A Korovkin theorem in multivariate modular function spaces

In this paper a modular version of the classical Korovkin theorem in multivariate modular function spaces is obtained and applications to some multivariate discrete and integral operators, acting in Orlicz spaces, are given.


Introduction
The class of modular function spaces was introduced, for the first time, by H. Nakano [29] and then extensively studied by J. Musielak [27] who developed a theory of approximation in this general frame for classes of linear and nonlinear operators ( [28]).An abstract approach to the theory of approximation was given in its definite form in [4].This book represents the first attempt at a comprehensive treatment of approximation theory in modular spaces for nets of nonlinear operators.The interest in working in such general spaces is mainly to ensure an unifying approach which includes, by a unique method, several results in various functional spaces.Indeed modular function spaces include L p -spaces, Orlicz spaces, Musielak-Orlicz particular we obtain, as a special case, a version of the Korovkin theorem in L p spaces and in Orlicz or Musielak-Orlicz spaces.In Section 4, we apply our general theory to some kind of discrete operators acting on multivariate functions defined on nonempty bounded subsets of IR n .Then in Section 5 we consider the case of Mellin type integral operators (see [10]) for one dimensional Mellin convolution operators.Our result can be applied to various classical operators like multivariate Bernstein operators ( [24]) and multivariate moment operators ( [11] and [17]).

Notations and definitions
Let A be a nonempty open set in a Hausdorff locally compact topological space H provided with a regular measure μ defined on the Borel sets of H.We will assume that A is compact.We will denote by X(A) the space of all real-valued Borel measurable functions f : A → IR provided with equality μ-a.e., by C(A) the space of all continuous and bounded real functions defined on A and by C u (A) the subset of C(A) whose elements have a continuous extension to A. A functional : We will say that a modular is Q-quasi convex if there is constant for every f, g ∈ X(A), α, β ≥ 0, α + β = 1.If Q = 1 we will say that is convex.By means of the functional , we introduce the vector subspace of X(A), denoted by L (A), defined by The subspace L (A) is called the modular space generated by .It is easy to see that when is Q -quasi-convex we have the following characterization of the modular space L (A) : see for example [27] and [4].The subspace of L (A) defined by is called the space of the finite elements of L (A), see [27].The following assumptions on modulars will be used a) is monotone, i.e. for f, g ∈ X(A) is finite, i.e. denoting by e 0 the function e 0 (t) = 1 for every t ∈ A, e 0 ∈ L (A).Note that clearly e 0 ∈ C u (A).c) is absolutely finite, i.e. is finite and for every ε > 0, λ > 0 there is δ > 0 such that [λχ B ] < ε for any measurable subset B ⊂ A with μ(B) < δ.Here χ B denotes the characteristic function of the set B. d) is strongly finite, i.e. e 0 ∈ E (A).e) is absolutely continuous, i.e. there exists α > 0 such that for every f ∈ X(A), with [f ] < +∞, the following condition is satisfied: for every ε > 0 there is δ > 0 such that [αf χ B ] < ε, for every measurable subset B ⊂ A with μ(B) < δ.
Classical examples of modular spaces are given by the Orlicz spaces generated by a ϕ− function ϕ or, more generally, by any Musielak-Orlicz space generated by a ϕ-function ϕ depending on a parameter, satisfying some growth condition with respect to the parameter (see [27], [23], [4] in some special cases).The modular functionals generating the above spaces satisfy all the previous assumptions.
We say that a sequence of functions ( This notion extends the norm-convergence in L p − spaces.Moreover it is weaker than the F-norm-convergence induced by the Luxemburg F-norm generated by and defined by We recall that a sequence of functions (f n for every λ > 0. The two notions of convergence are equivalent if and only if the modular satisfies a Δ 2 − condition, i.e. there exists a constant M > 0 such that [2f ] ≤ M [f ], for every f ∈ X(A), see [27].For example, this happens for every L p -spaces and Orlicz spaces generated by ϕ-functions with the Δ 2 -regularity condition (see [27], [4]).The modular convergence induces a topology on L (A), called modular topology.Given a subset B ⊂ L (A), we will denote by B the closure of B with respect to the modular topology.Then f ∈ B if there is a sequence (f n ) n∈I N ⊂ B such that f n is modularly convergent to f.Let us remark that C(A) ⊂ L (A) whenever is monotone and finite.Indeed, for λ > 0 we have [λf ] ≤ [λ f ∞ e 0 ], and so, since e 0 ∈ L (A), we have lim λ→0 + [λf ] = 0, that is f ∈ L (A).Analogously, if is monotone and strongly finite, then C(A) ⊂ E (A).We have the following (see [25] and [4]).

A Korovkin theorem in modular function spaces
Let e 1 , . . .e m be m functions in C u (A) such that the following property (P) holds: there exist continuous functions a i ∈ C u (A), i = 1, . . .m such that the function is positive and equal to zero if and only if s = t.
Let T = (T n ) n∈I N be a family of positive linear operators T n : D → X(A), where C u (A) ⊂ D ⊂ X(A).Here D is the domain of the operators T n .We will assume that the family (T n ) n∈I N satisfies the following property ( * ) : there exists a subset X T ⊂ D ∩ L (A) with C u (A) ⊂ X T and a constant R > 0 such that for every function f ∈ X T we have for an absolute constant R > 0 for every λ > 0 and for every f ∈ D ∩ L (A), then clearly we can take X T = L (A) ∩ D. We will provide an example of T n for which property ( * ) holds for a suitable subspace In what follows, we will assume that lim n→+∞ T n e i = e i , i = 1, . . .m modularly in L (A). (2)

Lemma 1. Let be a monotone modular. Let the assumption (2) be satisfied and let us consider the function
where a i are constants.Then lim n→+∞ T n P = P modularly in L (A).
Let M be such that |a i | ≤ M for every i = 1, . . .m and let α > 0 be such that αmM ≤ λ.Then, using the property of the modular, we get and so the assertion follows.
Lemma 2. Let be a monotone modular.Let the assumptions (P) and (2) be satisfied.Then for the function P s (t) in (1) there holds Proof.Let M > 0 be so large that |a i (s)| ≤ M for every i = 1, . . ., m and for every s ∈ A. From (2) we can find a constant λ > 0 such that and so the assertion follows.
Lemma 3. Let be a finite, monotone and Q -quasi-convex modular.Let the assumptions (P) and (2) be satisfied.Let Proof.Firstly, note that there exists a function P of the form P (t) = m i=1 a i e i (t), such that P (t) > 0 for all t ∈ A. Indeed, given two points s 1 = s 2 of A we can take P = P s1 + P s2 .Let us consider the diagonal Applying the operators T n we have Then, for γ > 0 we have Let us consider I 1 .We can choose a positive constant a > 0 such that 1 = e 0 (t) ≤ a P (t), t ∈ A. So applying the modular we have Let us consider I 1,1 .By Lemma 1, there exists α > 0 such that [α(( for sufficiently large n.For I 1,2 since the functions e 1 , . . ., e m ∈ L (A), there exists ν > 0 such that [νe i ] < +∞ for every i = 1, . . ., m.Now, putting M = max i=1,...m |a i | and taking γ such that 4γQamM < ν and 4γQa < α, we have Thus we get I 1 ≤ εW, for an absolute constant W > 0. For I 2 , by Lemma 2, we can take γ such that lim n→+∞ I 2 = 0 modularly.Thus, for sufficiently small γ > 0 we get lim n→+∞ [γ(( Lemma 4. Let be a finite, monotone and Q -quasi-convex modular.Let the assumptions (P) and (2) be satisfied.Then for every f ∈ C u (A) we have lim n→+∞ Proof.Let f ∈ C u (A) be fixed.Let us take where the function P is strictly positive in A. By Lemma 3 there exists For a constant δ > 0 we have For J 1 , if 2δ < γ we have lim n→+∞ J 1 = 0. Moreover let Γ := max s∈A | f (s) P (s) |, then J 2 ≤ [2δΓ(T n P − P )] and so for sufficiently small δ > 0 we get lim n→+∞ J 2 = 0 and so the assertion follows.
Remark 1.We remark that if assumption (2) holds in strong sense in L (A) then using exactly the same proof as before we can show that lim n→+∞ T n f = f strongly in L (A) for every f ∈ C u (A).
The main theorem of this section is the following Theorem 1.Let be a monotone, absolutely finite, absolutely continuous and Q -quasi-convex modular on X(A).Let T = (T n ) n∈I N be a sequence of positive linear operators satisfying property ( * ).Let the assumption (P) be satisfied.Then if Passing to limsup, taking into account Remark 1 and property ( * ), we obtain lim sup n→+∞ [λ(T n f − f )] ≤ ε(R + 1) and the assertion follows from the arbitrariness of ε > 0.
Remark 2. Note that a similar result holds true in the case when A is compact replacing of course C u (A) with C(A).The proof is exactly the same.

Application to discrete operators
Let A ⊂ IR N be a bounded open set and let (r(n)) n∈I N be an increasing sequence of natural numbers.
For every fixed n ∈ IN, by ), we denote a finite sequence of points such that Γ n = A. Let us consider a sequence S = (S n ) n∈I N of positive operators of the form Note that the domain of the operator (3) contains the space X(A), due to the nature of the operator.Here X(A) is the space of all real valued measurable functions which are everywhere defined on A (i.e.we distinguish two equivalent but different functions).For every j = 1, . . ., N and s = (s 1 , . . ., s N ) we put We put e 0 (t) = 1, e i (t) = t i for i = 1, . . ., N and e N +1 (t) = |t| 2 , t = (t 1 , . . ., t N ) ∈ A. Note that these functions satisfy property (1) taking According to the above assumptions we have immediately S n e 0 = e 0 = 1, for every n ∈ IN.We have the following Proposition 2. Let be a finite and monotone modular on X(A).Then a necessary and sufficient condition that modularly (strongly) in L (A) is that lim n→+∞ S n e j = e j , j=1,. . ., N+1, modularly (strongly) in L (A).
Proof.We prove the proposition in case of strongly convergence.We can assume λ = 1.First we prove the necessary condition.It is obvious that (S n e j )(s) − e j (s) = m j (K n , s), j = 1, . . ., N. Moreover

Passing to the modular we have [S
that is the assertion.For the sufficient condition, note that and so applying the modular, as before, we obtain the assertion.
We have the following corollary Corollary 1.Let be a monotone, strongly finite, absolutely continuous and Q -quasi-convex modular on X(A).Assume that the family (S n ) n∈I N satisfies property ( * ) and (4) holds in the strong sense.Then where X S is the corresponding class given in property ( * ).
Here we will describe the class X S in some particular case.Let Φ be the class of all functions ϕ : IR + 0 → IR + 0 such that ϕ is a convex function, ϕ(0) = 0, ϕ(u) > 0 for u > 0 and lim u→+∞ ϕ(u) = +∞.For ϕ ∈ Φ, we define for every f ∈ X(A), the functional As it is well known, ϕ is a convex modular on X(A) and the subspace is the Orlicz space generated by ϕ, (see [27]).The subspace of L ϕ (A), defined by is called the space of finite elements of L ϕ (A).For example every bounded function belongs to E ϕ (A).Note that this modular satisfies all the assumptions listed in Section 2.
Let us consider the sequence of operator (3) and let us assume that where ξ n is a bounded sequence of positive numbers.For every n ∈ IN, we define Now, let us denote by A ϕ the class of all functions in L ϕ (A) such that lim sup for every λ > 0 and an absolute constant R > 0 independent of f and λ.We have the following Proof.Let λ > 0 be fixed.Using the Jensen inequality and the assumptions on the kernel (K n ) n∈I N , we get and so, passing to the limsup, we obtain immediately Let n ∈ IN be fixed and let r i (n), a finite sequence of positive integers, i = 1, . . .N. Let us consider a multi-index h = (h 1 , . . ., h N ) ∈ IN N , such that 0 ≤ h i ≤ r i (n), for every i = 1, . . ., N. For any choice of h we consider the vector ν n,h = (ν 1 n,h1 , . . ., ν N n,hN ), where for every i = 1, . . ., N, (ν i n,hi ), i = 0, . . .r i (n), is a finite partition of the interval I = [0, 1] of the i-axis.Putting let us assume that there exist two sequences (a n ), (b n ) of positive real numbers, such that 0 < a n ≤ Δ n h ≤ b n , for every h ∈ IN N and n ∈ IN, and b n → 0, n → +∞.By a renumbering of the vectors ν n,h into a sequence ν n,k , k = 0, 1, . . .r(n), let us consider a kernel K n (s, ν n,k ), satisfying the above assumptions and let ξ n be the corresponding sequences of numbers which dominate the integrals over A. Finally, let us assume that 0 ≤ ξ n /a n ≤ M, for a fixed constant M > 0 and any n ∈ IN.Thus, in this instance, the class A ϕ contains all the Riemann integrable functions over A. Indeed, we have, for λ = 1 lim sup The last sum is a Riemann sum of the function ϕ • |f | and so, if f is Riemann integrable, then the above limsup is dominated by the integral M A ϕ(|f (s)|)ds and from this the assertion follows.

Application to Mellin-type operators
Let us consider A = [0, 1] N and for any vectors t = (t 1 , . . ., t N ), s = (s 1 , . . ., s N ) ∈ A, we put ts = (t 1 s 1 , . . ., t N s N ).Let (K n ) n∈I N be a sequence of kernel functions K for every n ∈ IN and W is an absolute constant.Here, for a sake of simplicity, we consider an Orlicz space.Let ϕ ∈ Φ be fixed and let L ϕ (A) be the corresponding Orlicz space.For any function f ∈ L ϕ (A) we define the positive linear operator In this instance we can show that L ϕ (A) where DomT n is the subset of X(A) on which T n f is well defined as a measurable function of s ∈ A. A first result on these operators is given by the following proposition.
Proof.By the Jensen inequality and the Fubini-Tonelli theorem, we have As a consequence of the above proposition we get X T = L ϕ (A).We define the integral moments m i (K n , s) and m i,2 (K n , s) on putting, for i = 1, . . ., N, As in discrete case, according to the above assumptions, we have immediately Proof.The proof follows from the identities, m i (K n , s) = (T n e i − e i )(s) and m i,2 (K n , s) = (T n e 2 i − e 2 i )(s) − 2s i m i (K n , s), for i = 1, . . ., N. As a consequence we get the following corollary Corollary 2. If the moments m i (K n , •) and m i,2 (K n , •), i = 1, . . ., N, are strongly convergent to zero then lim n→+∞ T n f = f, modularly in L ϕ (A) for each f ∈ L ϕ (A).
Proof.We only remark that, putting e N +1 (t) = |t| 2 , we have also lim n→+∞ T n e N +1 = e N +1 strongly in L ϕ (A).
Remark 3 Note that the above results hold also in abstract modular function spaces.In this instance, besides the above assumptions on the generating modular , (monotonicity, absolute finiteness and absolute continuity), we have to assume some generalized Jensen convexity, in integral form and a notion of subboundedness (see e.g.[4]).In particular we have to assume an inequality of the form [f (t•)] ≤ F (t) [f (•)] where F is a measurable function such that A K n (t)F (t)dt ≤ W for every n ∈ IN and an absolute constant W > 0. These assumptions are automatically satisfied in Orlicz spaces and are fundamental in order to obtain the modular continuity of the operators T n (Proposition 4).
Aknowledgments.We wish to express our deep gratitude to the Referees for their valuable and interesting suggestions which improved the paper.