Metric approximation of set-valued functions of bounded variation by integral operators

We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$. All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a set-valued function $F$, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to $F$. At points of discontinuity of $F$, we derive estimates, which yield the convergence to a set, first described in our previous work on the metric Fourier operator. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction $F$ is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $L^1$ provides our global estimates. The theory is illustrated by presenting the examples of two concrete operators: the Bernstein-Durrmeyer operator and the Kantorovich operator.


Introduction
We study set-valued functions (SVFs, multifunctions) that map a compact interval [a, b] ⊂ R into the space of compact non-empty subsets of R d .These functions appear in different fields such as dynamical systems, control theory, optimization, game theory, differential inclusions, economy, geometric modeling.See the book [2] for foundations of set-valued analysis.
Approximation methods for SVFs has been developing in the last decades.Older works deal mostly with convex-valued multifunctions and their approximation based on Minkowski linear combinations, e.g., [24,13,15,23,5,9].In [28], such an adapation of the classical Bernstein polynomial operator is proved to converge to SVFs with convex compact images (values).Yet, it is shown that this adaptation fails to approximate general SVFs (with general compact not necessarily convex images).In general, approximation methods developed for multifunctions with convex images usually are not suitable for general SVFs.
A first successful attempt to approximate general SVFs from their samples is accomplished by Z. Artstein in [1], where piecewise linear approximants are constructed.This is done by replacing binary Minkowski average of two sets with the metric average, which is further extended in [16] to the metric linear combination of several sets.Based on the metric linear combination, N. Dyn, E. Farkhi and A. Mokhov developed in a series of works [14,20,16,17,18] adaptation of classical sample-based approximation operators to continuous general SVFs.For these adapted operators, termed metric operators, error estimates are obtained, which for most operators are similar to those obtained in the real-valued case.Special attention is given to Bernstein polynomial operators, Schoenberg spline operators and polynomial interpolation operators.Later in [6], the above metric approach is extended to SVFs of bounded variation.
The metric approach is applied in [19] to introduce and study the metric integral for general SVFs of bounded variation.The metric integral is not necessarily convex in contrast to the Aumann integral, which is always convex, even if the integrand is not convex-valued [3].In [7] the metric integral is extended to the weighted metric integral, which is used, with the Dirichlet kernels as weight functions, to define metric Fourier partial sums for SVFs of bounded variation.The convergence of these partial sums is analyzed at points of continuity of a multifunction as well as at points of discontinuity.An important tool in the analysis at points of discontinuity is the notion of one-sided local quasi-moduli of a function of bounded variation.
In this paper we adapt integral approximation operators for real-valued functions to general SVFs of bounded variation.Previous adaptations of integral operators to SVFs are limited to convex-valued multifunctions and are based on the Aumann integral (see e.g.[4,10]).Our adaptation is based on the weighted metric integral, and its analysis applies and extends the techniques developed in [7].
The outline of the paper is as follows.Section 2 gives a short overview of notions we use in the paper, and also discusses different regularity properties of functions with values in a metric space.In Section 3 we refine known results concerning approximation of real-valued functions by sequences of integral approximation operators, which are necessary for the adaptation of these operators to SVFs.
The core part of the paper is Section 4 where we construct an adaptation of integral approximation operators to general SVFs.For set-valued functions of bounded variation with compact graphs, we study pointwise convergence, in the Hausdorff metric, of sequences of such operators at points of continuity of the function as well as at points of discontinuity, and derive estimates for the rate of convergence.In Section 5 we illustrate our theory by considering examples of two particular integral approximation operators, the Bernstein-Durrmeyer operator and the Kantorovich operator.
In the final Section 6 we provide global error bounds.The multifunction F is represented by the set of all its metric selections (see [18] for more information on representations of SVFs), while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator.A bound of the Hausdorff distance between these two sets of single-valued functions in L 1 is obtained using results from [8].

Preliminaries
In this section we introduce some notation and basic notions related to sets and set-valued functions.We discuss notions of regularity of functions in metric spaces.We review the notions of metric selections and the metric integral of set-valued functions.

On sets
All sets considered from now on are sets in R d .We denote by K(R d ) the collection of all compact non-empty subsets of R d .The metric in R d is of the form ρ(u, v) = |u − v|, where | • | is any fixed norm on R d .Recall that R d endowed with this metric is a complete metric space and that all norms on R d are equivalent.To measure the distance between two non-empty sets A, B ∈ K(R d ), we use the Hausdorff metric based on ρ where the distance from a point c to a set D is dist(c, D)ρ = inf d∈D ρ(c, d).It is well known that K(R d ) endowed with the Hausdorff metric is a complete metric space [25,27].
In the following, we keep the metric in R d fixed, and omit the notation ρ as a subscript.We denote by |A| = haus(A, {0}) the "norm" of the set Using metric pairs, we can rewrite We recall the notions of a metric chain and of a metric linear combination [19].
Note that the metric linear combination depends on the order of the sets, in contrast to the Minkowski linear combination of sets which is defined by The upper Kuratowski limit of a sequence of sets {An} ∞ n=1 is the set of all limit points of converging subsequences

Notions of regularity of functions with values in a metric space
In this paper we consider functions defined on a fixed compact interval [a, b] ⊂ R with values in a complete metric space (X, ρ), where X is either We recall the notion of the variation of f : The variation of f on the partition χ is defined as where the supremum is taken over all partitions of [a, b].
We call functions of bounded variation BV functions and write and that v f is monotone non-decreasing.
For a BV function f : R → X the following property holds (see e.g.Lemma 2.4 in [6]), b a We recall the notion of the local modulus of continuity [26], which is central to the approximation of functions at continuity points.
It follows from the definition of the variation that A function f : [a, b] → X of bounded variation with values in a complete metric space (X, ρ) is not necessarily continuous, but has right and left limits at any point x [11].We denote the one-sided limits by In [7], we introduced the notion of the left and right local quasi-moduli.For a function f : [a, b] → X of bounded variation, the left local quasi-modulus at point x * is Similarly, the right local quasi-modulus is Clearly, for f ∈ BV[a, b] the local quasi-moduli satisfy and lim we obtain lim , respectively, and δ > 0 we have Proof.The first inequality follows from the fact that for x * ∈ (a, b] and max{x * − δ, a} ≤ x < x * we have Similarly one can show the second inequality.

Below we discuss several notions of Lipschitz regularity. A functions
(a) We say that f is locally Lipschitz around a point x with the Lipschitz constant L > 0 if there exists δ > 0 such that We denote by Lip{x, L} the collection of all functions f satisfying (7).
Remark 2.7.Note that if f is locally Lipschitz around a point x with the Lipschitz constant L, then f ∈ Lip{x, L}, but the inverse implication does not hold.For example, the function f (x) = x sin(1/x) for x = 0 and f (0) = 0 is not locally Lipschitz around x = 0, while f ∈ Lip{0, 1}.
We say that f : The following lemmas deal with relations between the above notions.
2 we obtain from the above inequality the estimate ρ(f Proof.Assume that f satisfies ( 6) with some δ > 0. To prove (i where the supremum is taken over all partitions By (i) and Remark 2.7 we obtain (ii).(i) If v f is locally Lipschitz around a point x with some L > 0 and δ > 0, then f is locally Lipschitz around x with the same L and δ.
To prove (ii) we replace either y or z in (9) by x.
Remark 2.11.The inverse implication of Lemma 2.10 (ii) does not hold.Consider, for example, the function f (x) = x sin(1/x) for x = 0 and f (0 In the following we consider f : Proof.Consider a sequence of partitions χn Using the triangle inequality in the estimate for the Riemann sums we obtain We recall the notion of integral moduli of continuity for functions with values in R d .We extend a function The distance between two functions f, g ∈ L 1 [a, b] is given by We denote The first order integral modulus of continuity of f : The second order integral modulus of continuity is Using ( 2) one can easily obtain that for

Metric selections and the weighted metric integral of multifunctions
We consider set-valued functions (SVFs, multifunctions It is easy to see that if F ∈ BV[a, b] then Graph(F ) is a bounded set and We denote the class of SVFs of bounded variation with compact graphs by The notions of the metric selections and of the weighted metric integral are central in our work.We recall their definitions.
Given a multifunction A selection s of F is called a metric selection, if there is a sequence of chain functions We denote the set of all metric selections of F by S(F ).
Note that the definitions of chain functions and metric selections imply that a metric selection s of a multifunction F is constant in any open interval where the graph of s stays in the interior of Graph(F ).
Below we quote some results from [19] and [7] which are used in this paper.In particular, if F is continuous at x * , then s is continuous at x * . Result A direct consequence of Lemma 2.5 and Result 2.17 is Now we show that the metric selections of F inherit the Lipschitz regularity property at x from vF .
The metric integral of SVFs is introduced in [19] and extended to the weighted metric integral in [7].We recall its definition.Here and in such occasions below we understand that the integral is applied to each component of s =    s1 . . .
3 Rate of pointwise convergence of integral operators for realvalued functions , be a sequence of functions that are integrable with respect to t for each x.We term the functions Kn kernels.With the help of the sequence of the kernels we define a sequence of linear integral operators {Tn} n∈N on real-valued functions in L ∞ [a, b] by Furthermore, for each

The case of continuity points
The theorem below can be considered as a refinement of Theorem 2. x ∈ [a, b] and let M (x) be as in (14).
(ii) If lim n→∞ αn(x) = 0, lim n→∞ βn(x, δ) = 0 for any sufficiently small δ > 0, M (x) < ∞, and if x is a point of continuity of f , then uniformly in x ∈ I for any sufficiently small δ > 0, and if M (x) is bounded on I, then the convergence is uniform in I. Proof.
(i) We have Using the fact that |f (t) − f (x)| ≤ ω(f, x, 2δ) in the first integral and that |f (t which gives (15).
(iii) By Lemma 3.2, ω(f, x, 2δ) tends to zero uniformly in x ∈ I when δ → 0+ and it follows from ( 15) that the convergence is uniform in I.
In view of Lemma 2.12, estimates in the proof of Theorem 3.3 can be repeated verbatim for f : where C depends only on the underlying norm | • | in R d .

The case of discontinuity points
In this section we follow and refine the analysis in [21] for real-valued functions. Let Note that f ∈ BV[a, b] implies that gx is well defined and is continuous at x.It is easy to check that where χA denotes the characteristic function of a set A where Inserting this representation into the operator (13), we obtain Taking into account that gx(x) = 0, the estimate (15) for Tngx(x) takes the form By the definition of the function gx we easily see that gx ∞ ≤ 2 f ∞ .By the definition of the local modulus of continuity and the local quasi-moduli (see Section 2.2), we get Clearly, Combining this with ( 17), ( 18), ( 19) we arrive at Thus we obtain the following result, (i) For all δ > 0 and n ∈ N we have (ii) If lim n→∞ αn(x) = 0, lim n→∞ βn(x, δ) = 0 for any sufficiently small δ > 0, lim n→∞ Tn(sign(• − x))(x) = 0, and Proof.By the triangle inequality and ( 20) we have which leads to the first claim.The second claim follows directly from it by applying (5).
Similarly to Corollary 3.4 for vector-valued functions we obtain 4 Rate of pointwise convergence of integral operators for setvalued functions Denoting κn,x(t) = Kn(x, t) and using the concept of the weighted metric integral (see Section 2.3), we define By Result 2.20 and since κn,x(t) = Kn(x, t) we have for where S(F ) is the set of metric selections of F .By Result 2.14 we have as well It is easy to obtain from ( 22) and ( 23) that haus (TnF (x), The arguments leading to (24) are similar to those in the proof of Lemma 6.3 below.
This approach was used for operators T defined on continuous real-valued functions in [18, Section 8.2].

The case of continuity points
Recall that by Results 2.13 and 2.15 for each selection s ∈ S(F ) we have Thus for each s ∈ S(F ) the estimate in Corollary 3.4 turns into We arrive at Proof.The statements (i) and (ii) follow from the first two claims of Theorem 3.3 combined with (24) and (25).The proof of the statements (iii) is based on Result 2.3, which implies that vF is uniformly continuous on compact intervals, and therefore ω(vF , x, 4δ) → 0 uniformly for x ∈ I.
By Corollary 3.7 and the above inequality we have for any s ∈ S(F ) This together with (26), by arguments as in the proof of (ii) of Theorem 3.3, leads to the following result

Specific operators
Here we apply the results of the previous sections to two specific integral operators.In particular, at points of discontinuity we obtain error estimates that combine ideas from [21,29] with the local quasi-moduli of continuity ( 3), (4).

Bernstein-Durrmeyer operators
For x ∈ [0, 1], n ∈ N the Bernstein basis polynomials are defined as p n,k (x) = 1 and The Bernstein-Durrmeyer operator is defined for where Kn(x, t) = (n + 1) The Bernstein-Durrmeyer operator for a set-valued function with κn,x(t) = Kn(x, t), where Kn(x, t) is given in (28).The properties of the Bernstein basis polynomials p n,k yield Kn(x, t) ≥ 0, A direct calculation shows that (see e.g.[21])

The case of continuity points
The assumptions of Theorem 3.3 are fulfilled for the Bernstein-Durrmeyer operator.Thus we obtain the following result, where part (i), which provides rate of convergence, is new in this form, while parts (ii) and (iii) are already known.
From Theorem 4.2 for the Bernstein-Durrmeyer operator we get (ii) If x is a point of continuity of F , then limn→∞ haus (MnF (x), F (x)) = 0.
(iii) If F is continuous at all points of a closed interval I ⊆ [0, 1], then the convergence is uniform in I.
By formula (21) we can extend this operator to set-valued functions F ∈ F[0, 1].Using the properties of p n,k (x), we obtain, as in the case of the Bernstein-Durrmeyer operator, that Kn(x, t) ≥ 0, Calculating Knei(x), i = 0, 1, 2 one can get (see e.g.[29]) and Thus, estimating βn(x, δ) as in the case of the Bernstein-Durrmeyer operator, we obtain The case of continuity points Theorem 3.3 takes for the Kantorovich operator the following form, where part (i) is new in this form (providing rate of convergence), while parts (ii) and (iii) are well-known.(ii) If x is a point of continuity of F , then limn→∞ haus (KnF (x), F (x)) = 0.
(iii) If F is continuous at all points of a closed interval I ⊆ [0, 1], then the convergence is uniform in I.

The case of discontinuity points
Theorem 3.6 for the Kantorovich operator gives the estimate for x ∈ (0, 1).An estimate for Kn(sign(• − x))(x) is given in [29].Closely following the consideration of Zeng and Piriou in [29], we derive here a very similar estimate in a slightly different form.We also replace the estimate used in [29] and quoted there from an unpublished paper, by the estimate of Guo, which is published in [21] with a full proof.
Theorem 4.3 for the Kantorovich operator takes the following form. .

Rate of convergence of the Bernstein-Durrmeyer operators and the Kantorovich operators for locally Lipschitz functions
In this section the sequence {Tn} is the sequence of Bernstein-Durrmeyer operators {Mn} or the sequence of Kantorovich operators {Kn}.
Using the notion of locally Lipschitz functions we can state the following result.
Since F is Lipschitz around the point x with some L > 0, by (ii) of Lemma 2.9, vF ∈ Lip{x, L} and by Lemma 2.19 each metric selection s ∈ S(F ) satisfies s ∈ Lip{x, 4L}.By Lemma 2.8 there exists L > 0 such that Using Lemma 2.12, (32), the Cauchy-Schwarz inequality and (29) for {Mn} or (30) for {Kn}, we get Motivated by (22), we regard the set of functions TnS(F ) as an approximant to F , represented, in view of Result 2.14, by the set of functions S(F ).We show below that S(F ) and TnS(F ) are elements of the metric space H of compact non-empty subsets of L 1 [a, b], endowed with the Hausdorff metric.We use this metric to measure the approximation error of S(F ) by TnS(F ).

Two compact sets of functions
In the the next two lemmas we prove that the sets S(F ), TnS(F ) are compact in L 1 [a, b], thus they are elements of H.It is enough to show that they are sequentially compact, i.e. that every sequence in S(F ) (in TnS(F ), respectively) has a convergent subsequence with a limit in S(F ) (in TnS(F ), respectively).Proof.We have to show that any sequence {σ k } ∞ k=1 ⊂ TnS(F ) has a subsequence converging in the L 1 -norm and that its L 1 -limit is in TnS(F ).
By definition  Now the statement follows easily.

Error estimates
For ei(x) = x i , denote λn = (max i=0,1 Tnei − ei L 1 ) 1/2 .Here we use the integral moduli ϑ and ϑ2 defined in Section 2.2.In light of the Theorem in [8] we get for real-valued functions (as a special case of the statement in [8] for p = 1)
then Tnf is obtained by the operation of Tn on each component of f .Next we introduce the following notation.For x ∈ [a, b] let αn(x) = b a Kn(x, t)dt − 1 and βn(x, δ) = |x−t|≥δ |Kn(x, t)|dt, δ > 0.

1 in [ 12 ,Lemma 3 . 2 .Theorem 3 . 3 .
Chapter 1].For that we need the following result[22,  page 12] If the function f is defined on [a, b] and I ⊆ [a, b] is a closed interval such that f is continuous at all the points of I, then for any ǫ > 0 there exists δ > 0 such that |f (y) − f (x)| < ǫ for all x ∈ I, y ∈ [a, b] and |y − x| < δ.Let f : [a, b] → R be bounded and measurable on [a, b],

Remark 4 . 1 .
Any bounded linear operator defined for single-valued functions can be extended to set-valued functions from the class F[a, b] by

and the statement follows. 6
Approximation of the set of metric selections in L 1 [a, b] In this section we consider the sequence of operators {Tn} defined in (13) and study two sets of functions, S(F ) and the set TnS(F ) = {Tns : s ∈ S(F )}.Note that for F ∈ F[a, b], the set S(F ) ⊂ L 1 [a, b], since it consists of functions of bounded variation.We assume that Kn(•, •) ∈ C([a, b] 2 ).This condition guaranties that Tn : S(F ) → C[a, b] ⊂ L 1 [a, b].

Lemma 6 . 1 .
For F ∈ F[a, b] the set S(F ) is compact in L 1 [a, b].Proof.By Result 2.13 any metric selection s ∈ S(F ) satisfies s ∞ ≤ F ∞ and V b a (s) ≤ V b a (F ).Applying Helly's selection principle, we conclude that for any sequence of metric selections there is a subsequence converging pointwisely at all points x ∈ [a, b].By Result 2.16 the pointwise limit function of such a subsequence is a metric selection.Thus by Lebesgue Dominated Convergence Theorem this subsequence converges in the L 1 -norm to the same limit metric selection.Lemma 6.2.Let F ∈ F[a, b] and assume that Kn(•, •) ∈ C([a, b] 2 ).Then the set TnS(F ) is compact in L 1 [a, b].
2.16.[7,Theorem 4.13]For F ∈ F[a, b], the pointwise limit (if exists) of a sequence of metric selections of F is a metric selection of F .