 < +∞;

(L3) there exists α0 > 0 such that, for every M > 0, Z w L(wx) dx = O(w−α0 ), as |x|>M

w → +∞.

Remark 3.1 (a) Note that, if L(x) = O(x−1−β0 −ε ) as x → ±∞, for some ε > 0, condition (L2) holds for β0 , see [22, 3, 31]. (b) If the function L has compact support, i.e., supp L ⊂ [−B, B], B > 0, condition (L2) is satisfied for every β0 > 0. (c) If supp L ⊂ [−B, B], B > 0, we obtain, for every M > 0, Z Z w L(wx) dx = L(u) du = 0, |x|>M

|u|>wM

for sufficiently large w > B/M ; then condition (L3) is fulfilled for every α0 > 0. (d) Condition (L3) is used only to study the nonlinear sampling Kantorovich operators in the setting of Orlicz spaces. We now recall the definition of the nonlinear sampling Kantorovich operators for a given kernel χ. We define ! Z tk+1 /w X w f (u)du , x ∈ R, (Sw f )(x) := χ wx − tk , ∆k tk /w k∈Z

where f : R → R is a locally integrable function such that the series is convergent for every x ∈ R. From now on, we will consider functions on the domain of the operators (Sw )w>0 .

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Order of approximation for nonlinear sampling Kantorovich operators

Remark 3.2 Note that, if the kernel χ is of the form χ(x, u) = L(x) u, with L satisfying conditions (L1), (L2) and (L3), our operators reduces to the linear ones introduced in [3]. In this case condition (χ4) becomes X as w → +∞, L(wx − tk ) − 1 = O(w−θ0 ), (1) Tw (x) = k∈Z

uniformly with respect to x ∈ R, for some θ0 > 0. Sometimes, in place of condition (1) we require a stronger condition, i.e., X (2) L(u − tk ) = 1, k∈Z

for every u ∈ R. In this case condition (χ4) holds for every θ0 > 0. When tk = k (uniform sampling scheme), (2) is equivalent to  0, k ∈ Z \ {0} , b L(k) := 1, k = 0, Z b where L(v) := L(u)e−ivu du, v ∈ R, is the Fourier transform of L; see [22, 25, 3, 29, 31].

R

We now recall the following lemma, that will be very useful in the paper. Lemma 3.3 ([31]) LetP L be a function satisfying conditions (L1) and (L2). Then, (i) m0,π (L) := supx∈R k∈Z L(wx − tk ) < +∞; (ii) for every γ > 0 X L(wx − tk ) = O(w−β0 ), as w → +∞, |wx−tk |>γw

uniformly with respect to x ∈ R, where β0 > 0 is the constant of condition (L2). Remark 3.4 Note that, (i) of Lemma 3.3, together with assumptions (χ2) and (χ3) ensures that the functionals (Sw (·))w>0 are well-defined for functions f ∈ L∞ (R). Indeed, X |(Sw f )(x)| ≤ L(wx − tk ) ψ(kf k∞ ) ≤ m0,π (L) ψ(kf k∞ ) < +∞, k∈Z

for every x ∈ R, i.e., Sw : L∞ (R) → L∞ (R). 4. Order of approximation in C(R). We will now study the order of approximation of the family of nonlinear sampling Kantorovich operators in C(R). We have the following.

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D. Costarelli, G. Vinti

Theorem 4.1 Let χ be a kernel and suppose in addition that ψ of condition (χ3) satisfies the following assumption: ψ(u) = O(up ),

(3)

u → 0+ ,

as

for some 0 < p ≤ 1. Then, for every f ∈ Lip∞ (ν), 0 < ν ≤ 1, we have: kSw f − f k∞ = O(w− ),

w → +∞,

as

with  := min {νp, β0 , θ0 }, where β0 and θ0 are the positive constants of conditions (L2) and (χ4), respectively. Proof We first consider the case of kernel χ satisfying condition (χ3) with L such that condition (L2) is satisfied for 0 < β0 ≤ p.

By condition (3) there exists M > 0 and 0 < u < 1 such that ψ(u) ≤ M up , for every u ∈ R+ 0 with u ≤ u. Let now f ∈ Lip∞ (ν), with 0 < ν ≤ β0 /p, be fixed. We know that there are C1 > 0 and γ > 0 such that sup |f (x) − f (x + t)| ≤ C1 |t|ν , x∈R

for every |t| ≤ γ. Without loss of generality, we can assume γ > 0 such that (γ/2)ν ≤ u/2. Now, let x ∈ R be fixed. We can write |(Sw f )(x) − f (x)| X X ≤ |(Sw f )(x) − χ (wx − tk , f (x)) | + | χ (wx − tk , f (x)) − f (x)| k∈Z

k∈Z

! w Z tk+1 /w X X ≤ L (wx − tk ) ψ f (u)du − f (x) + χ (wx − tk , f (x)) − f (x) ∆k tk /w k∈Z

≤

k∈Z

  

X

+

|wx−tk |≤wγ/2

X

  

|wx−tk |>wγ/2

L (wx − tk ) ψ

w ∆k

Z

tk+1 /w

tk /w

!

|f (u) − f (x)| du

X + χ (wx − tk , f (x)) − f (x) =: I1 + I2 + I3 , k∈Z

for every w > 0. By the change of variable u = x + t in the integrals of I1 , we have, ! Z (tk+1 /w)−x X w I1 = L (wx − tk ) ψ |f (x + t) − f (x)| dt . ∆k (tk /w)−x |wx−tk |≤wγ/2

Now, for every t ∈ [tk /w − x, tk+1 /w − x], if |wx − tk | ≤ wγ/2, we have |t| ≤ |t − tk /w + x| + |tk /w − x| ≤

∆ γ + < γ, w 2

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Order of approximation for nonlinear sampling Kantorovich operators

for sufficiently large w > 0, then, since f ∈ Lip∞ (ν), we obtain (4)

I1 ≤

X

|wx−tk |≤wγ/2

w L (wx − tk ) ψ C1 ∆k

Z

(tk+1 /w)−x

ν

!

|t| dt .

(tk /w)−x

In order to estimate (4), we introduce the following notations. Denote by L = {k ∈ Z : |wx − tk | ≤ wγ/2}, L1 = {k ∈ L : tk /w ≥ x} and L2 = {k ∈ L : tk+1 /w ≤ x}, with w > 0. In general L = L1 ∪ L2 ∪ L3 , where L3 = {k} if there exists k ∈ Z such that tk /w < x < tk+1 /w, otherwise L3 = ∅. In what follows we consider only the case of L3 = {k} (the case L3 = ∅ is similar). Now, we rewrite the right-hand side of (4) as follows X

|wx−tk |≤wγ/2

=

 

X

+

k∈L1

=:

L(wx − tk ) ψ

X

X

+

k∈L2

k∈L3

J1 + J2 + J3 .

C1



w ∆k

Z

(tk+1 /w)−x (tk /w)−x

L(wx − tk ) ψ

C1

w ∆k

|t|ν dt

Z

!

(tk+1 /w)−x

(tk /w)−x

!

|t|ν dt

We first estimate J1 . We obtain J1 ≤

X

k∈L1



L(wx−tk ) ψ C1

sup tk t∈[ w

t −x, k+1 w

−x]



|t|ν  ≤

X

k∈L1

L(wx−tk ) ψ (C1 |tk+1 /w − x|ν ) .

Now, since k ∈ L1 ⊂ L and | · |ν , 0 < ν ≤ 1, is concave and then subadditive, we have  ν tk + ∆k − wx ν tk − wx ν ∆ ν |tk+1 /w − x| = ≤ w + w w  ν  γ ν  ∆  ν ∆ u + ≤ + ≤ ≤ u 2 w 2 w for sufficiently large w > 0; then by condition (3) (5)

J1 ≤

M C1p X νp L(wx − tk ) |tk + ∆k − wx| , wνp k∈L1

for sufficiently large w > 0. Observing again that νp ≤ β0 ≤ p ≤ 1, | · | and then subadditive; hence it turns out that J1 ≤

≤

νp

is concave

M C1p X νp L(wx − tk ) [ |tk − wx| + ∆νp k ] wνp k∈L1

M C1p X M C1p ∆νp X νp L(wx − t ) |wx − t | L(wx − tk ). + k k wνp wνp k∈L1

k∈L1

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D. Costarelli, G. Vinti

We now estimate J2 . Since k ∈ L2 ⊂ L, it follows that |tk /w − x|ν ≤ (γ/2)ν ≤ u. Then, by condition (3) we obtain   X ν L(wx − tk ) ψ C1 sup J2 ≤ |t|  t∈[

k∈L2

≤

X

k∈L2

ν

L(wx − tk ) ψ (C1 |tk /w − x| ) ≤

tk w

−x,

tk+1 w

−x]

M C1p X νp L(wx − tk ) |wx − tk | , wνp k∈L2

for sufficiently large w > 0. Finally, we estimate J3 . Using again the subadditivity of | · |νp and condition (3), we obtain   J3 ≤ L(wx − tk ) ψ C1

t∈

ht

k w

sup

−x,

tk+1 w

i −x

ν |t| 

ν
ν  ≤ L(wx − tk ) ψ C1 max tk /w − x , tk+1 /w − x

νp   M C1p L(wx − tk ) tk − wx + ∆νp νp w p νp M C1p wx − tk νp + M C1 ∆ L(wx − tk ), = ) L(wx − t k wνp wνp for sufficiently large w > 0. Then, by the above estimates we can state that ( ) X M C1p X νp νp J1 + J2 + J3 ≤ L(wx − tk ) |wx − tk | + ∆ L(wx − tk ) wνp ≤

k∈L

(6)

≤

k∈L1 ∪L3

M C1p {mνp,π (L) + ∆νp m0,π (L)} , wνp

for every x ∈ R and for sufficiently large w > 0. Now, since if mβ0 ,π (L) < +∞ then mα,π (L) < +∞ for every 0 < α ≤ β0 and since νp ≤ β0 , we obtain from (6) that: J1 + J2 + J3 ≤

M C1p e w−νp < +∞, {mνp,π (L) + ∆νp m0,π (L)} =: C wνp

for every sufficiently large w > 0; then I1 = O(w−νp ), as w → +∞. We now estimate I2 . We have, as a consequence of Lemma 3.3 (ii), X I2 ≤ ψ (2kf k∞ ) L (wx − tk ) = O(w−β0 ), as w → +∞. |wx−tk |>wγ/2

Finally, we estimate I3 . Obviously, since χ(x, 0) = 0, if f (x) = 0 it turns out that I3 = 0. While, for f (x) 6= 0 we have X I3 = χ(wx − tk , f (x)) − f (x) ≤ |f (x)| Tw (x); k∈Z

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Order of approximation for nonlinear sampling Kantorovich operators

then we obtain by condition (χ4) I3 ≤ kf k∞ Tw (x) = O(w−θ0 ),

w → +∞,

as

uniformly with respect to x ∈ R. In conclusion, we have kSw f − f k∞ = O(w− ),

as

w → +∞,

with  := min {νp, β0 , θ0 }. Let now consider the case f ∈ Lip∞ (ν) with β0 /p ≤ ν ≤ 1 be fixed. It is easy to observe that, since Lip∞ (ν) ⊆ Lip∞ (β0 /p), we can immediately reduces to the above case, from which we obtain that kSw f − f k∞ = O(w− ),

as

w → +∞,

with  := min {β0 , θ0 } = min {νp, β0 , θ0 }. Finally, we study the case of χ satisfying condition (χ3) with L such that, condition (L2) is satisfied for β0 > p. As noted above, mβ0 ,π (L) < +∞ implies mp,π (L) < +∞, then L satisfies condition (L2) also for β0 = p and thus we can reduces to the previous case. This completes the proof.  5. Order of approximation in Orlicz spaces. We will now study the order of approximation for the nonlinear sampling Kantorovich operators in the general setting of Orlicz spaces. In order to obtain results in this direction, we will need of a growth condition on the composition of the function ϕ, which generates the Orlicz space and the function ψ of condition (χ3). Namely, given a ϕ-function ϕ we require that there exists a ϕ-function η such that, for every λ ∈ (0, 1), there exists a constant Cλ ∈ (0, 1) satisfying (H)

ϕ(Cλ ψ(u)) ≤ η(λu),

u ∈ R+ 0,

where ψ is the ϕ-function of condition (χ3) (see [10, 57, 30]). In what follows, we will write ϕ ∈ Hη to denote that the above condition (H) is satisfied for ϕ and η convex. First of all, we recall a modular continuity property for our operators. For a proof see [57]. Theorem 5.1 ([57]) Let ϕ ∈ Hη and f , g ∈ Lη (R). Then, for every λ ∈ (0, 1), there exists µ > 0 such that I ϕ [µ(Sw f − Sw g)] ≤

1 kLk1 I η [λ(f − g)]. δm0,π (L)

In particular, for g ≡ 0, we have that Sw : Lη (R) → Lϕ (R).

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D. Costarelli, G. Vinti

Now, we are ready to investigate the rate of convergence for the family of our nonlinear operators in Orlicz spaces. Theorem 5.2 Let χ be a kernel, ϕ ∈ Hη and f ∈ Lϕ+η (R) ∩ Lipη (ν), 0 < ν ≤ 1. Suppose in addition that, there exist α1 , γ > 0 such that Z (7) w L(wt) |t|ν dt = O(w−α1 ), as w → +∞, |t|≤γ

where L is the function of condition (χ3). Then, there exists µ > 0 such that I ϕ [µ(Sw f − f )] = O(w−θ ),

w → +∞,

as

with θ := min {ν, α0 , α1 , θ0 }, where α0 , θ0 > 0 are the constants of conditions (L3) and (χ4), respectively. Proof First of all, since f ∈ Lϕ+η (R) ∩ Lipη (ν), we have that there exist λ1 , λ2 > 0 such that I ϕ+η [λ1 f ] < +∞ and I η [λ2 (f (·) − f (· + t))] = O(|t|ν ),

t → 0,

as

i.e., there exist M1 , γ > 0 such that I η [λ2 (f (·) − f (· + t))] ≤ M1 |t|ν , for every |t| ≤ γ. Now, we fix λ > 0 such that λ < min {1, λ1 /2, λ2 } . In correspondence to λ, by condition (H) there exists Cλ ∈ (0, 1) such that ϕ(Cλ ψ(u)) ≤ η(λu), u ∈ R+ 0 , while by condition (χ4) there exists M2 > 0 such that Tw (x) ≤ M2 w−θ0 , uniformly with respect to x ∈ R, for sufficiently large w > 0. Now we choose µ > 0 such that µ < min {Cλ /(3m0,π (L)), λ1 /(3M2 )} .

By the properties of I ϕ we can write: Z I ϕ [µ(Sw f − f )] = ϕ(µ|(Sw f )(x) − f (x)|) dx R

(Z ! ! Z tk+1 /w X 1 w tk ≤ ϕ 3µ (Sw f )(x) − f (u + x − )du dx χ wx − tk , 3 ∆k tk /w w R k∈Z ! ! Z tk+1 /w Z X X tk w + ϕ 3µ χ wx − tk , f (u + x − )du − χ (wx − tk , f (x)) dx ∆ w k tk /w R k∈Z

k∈Z

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Order of approximation for nonlinear sampling Kantorovich operators

! ) X 1 + ϕ 3µ χ (wx − tk , f (x)) − f (x) dx =: {J1 + J2 + J3 } . 3 R Z

k∈Z

First, we estimate J1 . By condition (χ3), and applying Jensen’s inequality and Fubini-Tonelli theorem, we obtain ! ! Z tk+1 /w X tk w f (u + x − )du dx J1 = ϕ 3µ (Sw f )(x) − χ wx − tk , ∆k tk /w w R Z

≤ ≤

Z

R

1 m0,π (L)

(8) =

k∈Z

ϕ 3µ

m0,π (L)

Z

w ∆k

L (wx − tk ) ψ

k∈Z

tk+1 /w

tk /w

Z

w L(wx − tk )ϕ 3µm0,π (L)ψ ∆k R

Z

L(wx − tk )ϕ 3µm0,π (L) ψ

R k∈Z

!!

dx

tk+1 /w

tk /w

XZ k∈Z

tk |f (u) − f (u + x − )|du w

w ∆k

Z X

1

X

tk |f (u) − f (u + x − )|du w

tk+1 /w

tk /w

tk |f (u) − f (u + x − )|du w

!!

!!

dx

dx.

Putting t = x − tk /w in (8), using condition (H), and applying again Fubini-Tonelli theorem and Jensen’s inequality, we may write !! Z tk+1 /w XZ 1 w J1 ≤ L(wt)ϕ 3µm0,π (L)ψ |f (u) − f (u + t)| du dt m0,π (L) ∆k tk /w R k∈Z

≤ ≤ ≤ ≤ ≤

1 m0,π (L) 1 m0,π (L) 1 m0,π (L)

R

k∈Z

k∈Z

w L(wt) η λ ∆ k R

XZ

k∈Z

δm0,π (L)

Z

R

Z

R

w ∆k

L(wt) ϕ Cλ ψ

XZ

1

1

XZ

L(wt)

w L(wt)

w ∆k

Z

Z

tk /w

!!

tk+1 /w

tk /w

|f (u) − f (u + t)| du

tk+1 /w

dt

!

|f (u) − f (u + t)|du dt

tk+1 /w

tk /w

XZ

Z

η (λ|f (u) − f (u + t)|) dudt

tk+1 /w

η (λ|f (u) − f (u + t)|) dudt

k∈Z

tk /w

Z

 η (λ|f (u) − f (u + t)|) du dt

w L(wt) δm0,π (L) R R (Z Z  1 = w L(wt) η (λ|f (u) − f (u + t)|) du dt + δm0,π (L) |t|≤e γ R ) Z  Z 1 + w L(wt) η (λ|f (u) − f (u + t)|) du dt =: {J1,1 + J1,2 } , δm0,π (L) |t|>e γ R

283

D. Costarelli, G. Vinti

with γ e := min {γ, γ}, where γ > 0 is the constant of condition (7). Now, since f ∈ Lipη (ν), and using (7), we obtain that: Z J1,1 ≤ M1 as w → +∞. w L(wt) |t|ν dt = O(w−α1 ), |t|≤e γ

For J1,2 , by the convexity of η, we have Z  Z Z 1 J1,2 ≤ η(2λ|f (u)|)du + wL(wt) η(2λ|f (u + t)|)du dt. 2 R |t|>e γ R Noting that

Z

R

η(2λ|f (u + t)|) du =

Z

R

η(2λ|f (u)|) du,

for every t ∈ R, we obtain Z  Z Z η J1,2 ≤ w L(wt) η(2λ|f (u)|)du dt ≤ I [λ1 f ] w R

|t|>e γ

L(wt)dt < +∞,

|t|>e γ

for w > 0; hence J1,2 = O(w−α0 ), as w → +∞, by condition (L3) with α0 > 0. Now, we estimate J2 . By condition (χ3), putting t = u−tk /w and using Jensen’s inequality, we have !! Z w Z tk+1 /w X tk dx f (u + x − )du − f (x) J2 ≤ ϕ 3µ L (wx − tk ) ψ ∆ w k tk /w R k∈Z

!! w Z ∆k /w = ϕ 3µ L (wx − tk ) ψ dx f (t + x) dt − f (x) ∆k 0 R k∈Z !! Z Z ∆k /w X w ≤ ϕ 3µ L (wx − tk ) ψ |f (t + x) − f (x)| dt dx ∆k 0 R Z

X

k∈Z

≤

1 m0,π (L) ≤

Z X

R k∈Z

1 m0,π (L) ≤

L (wx − tk ) ϕ 3µm0,π (L)ψ

Z X

R k∈Z

1 m0,π (L)

w ∆k

L (wx − tk ) ϕ Cλ ψ

Z X

R k∈Z

w L (wx − tk ) η λ ∆k

w ∆k Z

0

Z

Z

0

∆k /w

0

∆k /w

!!

∆k /w

|f (t + x) − f (x)|dt !!

|f (t + x) − f (x)|dt

dx

dx

!

|f (t + x) − f (x)| dt dx.

Applying again Jensen’s inequality and Fubini-Tonelli theorem and since f ∈ Lipη (ν), we have " # Z X Z ∆k /w 1 w η (λ|f (t + x) − f (x)|) dt dx. J2 ≤ L (wx − tk ) m0,π (L) R ∆k 0 k∈Z

284

Order of approximation for nonlinear sampling Kantorovich operators

≤

1 δ m0,π (L) ≤

Z

w

R

k∈Z

Z

1 δ m0,π (L)

w = δ

Z

∆/w

0

X

R

L (wx − tk )

m0,π (L) w

"Z

"Z

#

∆/w

η (λ|f (t + x) − f (x)|) dt dx.

0

#

∆/w

η (λ|f (t + x) − f (x)|) dt dx.

0

Z

 Z ∆/w M1 η (λ|f (t + x) − f (x)|) dx dt ≤ w |t|ν dt, δ R 0

for sufficiently large w > 0. Now, we obtain w

Z

0

∆/w ν

|t| dt =

Z

∆

0

u | |ν du = w−ν w

Z

∆

0

|u|ν du = O(w−ν ),

as w → +∞,

then, J2 = O(w−ν ),

w → +∞.

as

Finally, denoted by A0 ⊆ R the set of all points of R for which f 6= 0 a. e., we have ! Z Z X ϕ (3µ |f (x)| Tw (x)) dx. χ (wx − tk , f (x)) − f (x) dx ≤ J3 = ϕ 3µ A0 A0 k∈Z

Now, using the convexity of ϕ and condition (χ4), we obtain Z Z
J3 ≤ ϕ (3µ |f (x)| Tw (x)) dx ≤ ϕ 3M2 µ w−θ0 |f (x)| dx A0

≤ w−θ0

A0

Z

A0

ϕ (3M2 µ |f (x)|) dx ≤ w−θ0 I ϕ [λ1 f ] < +∞,

for every sufficiently large w > 0, i.e., J3 = O(w−θ0 ), as w → +∞. In conclusion, we have I ϕ [µ(Sw f − f )] = O(w−θ ), as w → +∞, where θ := min {ν, α0 , α1 , θ0 }.



Note that, under suitable assumptions on the kernels, condition (7) is satisfied. For instance, in the special case of kernels χ satisfying condition (χ3) with L having compact support, e.g. supp L ⊂ [−B, B], B > 0, we have that Z Z u ν (9) w L(wt) |t|ν dt ≤ L(u) du =: Kw−ν , w |t|≤γ |u|≤B

for sufficiently large w > 0, i.e., α1 = ν. Moreover, as noted in Remark 3.1 (c), in the above case L satisfies condition (L3) for every α0 > 0. Hence, we obtain the following.

285

D. Costarelli, G. Vinti

Corollary 5.3 Let χ be a kernel satisfying condition (χ3) with L having compact support and let ϕ ∈ Hη . Moreover, let f ∈ Lϕ+η (R) ∩ Lipη (ν), 0 < ν ≤ 1. Then, there exists a constant µ > 0 such that I ϕ [µ(Sw f − f )] = O(w−θ ),

as

w → +∞,

with θ := min {ν, θ0 }, where θ0 > 0 is the constant of condition (χ4). In case that L has not compact support, we may require that the absolute moment of order ν > 0, defined by Z mν (L) := L(u) |u|ν du, 0 < ν ≤ 1, R

are finite. Also in this case, we can prove that condition (7) holds for α1 = ν. Indeed, we have Z Z u ν ν w L(wt) |t| dt = L(u) du ≤ mν (L) w−ν < +∞, w |t|≤γ |u|≤wγ

for every fixed w > 0. Then we can obtain the following.

Corollary 5.4 Let χ be a kernel and ϕ ∈ Hη . Suppose in addition that mν (L) < +∞, for some 0 < ν ≤ 1 and let f ∈ Lϕ+η (R) ∩ Lipη (ν). Then, there exists a constant µ > 0 such that I ϕ [µ(Sw f − f )] = O(w−θ ),

as

w → +∞,

with θ := min {ν, α0 , θ0 }, where α0 , θ0 > 0 are the constants of conditions (L3) and (χ4), respectively. Now, we give some examples of Orlicz spaces to which the above theory can be applied. As a first example, we consider the Orlicz space generated by the convex ϕ p ϕ-function ϕ(u) := up , u ∈ R+ 0 , 1 ≤ p < ∞, i.e., the Orlicz space L (R) = L (R), q/p very important for its applications in Signal Processing. If ψ(u) = u , 1 ≤ q ≤ p, condition (H) is satisfied with η(u) = uq and Cλ = λq/p . Furthermore, Lϕ+η (R) = Lp (R) ∩ Lq (R). Clearly, if q = p we are in the case of strongly Lipschitz kernel χ. Other important spaces in the applications are the so-called interpolation spaces Lα logβ L(R) (or Zygmund spaces), generated by the convex ϕ-function ϕα,β (u) := uα lnβ (e + u), u ∈ R+ 0 , α ≥ 1, β > 0. In this setting, if for example ψ(u) = u, u ∈ R+ , for (H) we have η(u) = ϕα,β (u) and Cλ = λ. Finally, others interesting 0 examples can be furnished by the exponential spaces, generated by the convex ϕγ function ϕγ = eu − 1, u ∈ R+ 0 and γ > 0. 6. Applications to special kernels. We now describe a natural procedure to construct kernels for the nonlinear sampling Kantorovich operators. We consider kernels of the following form χ(wx − tk , u) = L(wx − tk ) gw (u),

286

Order of approximation for nonlinear sampling Kantorovich operators

where (gw )w>0 is a family of functions gw : R → R satisfying gw (u) → u uniformly on R as w → +∞ and such that there exists a ϕ-function ψ with (10)

|gw (u) − gw (v)| ≤ ψ(|u − v|),

for every u, v ∈ R and w > 0. In this case, assumptions (χi), i = 1, . . . , 4, and (Lj), j = 1, 2, 3 could be translated to require that: (L1) the map k 7→ L(wx − tk ) ∈ `1 (Z) for every x ∈ R and w > 0, L is locally bounded in a neighborhood of the origin and there exists β0 > 0 such that X mβ0 ,π (L) := sup L(wx − tk )|wx − tk |β0 < +∞; x∈R

k∈Z

(L2) gw (0) = 0, for every w > 0; (L3) there exist θ0 > 0 such that g (u) X w L(wx − tk ) − 1 = O(w−θ0 ), Tw (x) := sup u u6=0 k∈Z

as w → +∞, uniformly with respect to x ∈ R;

(L4) there exists α0 > 0 such that, for every M > 0 Z w L(wx) dx = O(w−α0 ), |x|>M

as

w → +∞.

To construct a first example of kernel, we consider the well-known Fejér’s kernel of the form x 1 F (x) := sinc2 , (x ∈ R), 2 2 where the sinc-function and its Fourier transform are given by (  sin(πx) 1, |v| ≤ π, , x ∈ R \ {0} , d sinc(x) := sinc(v) := πx 0, |v| > π. 1, x = 0,

The Fejér’s kernel F play now the role of the function L. Clearly, F is bounded, belongs to L1 (R) and satisfies the moment conditions (L1) for every 0 < β0 ≤ 1 by Remark 3.1 (a) and in addition we have mν (F ) < +∞ for every 0 < ν ≤ 1 (see [22, 3, 29, 30, 31]). Moreover, for every M > 0 Z Z 2 1 e −1 , w F (wu) du ≤ 2 w−1 du =: Kw 2 π |u|>M |u|>M u for every w > 0 and therefore condition (L4) holds for α0 = 1. Furthermore, taking into account that the Fourier transform of F is given by (see [22])  1 − |v/π|, |v| ≤ π, b F (v) := 0, |v| > π,

287

D. Costarelli, G. Vinti

P we have that k∈Z F (u − k) = 1, for every u ∈ R, by Remark 3.2. Then, condition (L3) reduces to sup |gw (u)/u − 1| = O(w−θ0 ), as w → +∞, for some θ0 > 0. An u6=0

example of (gw )w>0 satisfying (L3) can be found at the end of this section. Then, the nonlinear sampling Kantorovich operators based on Fejér’s kernel take now the form !   Z (k+1)/w wx − k 1X 2 F sinc gw w f (u) du , (x ∈ R), (Sw f )(x) = 2 2 k/w k∈Z

for every w > 0, where f : R → R is a locally integrable function such that the above series is convergent for every x ∈ R. Note that in the above operators we F used a uniform sampling scheme, i.e., tk = k, k ∈ Z. For Sw f , from Theorem 4.1 and Corollary 5.4 we obtain respectively the following. Corollary 6.1 (i) If ψ of condition ( 10) satisfies the assumption ( 3) for 0 < p ≤ 1, then for every f ∈ Lip∞ (ν), 0 < ν ≤ 1, we have F kSw f − f k∞ = O(w− ),

as w → +∞,

where  := min {νp, θ0 }. (ii) Let ϕ ∈ Hη and f ∈ Lϕ+η (R) ∩ Lipη (ν), 0 < ν ≤ 1. Then, there exists a constant µ > 0 such that F I ϕ [µ(Sw f − f )] = O(w−θ ),

as

w → +∞,

where θ := min {ν, θ0 }. We observe that, since Fejér’s kernel has unbounded support, for the reconstruction F f , we need an infinite number of mean values of a given signal f by means of Sw R (k+1)/w w k/w f (u)du in order to evaluate our operators at any fixed point x ∈ R. Therefore, for a practical application of the theory of nonlinear sampling Kantorovich operators based upon kernel of the type considered above, with L having unbounded support, the sampling series must be truncated and this produce truncation errors. In order to avoid this problem, we can take into consideration kernels with L having compact support. In this case, the infinite sampling series computed at any x ∈ R, reduces to a finite one. Remarkable examples of such kernels are given by the well-known B-spline of order n ∈ N, defined by Mn (x) :=

  n n−1 X 1 n n (−1)i +x−i , (n − 1)! i=0 i 2 +

where the function (x)+ := max {x, 0} denotes the positive part of x ∈ R (see [22, 3, 57]). We have that, the Fourier transform of Mn is given by  v  n d M , (v ∈ R), n (v) := sinc 2π

288

Order of approximation for nonlinear sampling Kantorovich operators

P and then, as happens for Fejér’s kernel, we have k∈Z Mn (u−k) = 1, for every u ∈ R (see Remark 3.2); therefore condition (L3) reduces to sup |gw (u)/u − 1| = O(w−θ0 ), u6=0

as w → +∞, for some θ0 > 0. Clearly, Mn are bounded on R, with compact support on [−n/2, n/2] and hence Mn ∈ L1 (R), for all n ∈ N. Moreover, it easy to deduce that conditions (L1) and (L4) are fulfilled for every β0 > 0 and α0 > 0, as noted in Remark 3.1 (b) and (c), respectively. Then, the nonlinear sampling Kantorovich operators based upon the B-spline kernel of order n, with tk = k, k ∈ Z, take now the form ! Z (k+1)/w X Mn (Sw f )(x) = Mn (wx − k) gw w f (u) du , (x ∈ R), k∈Z

k/w

for every w > 0, where f : R → R is a locally integrable function such that the above series is convergent for every x ∈ R. From Theorem 4.1 and Corollary 5.3 we obtain respectively the following. Corollary 6.2 (i) If ψ of condition ( 10) satisfies the assumption ( 3) for 0 < p ≤ 1 then, for any f ∈ Lip∞ (ν), 0 < ν ≤ 1, we have Mn kSw f − f k∞ = O(w− ),

as

w → +∞,

where  := min {νp, θ0 }. (ii) Let ϕ ∈ Hη and f ∈ Lϕ+η (R) ∩ Lipη (ν), 0 < ν ≤ 1. Then, there exists a constant µ > 0 such that Mn f − f )] = O(w−θ ), I ϕ [µ(Sw

as

w → +∞,

with θ := min {ν, θ0 }. Finally, we show examples of family (gw )w>0 satisfying all the assumptions of our theory. Example 6.3 Define gw (u) = u1−1/w for every u ∈ (a, 1), with 0 < a < 1/e and gw (u) = u otherwise, for w > 0. Clearly, gw (u) → u uniformly on R, as w → +∞. In this instance, condition (L3) holds for θ0 = 1. Indeed, the function w gw (u) − u on (a, 1) achieve the maximum at u0 := ( w−1 w ) , for sufficiently large w > 0 (gw (u) − u = 0 otherwise), then we have, for every u ∈ R  w   1 C w−1 |gw (u) − u| ≤ |gw (u0 ) − u0 | = ≤ , w w−1 w−1 for sufficiently large w > 0 and for some constant C > 0; then gw (u)
C 1 sup − 1 = sup |gw (u) − u| ≤ a−1 = O w−1 , u |u| w − 1 u6=0 u∈(a, 1) as w → +∞.

D. Costarelli, G. Vinti

289

Moreover, it easy to see that the functions gw , w > 0, above defined, satisfy condition (10) for sufficiently large w > 0 with a concave ψ for which assumption (3) of Theorem 4.1 holds. Furthermore, if we consider the case gw (u) = u, u ∈ R, w > 0, we have that the function ψ of the kernel χ(x, u) = L(x) u is ψ(u) = u, u ∈ R+ 0 . In this instance, the F Mn operators Sw and Sw reduce to the linear ones studied in [3, 31] and as noted in Remark 3.2, condition (L3) becomes X as w → +∞, L(wx − k) − 1 = O(w−θ0 ), Tw (x) = k∈Z

uniformly with respect to x ∈ R, that is fulfilled for every θ0 > 0. References

[1] L. Angeloni, G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing, Differential and Integral Equations 18 (8) (2005), 855-890.

[2] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation of the Whittaker Sampling

Series in terms of an Average Modulus of Smoothness covering Discontinuous Signals, J. Math. Anal. Appl. 316 (2006), 269-306.

[3] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Kantorovich-Type Generalized Sampling Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing 6 (1) (2007), 29-52.

[4] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Prediction by samples from the past with error

estimates covering discontinuous signals, IEEE, Transaction on Information Theory 56 (1) (2010), 614-633.

[5] C. Bardaro, I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim. 33 (4) (2012), 374-396.

[6] C. Bardaro, J. Musielak, G.Vinti, Modular estimates and modular convergence for a class of nonlinear operators, Math. Japon. 39 (1994), 7-14.

[7] C. Bardaro, J. Musielak, G.Vinti, On absolute continuity of a modular connected with strong summability, Comment. Math. Prace Mat. 34 (1994), 21-33.

[8] C. Bardaro, J. Musielak, G.Vinti, Approximation by nonlinear integral operators in some modular function spaces, Ann. Polon. Math. 63 (2) (1996), 173-182.

[9] C. Bardaro, J. Musielak, G. Vinti, Nonlinear operators of integral type in some function

spaces. Fourth International Conference on Function Spaces (Zielona Góra, 1995) Collect. Math. 48 (1997), 409-422.

[10] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, New York, Berlin, 9, 2003.

[11] C. Bardaro, G. Vinti, Some Inclusion Theorems for Orlicz and Musielak-Orlicz Type Spaces, Annali di Matematica Pura e Applicata 168 (1995), 189-203.

[12] C. Bardaro, G. Vinti, Modular approximation by nonlinear integral operators on locally compact groups, Comment. Math. Prace Mat. 35 (1995), 25-47.

290

Order of approximation for nonlinear sampling Kantorovich operators

[13] C. Bardaro, G. Vinti, A general approach to the convergence theorems of generalized sampling series, Applicable Analysis 64 (1997), 203-217.

[14] C. Bardaro, G. Vinti, Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator, Results in Mathematics, special volume dedicated to Professor P.L. Butzer, 34 (3/4) (1998), 224-240.

[15] C. Bardaro, G.Vinti, On the order of modular approximation for nets of integral operators in modular Lipschitz classes, Funct. Approx. Comment. Math., special issue dedicated to Prof. Julian Musielak, 26 (1998), 139-154.

[16] C. Bardaro, G. Vinti, Nonlinear sampling type operators: approximation properties and regular methods of summability, Nonlinear Anal. Forum 6 (1) (2001), 15-26.

[17] C. Bardaro, G. Vinti, An Abstract Approach to Sampling Type Operators Inspired by the Work of P.L. Butzer - Part II - Nonlinear Operators, Sampling Theory in Signal and Image Processing 3 (1) (2004), 29-44.

[18] L. Bezuglaya, V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Zeitschrift für Analysis und ihre Anwendungen 12 (1993), 511-534.

[19] P.L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), 185-212.

[20] P.L. Butzer, W. Engels, S. Ries, R.L. Stens, The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM J. Appl. Math. 46 (1986), 299-323.

[21] P.L. Butzer, G. Hinsen, Reconstruction of bounded signal from pseudo-periodic, irregularly spaced samples, Signal Processing 17 (1989), 1-17.

[22] P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation I, Academic Press, New YorkLondon, 1971.

[23] P.L. Butzer, S. Ries, R.L. Stens, Shannon’s sampling theorem, Cauchy’s integral formula,

and related results, In: Anniversary Volume on Approximation Theory and Functional Analysis, (Proc. Conf., Math. Res. Inst. Oberwolfach, Black Forest, July 30-August 6, 1983), P.L. Butzer, R.L. Stens and B.Sz.-Nagy (Eds.), Internat. Schriftenreihe Numer. Math. 65, Birkhäuser, Basel, 1984, 363-377.

[24] P.L. Butzer, S. Ries, R.L. Stens, Approximation of continuous and discountinuous functions by generalized sampling series, J. Approx. Theory 50 (1987), 25-39.

[25] P.L. Butzer, W. Splettstößer, R.L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein 90 (1988), 1-70.

[26] P.L. Butzer, R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Review 34 (1) (1992), 40-53.

[27] P.L. Butzer, R.L. Stens, Linear prediction by samples from the past, Advanced Topics in

Shannon Sampling and Interpolation Theory, (editor R.J. Marks II), Springer-Verlag, New York, 1993.

[28] F. Cluni, D. Costarelli, A.M. Minotti, G. Vinti, Multivariate sampling Kantorovich operators:

approximation and applications to civil engineering, EURASIP Open Library. Proceedings of SampTA 2013, 10th International Conference on Sampling Theory and Applications, July 1st - July 5th, 2013, Jacobs University, Bremen, 400-403.

D. Costarelli, G. Vinti

291

[29] D. Costarelli, G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special volume dedicated to Prof. Giovanni Prodi 9 (IV) (2011), 445-468.

[30] D. Costarelli, G. Vinti, Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing, Numerical Functional Analysis and Optimization 34 (8) (2013), 819-844.

[31] D. Costarelli, G. Vinti, Order of approximation for sampling Kantorovich operators, to appear in: Journal of Integral Equations and Applications (2013).

[32] M.M. Dodson, A.M. Silva, Fourier Analysis and the Sampling Theorem, Proc. Ir. Acad. 86 A (1985), 81-108.

[33] C. Donnini, G. Vinti, Approximation by Means of Kantorovich Generalized Sampling Operators in Musielak-Orlicz spaces, PanAmerican Mathematical Journal 18 (2) (2008), 1-18.

[34] J.R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12 (1985), 45-89.

[35] J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Univ. Press, Oxford, 1996.

[36] J.R. Higgins, R.L. Stens, Sampling Theory in Fourier and Signal Analysis: advanced topics, Oxford Science Publications, Oxford Univ. Press, Oxford, 1999.

[37] A.J. Jerry, The Shannon sampling-its various extensions and applications: a tutorial review, Proc. IEEE 65 (1977), 1565-1596.

[38] W.M. Kozlowski, Modular Function Spaces, (Pure Appl. Math.) Marcel Dekker, New York and Basel, 1988.

[39] M.A. Krasnosel’skiˇi, Ya.B. Rutickiˇi, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd. - Groningen - The Netherlands, 1961.

[40] L. Maligranda, Orlicz Spaces and Interpolation, Seminarios de Matematica, IMECC, Campinas, 1989.

[41] I. Mantellini, G. Vinti, Approximation results for nonlinear integral operators in modular spaces and applications, Ann. Polon. Math. 81 (1) (2003), 55-71.

[42] J. Musielak, On some approximation problems in modular spaces, in Constructive Function

Theory, Proc. Int.Conf. Varna, June 1-5, 1981, pp. 455-461, Publ. House Bulgarian Acad. Sci., Sofia 1983.

[43] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math. (1034), 1983.

[44] J. Musielak, Approximation by nonlinear singular integral operators in generalized Orlicz spaces, Comment. Math. Prace Mat. 31 (1991), 79-88.

[45] J. Musielak, Nonlinear approximation in some modular function spaces I, Math. Japon. 38 (1993), 83-90.

[46] J. Musielak, On the approximation by nonlinear integral operators with generalized Lipschitz kernel over a compact abelian group, Comment. Math. Prace Mat. 33 (1993), 99-104.

[47] J. Musielak, On nonlinear integral operators, Atti Sem. Mat. Fis. Univ. Modena 47 (1999), 183-188.

292

Order of approximation for nonlinear sampling Kantorovich operators

[48] J. Musielak, Approximation by a nonlinear convolution operator in modular function spaces, Rocznik Nauk.-Dydakt. Prace Mat. 17 (2000), 181-190.

[49] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 28 (1959), 49-65. [50] M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Pure and Appl. Math., Marcel Dekker Inc. New York-Basel-Hong Kong, 1991.

[51] M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002.

[52] S. Ries, R.L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions’84, Sofia, 1984, 746-756.

[53] C.E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10-21. [54] C. Vinti, A Survey on Recent Results of the Mathematical Seminar in Perugia, inspired by the Work of Professor P.L. Butzer, Result. Math. 34 (1998), 32-55.

[55] G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Applicable Analysis 79 (2001), 217-238.

[56] G. Vinti, Approximation in Orlicz spaces for linear integral operators and applications, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 76 (2005), 103-127.

[57] G. Vinti, L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, Journal of Approximation Theory 161 (2009), 511-528.

[58] G. Vinti, L. Zampogni, A Unifying Approach to Convergence of Linear Sampling Type Operators in Orlicz Spaces, Advances in Differential Equations 16 (5-6) (2011), 573-600.

Danilo Costarelli Department of Mathematics and Physics, Section of Mathematics, University of Roma Tre, Largo S. Leonardo Murialdo, 1, 00146 Rome, Italy E-mail: [email protected] Gianluca Vinti Department of Mathematics and Computer Sciences, University of Perugia Via Vanvitelli, 1, 06123 Perugia, Italy E-mail: [email protected] (Received: 14.10.2013)