DOI: 10.24193/subbmath.2019.2.07
Approximation by max-product operators of Kantorovich type
Lucian Coroianu and Sorin G. Gal
Dedicated to Professor Heiner Gonska on the occasion of his 70th anniversary.
Abstract. We associate to various linear Kantorovich type approximation opera- tors, nonlinear max-product operators for which we obtain quantitative approxi- mation results in the uniform norm, shape preserving properties and localization results.
Mathematics Subject Classification (2010):41A35, 41A25, 41A20.
Keywords:Max-product operators, max-product operators of Kantorovich kind, uniform approximation, shape preserving properties, localization results, max- product Kantorovich-Choquet operators.
1. Introduction
The general form of a linear and positive discrete operator attached tof :I→[0,+∞) can be defined by
Dn(f)(x) = X
k∈In
pn,k(x)f(xn,k), x∈I, n∈N, wherepn,k(x) are various kinds of function basis onI withP
k∈Inpn,k(x) = 1,In are finite or infinite families of indices and{xn,k;k∈In}represents a division of I.
Based on the Open Problem 5.5.4, pp. 324-326 in [7], to eachDn(f)(x), can be attached the max-product type operator defined by
L(Mn )(f)(x) = W
k∈Inpn,k(x)·f(xn,k) W
k∈Inpn,k(x) , x∈I, n∈N. (1.1) HereW
k∈Aak= supk∈Aak.
This paper has been presented at the fourth edition of the International Conference on Numerical Analysis and Approximation Theory (NAAT 2018), Cluj-Napoca, Romania, September 6-9, 2018.
Thus, in a series of papers we have introduced and studied the so-called max-product operators attached to the Bernstein polynomials and to other linear Bernstein-type operators, like those of Favard-Sz´asz-Mirakjan operators (truncated and nontruncated case), Baskakov operators (truncated and nontruncated case), Meyer-K¨onig and Zeller operators and Bleimann-Butzer-Hahn operators. All these results were collected in the very recent research monograph [2].
Remark 1.1. The max-product operators can also be naturally called as possibilis- tic operators, since they can be obtained by analogy with the Feller probabilistic scheme used to generate positive and linear operators, by replacing the probability (σ-additive), with a maxitive set function and the classical integral with the possi- bilistic integral (see, e.g. [2], Chapter 10, Section 10.2). If, for example,pn,k(x),n∈N, k= 0, . . . , n is a polynomial basis, then the operators L(M)n (f)(x) become piecewise rational functions.
Now, to each max-product operatorL(Mn ), we can formally attach its Kantorovich variant, defined by
LKn(M)(f)(x) = W
k∈Inpn,k(x)·(1/(xn,k+1−xn,k))·Rxn,k+1 xn,k f(t)dt W
k∈Inpn,k(x) , (1.2)
with{xn,k;k∈In} a division of the finite or infinite intervalI.
The goal of this paper is to study these Kantorovich-type versions for various max-product operators. Firstly, we prove that these operators are subadditive, pos- itively homogeneous and monotone. For continuous functions we prove quantitative estimates, in most of the cases very good Jackson type estimates, shape preserving properties and localization results.
2. Uniform and pointwise approximation
Keeping the notations in the formulas (1.1) and (1.2), let us denote C+(I) ={f :I→R+;f is continuous onI},
where I is a bounded or unbounded interval and suppose that all pn,k(x) are con- tinuous functions on I, satisfying pn,k(x) ≥ 0, for all x ∈ I, n ∈ N, k ∈ In and P
k∈Inpn,k(x) = 1, for allx∈I, n∈N.
In many cases, for the Kantorovich max-product operatorKn(M)we could deduce quantitative estimates in approximation, by using the elaborated methods we used for the Bernstein-type max-product in the book [2]. However, here we will use a more simple method, which will be based on the already obtained estimates for the original type max-product operators denoted byL(M)n .
Firstly, we present the following result.
Lemma 2.1. (i) For any f ∈C+(I),LKn(M)(f)is continuous on I.
(ii) Iff ≤g then LKn(M)(f)≤LKn(M)(g).
(iii)LKn(M)(f+g)≤LKn(M)(f) +LKn(M)(g).
(iv) If f ∈C+(I)andλ≥0 thenLKn(M)(λf) =λLKn(M)(f).
(v) IfLKn(M)(e0) =e0, where e0(x) = 1, for all x∈I, then for anyf ∈C+(I), we have
LKn(M)(f)(x)−f(x) ≤
1 +1
δLKn(M)(ϕx)(x)
ω1(f;δ), for any x∈I andδ >0. Here,ϕx(t) =|t−x|,t∈I and
ω1(f;δ) = sup{|f(x)−f(y)|; x, y∈I, |x−y| ≤δ}.
(vi)
LKn(M)(f)−LKn(M)(g)
≤LKn(M)(|f−g|).
Proof. The proofs of (i)-(iv) are immediate from the definition ofKn(M). As for the proof of (v) and (vi), we exactly follow the proof of e.g., Theorem 1.1.2, pp. 16-17 in
[2].
Lemma 2.2. With the notations in (1.1) and (1.2), suppose that, in addition,
|xn,k+1−xn,k| ≤ C n+ 1
for all k ∈In, with C >0 an absolute constant. Then, for all x∈I and n∈N, we have
LKn(M)(ϕx)(x)≤L(M)n (ϕx)(x) + C n+ 1.
Proof. If f ∈ C+(I), then by the integral mean value theorem, there exists ξn,k ∈ (xn,k, xn,k+1), such that
Z xn,k+1
xn,k
f(t)dt= (xn,k+1−xn,k)·f(ξn,k), which immediately leads to
LKn(M)(f)(x) = W
k∈Inpn,k(x)·f(ξn,k) W
k∈Inpn,k(x) . (2.1)
Applying this form forf(t) =ϕx(t), we get LKn(M)(ϕx)(x) =
W
k∈Inpn,k(x)· |ξn,k−x|
W
k∈Inpn,k(x)
≤ W
k∈Inpn,k(x)· |ξn,k−xn,k| W
k∈Inpn,k(x) +L(M)n (ϕx)(x)≤ C
n+ 1+L(Mn )(ϕx)(x),
which proves the lemma.
Corollary 2.3. With the notations in (1.1) and (1.2) and supposing that, in addition,
|xn,k+1−xn,k| ≤ C n+ 1 for allk∈In, for anyf ∈C+(I), we have
LKn(M)(f)(x)−f(x) ≤2h
ω1(f;L(M)n (ϕx)(x)) +ω1(f;C/(n+ 1))i
(2.2) for any x∈I andn∈N.
Proof. By using Lemma 2.2, from the estimate in Lemma 2.1, (v), we immediately get
LKn(M)(f)(x)−f(x)
≤2ω1(f;L(Mn )(ϕx)(x) +C/(n+ 1))
≤2h
ω1(f;L(M)n (ϕx)(x)) +ω1(f;C/(n+ 1))i ,
which proves the corollary.
This corollary shows that knowing quantitative estimates in approximation by a given max-product operator, we can deduce a quantitative estimate for its Kan- torovich variant. Also, this method does not worsen the orders of approximation of the original operators. Let us exemplify below for several known max-product opera- tors.
Firstly, let us choosepn,k(x) = nk
xk(1−x)n−k,I= [0,1],In={0, . . . , n−1}and xn,k= n+1k . In this case,L(M)n in (1.1) become the Bernstein max-product operators.
Let us denote byBKn(M)their Kantorovich variant, given by the formula BKn(M)(f)(x) =
Wn k=0
n k
xk(1−x)n−k·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt Wn
k=0 n k
xk(1−x)n−k . (2.3)
We can state the following result.
Theorem 2.4. (i) If f ∈C+([0,1]), then we have
|BKn(M)(f)(x)−f(x)| ≤24ω1(f; 1/√
n+ 1) + 2ω1(f; 1/(n+ 1)), x∈[0,1], n∈N. (ii) Iff ∈C+([0,1]) is concave on [0,1], then we have
|BKn(M)(f)(x)−f(x)| ≤6ω1(f; 1/n), x∈[0,1], n∈N. (iii) Iff ∈C+([0,1]) is strictly positive on[0,1], then we have
|BKn(M)(f)(x)−f(x)| ≤2ω1(f; 1/n)·
nω1(f; 1/n) mf
+ 4
+ 2ω1(f; 1/n), for allx∈[0,1], n∈N, wheremf = min{f(x);x∈[0,1]}.
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 2.1.5, p.
30, in [2].
(ii) is immediate from Corollary 2.3 (withC= 1) and from Corollary 2.1.10, p.
36 in [2].
(iii) is immediate from Corollary 2.3 (withC= 1) and from Theorem 2.2.18, p.
63 in [2].
Now, let us choose pn,k(x) = (nx)k!k, I = [0,+∞), In = {0, . . . , n, . . . ,} and xn,k = n+1k . In this case, L(Mn ) in (1.1) become the non-truncated Favard-Sz´asz- Mirakjan max-product operators. Let us denote byF Kn(M)their Kantorovich variant
defined by
F Kn(M)(f)(x) = W∞
k=0 (nx)k
k! ·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0 (nx)k
k!
. (2.4)
We can state the following result.
Theorem 2.5. (i) If f : [0,+∞) → [0,+∞) is bounded and continuous on [0,+∞), then we have
|F Kn(M)(f)(x)−f(x)| ≤16ω1(f;√ x/√
n) + 2ω1(f; 1/n), x∈[0,+∞), n∈N. (ii) If f : [0,+∞)→ [0,+∞) is continuous, bounded, non-decreasing, concave func- tion on[0,+∞), then we have
|F Kn(M)(f)(x)−f(x)| ≤4ω1(f; 1/n), x∈[0,+∞), n∈N.
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 3.1.4, p.
162, in [2].
(ii) is immediate from Corollary 2.3 (with C = 1) and from Corollary 3.1.8, p.
168 in [2].
If we choose pn,k(x) = (nx)k!k, I = [0,1], In = {0, . . . , n} and xn,k = n+1k . In this case, L(Mn ) in (1.1) become the truncated Favard-Sz´asz-Mirakjan max-product operators. Let us denote byT Kn(M)their Kantorovich variant given by the formula
T Kn(M)(f)(x) = Wn
k=0 (nx)k
k! ·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt Wn
k=0 (nx)k
k!
. (2.5)
We can state the following result.
Theorem 2.6. (i) If f ∈C+([0,1]), then we have
|T Kn(M)(f)(x)−f(x)| ≤12ω1(f; 1/√
n) + 2ω1(f; 1/n), x∈[0,1], n∈N. (ii) Iff ∈C+([0,1]) is non-decreasing, concave function on [0,1], then we have
|T Kn(M)(f)(x)−f(x)| ≤4ω1(f; 1/n), x∈[0,1], n∈N.
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 3.2.5, p.
178, in [2].
(ii) is immediate from Corollary 2.3 (with C = 1) and from Corollary 3.2.7, p.
182 in [2].
Now, let us choose pn,k(x) = n+k−1k
xk/(1 +x)n+k, I = [0,+∞), In = {0, . . . , n, . . . ,}andxn,k =n+1k . In this case,L(Mn )in (1.1) become the non-truncated Baskakov max-product operators. Let us denote byV Kn(M)their Kantorovich variant defined by
V Kn(M)(f)(x) = W∞
k=0 n+k−1
k
xk
(1+x)n+k ·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0 n+k−1
k
xk (1+x)n+k
. (2.6)
We can state the following result.
Theorem 2.7. (i) If f : [0,+∞) → [0,+∞) is bounded and continuous on [0,+∞), then for all x∈[0,+∞)andn≥3, we have
|V Kn(M)(f)(x)−f(x)| ≤24ω1(f;p
x(x+ 1)/√
n−1) + 2ω1(f; 1/(n+ 1)).
(ii) If f : [0,+∞)→ [0,+∞) is continuous, bounded, non-decreasing, concave func- tion on[0,+∞), then forx∈[0,+∞)andn≥3 we have
|V Kn(M)(f)(x)−f(x)| ≤4ω1(f; 1/n).
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 4.1.6, p.
196, in [2].
(ii) is immediate from Corollary 2.3 (with C = 1) and from Corollary 4.1.9, p.
206 in [2].
If we choose pn,k(x) = n+k−1k
xk/(1 +x)n+k, I = [0,1], In = {0, . . . , n} and xn,k = n+1k , then in this case, L(M)n in (1.1) become the truncated Baskakov max- product operators. Let us denote byU Kn(M)their Kantorovich variant defined by
U Kn(M)(f)(x) = Wn
k=0 n+k−1
k
xk
(1+x)n+k ·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0 n+k−1
k
xk (1+x)n+k
. (2.7)
We can state the following result.
Theorem 2.8. (i) If f ∈C+([0,1]), then we have,
|U Kn(M)(f)(x)−f(x)| ≤48ω1(f; 1/√
n+ 1) + 2ω1(f; 1/(n+ 1)), x∈[0,1], n≥2.
(ii) Iff ∈C+([0,1]) is non-decreasing, concave function on [0,1], then we have
|U Kn(M)(f)(x)−f(x)| ≤6ω1(f; 1/n), x∈[0,1], n∈N.
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 4.2.6, p.
217, in [2].
(ii) is immediate from Corollary 2.3 (with C = 1) and from Corollary 4.2.9, p.
223 in [2].
Now, let us choosepn,k(x) = n+kk
xk,I= [0,1],In={0, . . . , n, . . .}andxn,k=
k
n+1+k. In this case,L(M)n in (1.1) become the Meyer-K¨onig and Zeller max-product operators. Also, it is easy to see that |xn,k+1−xn,k| ≤ n+11 , for all k ∈ In. Let us denote byZKn(M) their Kantorovich variant defined by
ZKn(M)(f)(x) = W∞
k=0 n+k
k
xk·(n+k+1)(n+k+2) n+1
R(k+1)/(n+k+2) k/(n+1+k) f(t)dt W∞
k=0 n+k
k
xk . (2.8)
The following result holds.
Theorem 2.9. (i) If f ∈C+([0,1]), then forn≥4,x∈[0,1], we have
|ZKn(M)(f)(x)−f(x)| ≤36ω1(f;√
x(1−x)/√
n) + 2ω1(f; 1/n).
(ii) If f ∈C+([0,1]) is non-decreasing concave function on [0,1], then forx∈[0,1]
andn≥2xwe have
|ZKn(M)(f)(x)−f(x)| ≤4ω1(f; 1/n).
Proof. (i) is immediate from Corollary 2.3 (withC= 1) and from Theorem 6.1.4, p.
248, in [2].
(ii) is immediate from Corollary 2.3 (with C = 1) and from Corollary 6.1.7, p.
256 in [2].
In what follows, let us choosepn,k(x) =hn,k(x)-the fundamental Hermite-Fej´er interpolation polynomials based on the Chebyshev knots of first kind
xn,k= cos
2(n−k) + 1 2(n+ 1) π
,
I= [−1,1], andIn={0, . . . , n}. In this case,L(M)n in (1.1) become the Hermite-Fej´er max-product operators. Also, applying he mean value theorem to cos, it is easy to see that|xn,k+1−xn,k| ≤ n+14 , for allk∈In. Let us denote byHKn(M)their Kantorovich variant defined by
HKn(M)(f)(x) = Wn
k=0hn,k(x)·x 1
n,k−xn,k+1·Rxn,k+1 xn,k f(t)dt W∞
k=0hn,k(x) , (2.9)
wherexn,k= cos2(n−k)+1
2(n+1) π . The following result holds.
Theorem 2.10. If f ∈C+([−1,1]), then forn∈N,x∈[−1,1], we have
|HKn(M)(f)(x)−f(x)| ≤30ω1(f; 1/n).
Proof. It is immediate from Corollary 2.3 (withC = 4) and from Theorem 7.1.5, p.
286, in [2].
Now, let us consider choose pn,k(x) = e−|x−k/(n+1)|, I = (−∞,+∞), In = Z- the set of integers and xn,k = n+1k . In this case, L(M)n in (1.1) become the Picard max-product operators. Let us denote by PKn(M) their Kantorovich variant defined by
PKn(M)(f)(x) = W∞
k=0e−|x−k/(n+1)|·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0e−|x−k/(n+1)| . (2.10) We can state the following result.
Theorem 2.11. If f :R→[0,+∞)is bounded and uniformly continuous on R, then we have
|PKn(M)(f)(x)−f(x)| ≤6ω1(f; 1/n), x∈R, n∈N.
Proof. It is immediate from Corollary 2.3 (withC= 1) and from Theorem 10.3.1, p.
423, in [2].
In what follows, let us choosepn,k(x) =e−(x−k/(n+1))2,I= (−∞,+∞),In=Z- the set of integers andxn,k =n+1k . In this case,L(Mn )in (1.1) become the Weierstrass max-product operators. Let us denote byWKn(M) their Kantorovich variant defined by
WKn(M)(f)(x) = W∞
k=0e−(x−k/(n+1))2·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0e−(x−k/(n+1))2 . (2.11) We can state the following result.
Theorem 2.12. If f :R→[0,+∞)is bounded and uniformly continuous on R, then we have
|WKn(M)(f)(x)−f(x)| ≤4ω1(f; 1/√
n) + 2ω1(f; 1/n), x∈R, n∈N.
Proof. It is immediate from Corollary 2.3 (withC= 1) and from Theorem 10.3.3, p.
425, in [2].
At the end of this section, let us choosepn,k(x) = n2(x−k/n)1 2+1,I= (−∞,+∞), In =Z-the set of integers and xn,k = n+1k . In this case, L(Mn ) in (1.1) become the Poisson-Cauchy max-product operators. Let us denote by CKn(M) their Kantorovich variant
CKn(M)(f)(x) = W∞
k=0
1
n2(x−k/(n+1))2+1 ·(n+ 1)R(k+1)/(n+1) k/(n+1) f(t)dt W∞
k=0
1 n2(x−k/(n+1))2+1
. (2.12) We can state the following result.
Theorem 2.13. If f :R→[0,+∞)is bounded and uniformly continuous on R, then we have
|CKn(M)(f)(x)−f(x)| ≤6ω1(f; 1/n), x∈R, n∈N.
Proof. It is immediate from Corollary 2.3 (withC= 1) and from Theorem 10.3.5, p.
426, in [2].
Remark 2.14. All the Kantorovich kind max-product operatorsLKn(M)given by (1.2) are defined and used for approximation of positive valued functions. But, they can be used for approximation of lower bounded functions of variable sign too, by introducing the new operators
Nn(M)(f)(x) =LKn(M)(f+c)(x)−c,
wherec >0 is such thatf(x) +c >0, for allxin the domain of definition off. It is easy to see that the operatorsNn(M)give the same approximation orders as LKn(M).
3. Shape preserving properties for the Bernstein-Kantorovich max-product operators
In this section we deal with the shape preserving properties of the Bernstein- Kantorovich max-product operatorsBKn(M)given by (2.3).
We can prove the following.
Theorem 3.1. Letf ∈C+([0,1]).
(i) If f is non-decreasing (non-increasing) on [0,1], then for all n ∈ N, BKn(M)(f)is non-decreasing (non-increasing, respectively) on [0,1].
(ii) Iff is quasi-convex on[0,1]then for alln∈N,BKn(M)(f)is quasi-convex on [0,1]. Here quasi-convexity on[0,1]means thatf(λx+ (1−λ)y)≤max{f(x), f(y)}, for allx, y, λ∈[0,1].
Proof. (i) By using the formula (2.1) forLKn(M), we can writeBKn(M)(f) under the form
BKn(M)(f)(x) = Wn
k=0 n k
xk(1−x)n−k·f(ξn,k) Wn
k=0 n k
xk(1−x)n−k , whereξn,k∈(xn,k, xn,k+1), for allk= 0, . . . , n.
Then, by analogy with the proofs for the Bernstein max-product operators (see [2], pp. 39-41, the proofs for the Bernstein-Kantorovich max-product operators, will be based on the properties of the functions
fk,n,j(x) =
n k
n j
· x
1−x k−j
·f(ξn,k).
Now, analyzing the proofs of Lemma 2.1.13, Corollary 2.1.14, Theorem 2.1.15 and Corollary 2.1.16 in [2], pp. 39-41, it is easy to see that they work identically for the abovefk,n,j too and we immediately obtain the required conclusions.
(ii) Since as in the case of the max-product Bernstein operators in Corollary 2.1.18, p. 41 in [2], this point is based on the properties from the above point (i) and on the properties in the above Lemma 2.1, (i)-(iv), we easily get the required
conclusion for this point too.
In what follows, we will prove thatBKn(M) preserves quasi-concavity too. This property holds in the case of the operatorB(Mn ) (By Theorem 5.1 in [5]). However, it is difficult to adapt the proof to our case. Instead, we can prove this property by finding a direct correspondence between the operatorsBn(M)andBKn(M).
Let us notice that the operatorBKn(M)can be obtained from the operatorBn(M). Suppose thatf is arbitrary inC+([0,1]). Let us consider
fn(x) = (n+ 1)
Z (nx+1)/(n+1) nx/(n+1)
f(t)dt (3.1)
It is readily seen thatB(Mn )(fn)(x) =BKn(M)(f)(x), for allx∈[0,1]. We also notice thatfn∈C+([0,1]). What is more, iff is strictly positive then so isfn.
A function f : [a, b]→R is quasi-concave if−f is quasi-convex. If f is contin- uous, quasi-concavity equivalently means that there exists c ∈ [a, b] such that f is nondecreasing on [a, c] and nonincreasing on [c, b].
We are now in position to prove thatBKn(M) preserves quasi-concavity too.
Theorem 3.2. Let f ∈ C+([0,1]). If fis quasi-concave on [0,1] then BKn(M)(f) is quasi-concave on[0,1].
Proof. For some arbitraryn≥1 let us consider the functionfn given by (3.1). More- over, letc ∈ [0,1] such that f is nondecreasing on [0, c] and nonincreasing on [c,1].
Then, letj(c)∈ {0, . . . , n} such that j(c)
n+ 1 ≤c≤j(c) + 1 n+ 1 .
Next, we consider the function gn which interpolates fn at all the knots kn, k = 0,1, . . . , n, and which is continuous on [0,1] and affine on any interval k
n,k+1n , k= 0,1, . . . , n−1. It means that gn is the continuous polygonal line which interpolates fn at all the knots nk,k= 0,1, . . . , n. This easily implies that
B(Mn )(fn)(x) =B(Mn )(gn)(x), x∈[0,1], hence,
BKn(M)(f)(x) =Bn(M)(gn)(x), x∈[0,1].
Let us now choose arbitrary 0≤k1< k2≤j(c)−1. We have gn
k1
n
= (n+ 1)
Z (k1+1)/(n+1) k1/(n+1)
f(t)dt and
gn
k2 n
= (n+ 1)
Z (k2+1)/(n+1) k2/(n+1)
f(t)dt.
As kn+11+1 ≤ n+1k2 andf is nondecreasing on [0,kn+12+1], we easily obtain (after applying the mean value theorem) that gn k1
n
≤gn k2
n
. The construction of gn easily im- plies that gn is nondecreasing on h
0,j(c)−1n i
. By similar reasoning we get that gn is nonincreasing on hj(c)+1
n ,1i
. Now, suppose that fj(c)
n+1
≥ fj(c)+1
n+1
. The quasi- concavity off implies that f(x)≥fj(c)+1
n+1
for any x∈hj(c)
n+1,j(c)+1n+1 i
. Since there existsx0∈hj(c)
n+1,j(c)+1n+1 i
such that (n+ 1)
Z (j(c)+1)/(n+1) j(c)/(n+1)
f(t)dt=f(x0) =gn
j(c) n
, and since fj(c)+1
n+1
≥ gn
j(c)+1
n
(this is true indeed as f is nondecreasing on hj(c)+1
n+1 ,1i
), we get that gn
j(c)
n
≥ gn
j(c)+1
n
. Therefore, gn is nonincreasing on hj(c)
n ,j(c)+1n i
. This implies thatgn is nondecreasing onh
0,j(c)−1n i
and nonincreasing
on hj(c)
n ,1i
. But f is affine on hj(c)−1
n ,j(c)n i
which means that it is monotone on this interval. Clearly this implies that gn is either nondecreasing on h
0,j(c)−1n i and nonincreasing onhj(c)−1
n ,1i
or, it is nondecreasing onh 0,j(c)n i
and nonincreasing on hj(c)
n ,1i
. It means thatgn is quasi-concave on [0,1]. By similar reasonings we get to the same conclusion if fj(c)
n+1
≤fj(c)+1
n+1
. The only difference is that now gn is either nondecreasing onh
0,j(c)n i
and nonincreasing onhj(c)
n ,1i
or, it is nondecreasing onh
0,j(c)+1n i
and nonincreasing onhj(c)+1
n ,1i
. Thus, we just proved thatgn is quasi- concave on [0,1]. By Theorem 5.1 in [5] (see also Theorem 2.2.22 in the book, it follows thatBn(M)(gn) is quasi-concave on [0,1]. AsBn(M)(gn) =BKn(M)(f), it follows
thatBKn(M)(f) is quasi-concave on [0,1].
As an important side remark, let us note that in Theorem 5.1 of paper [5](see also the book [2]), it is proved that iff is quasi-concave and c is a maximum point off then there exists a maximum point ofBn(M)(f) such that|c−c0| ≤ n+11 . By the construction of gn it follows that one maximum point of gn is between the values
j(c)−1
n , j(c)n or j(c)+1n . If we denote this value with cn then one can easily check that
|cn−c| ≤ n2. Now, applying the afore mentioned property obtained in [5], let c0 be a maximum point of B(Mn )(gn) =BKn(M)(f), such that|c0 −cn| ≤ n+11 . This easily implies that |c0 −c| ≤ n3. So, we obtained a quite similar result for the operator BKn(M)in comparison with the operator Bn(M).
4. Approximation of Lipschitz functions by Bernstein-Kantorovich max-product operators
Let us return to the functions fn given in (3.1) and let us find now an upper bound for the approximation of f by fn in terms of the uniform norm. For some x ∈ [0,1], using the mean value theorem, there exists ξx ∈ h
nx
n+1,nx+1n+1i
such that fn(x) =f(ξx). We also easily notice that |ξx−x| ≤n+11 . It means that
|f(x)−fn(x)| ≤ω1(f; 1/(n+ 1)), x∈R, n∈N. (4.1) In particular, if f is Lipschitz with constantC then fn is Lipschitz continuous with constant 3C. These estimation are useful to prove some inverse results in the case of the operator BKn(M) by using analogue results already obtained for the operator Bn(M).
Below we present a result which gives for the class of Lipschitz function the order of approximation 1/n in the approximation by the operatorBKn(M), hence an analogue result which holds in the case of the operatorBn(M).
Theorem 4.1. Suppose that f is Lipschitz on [0,1] with Lipschitz constant C and suppose that the lower bound of f ismf >0. Then we have
BKn(M)(f)−f ≤2C
C mf
+ 5
· 1
n,n≥1.
Proof. The estimation is immediate using the estimation from Corollary 2.4, (iii),
taking into account thatω1(f; 1/n)≤C/n.
5. Localization results for Bernstein-Kantorovich max-product operators
We firstly prove a very strong localization property of the operatorBKn(M). Theorem 5.1. Let f, g : [0,1]→ [0,∞)be both bounded on [0,1]with strictly positive lower bounds and suppose that there exist a, b ∈ [0,1], 0 < a < b < 1 such that f(x) =g(x) for all x∈[a, b]. Then for all c, d∈[a, b] satisfying a < c < d < bthere existsen∈N depending only onf, g, a, b, c, dsuch that BKn(M)(f)(x) =BKn(M)(g)(x) for allx∈[c, d]andn∈Nwithn≥en.
Proof. Let us choose arbitraryx∈[c, d] and for eachn∈Nletjx∈ {0,1, . . . , n}be such thatx∈[jx/(n+ 1),(jx+ 1)/(n+ 1)].Then by relation (4.17) in [1] we have
BKn(M)(f)(x) =B(Mn )(fn)(x) =
n
_
k=0
(fn)k,n,j
x(x), (5.1)
where fork∈ {0,1, . . . , n}we have (fn)k,n,j
x=
n k
n jx
x
1−x k−jx
fn k
n
. (5.2)
and eachfnis given by (3.1). Let us denote withmf, Mf andmfn, Mfnrespectively, the minimums and maximum values of the functions f and fn, respectively. By the mean value theorem, one can easily notice that for any x∈[0,1] there exists ξn,x ∈ [0,1] such that fn(x) = f(ξn,x). It means that 0 < mf ≤ mfn ≤ Mfn ≤ Mf. In what follows, the proof is very similar to the proof of Theorem 2.1 in [6] (see also Theorem 2.4.1 in [2]). However, as often we will use fn instead off, especially since the constants obtained in the proof of Theorem 2.1 in [6] depend onf, in our setting these constants would depend onfn, hence, they would depend onn, if we would apply directly the results in [6]. Therefore, there are some differences in the two proofs as our intention is to obtain constants that do not depend onfn.
We need the set In,x = {k ∈ {0,1, . . . , n} : jx−an ≤ k ≤ jx+an}, where an=h√3
n2i
(here [a] denotes the integer part ofa). Now, suppose thatk /∈In,x, and let us discuss first the case whenk < jx−an. If we look over the proof of Theorem 2.1 in [6], we observe that this proof is split in cases i) and ii). Case i) corresponds to
the case whenk < jx−an. Furthermore this case is divided in two subcases ia) and ib). In subcase ia) the inequality
fjx,n,,jx(x) fk,n,jx(x) ≥
1 + an nb−an
an
· f(jx/n) f(k/n) is obtained which then gives
fjx,n,jx(x) fk,n,jx(x) ≥
1 + an
nb−an an
· mf
Mf. Applying this reasoning but consideringfn instead off, we get
(fn)j
x,n,jx(x) (fn)k,n,j
x(x) ≥
1 + an
nb−an
an
·fn(jx/n) fn(k/n). But sincemf ≤mfn≤Mfn ≤Mf, we get
(fn)j
x,n,jx(x) (fn)k,n,j
x(x) ≥
1 + an
nb−an
an
· mf
Mf
.
We get the same conclusion all cases and subcases, that is, any lower bound for
fjx,n,,jx(x)
fk,n,jx(x) is also a lower bound for (f(fn)jx,n,jx(x)
n)
k,n,jx(x), for any k outside of In,x. Since in..., we proved that there existsN0∈Nwhich may depend only on f, a, b, c, d, such that for any n ≥N0, k ∈ {0,1, . . . , n}, with k < jx−an or k > jx+an, we have
fjx,n,,jx(x)
fk,n,jx(x) ≥ 1, it follows that (f(fn)jx,n,jx(x)
n)k,n,jx(x) ≥ 1, for any n ≥N0, k ∈ {0,1, . . . , n}, with k < jx−an or k > jx+an. Combining this fact with relations (5.1)-(5.2), we get that
BKn(M)(f)(x) = _
k∈In,x
(fn)k,n,j
x(x), x∈[c, d], n≥N0.
Using a similar reasoning as in the proof of Theorem 2.1 in [6], in what follows, we will prove that N0 can be replaced if necessary with a larger value Ne1 such that [n+1k ,k+1n+1]⊆[a, b] for any k∈In,x. Let us choose arbitrary x∈[c, d] and n∈Nso that n≥N0. If there existsk∈In,xsuch thatk/(n+ 1)∈/ [c, d] then we distinguish two cases. Either n+1k < cor n+1k > d.In the first case we observe that
0< c− k
n+ 1 ≤x− k
n+ 1 ≤ jx+ 1 n+ 1 − k
n+ 1 ≤jx+ 1 n+ 1 − k
n+ 1 ≤ an+ 1 n+ 1 . Since lim
n→∞
an+1
n+1 = 0,it results that for sufficiently largenwe necessarily have an+1n+1 <
c−a which clearly implies that n+1k ∈ [a, c]. In the same manner, when n+1k > d, for sufficiently largenwe necessarily have n+1k ∈[d, b].By similar reasoning it results that for sufficiently largenwe necessarily have n+1k ∈[a, b].Summarizing, there exists a constantNe1∈Nindependent of anyx∈[c, d] such that
BKn(M)(f)(x) = _
k∈In,x
(fn)k,n,j
x(x),x∈[c, d], n≥Ne1
and in addition for any x∈[c, d], n≥Ne1 andk ∈In,x, we have [n+1k ,n+1k+1]⊆[a, b].
Also, it is easy to check thatNe1depends only on a, b, c, d, f.
Now, fork∈ {0,1, . . . , n}taking (gn)k,n,j
x=
n k
n jx
x
1−x k−jx
gn
k n
,
applying the same reasoning, there exists Ne2 ∈ N which may depend only on a, b, c, d, g, such that
BKn(M)(g)(x) = _
k∈In,x
(gn)k,n,j
x(x),x∈[c, d], n≥Ne2
and in addition for any x∈[c, d], n≥Ne2 andk ∈In,x, we have [n+1k ,n+1k+1]⊆[a, b].
Since f(x) = g(x), x∈ [a, b], we get that for any n ≥ne = max{Ne1,Ne2}, k ∈ In.x
and x∈ [c, d], it holds that (fn)k,n,j
x(x) = (gn)k,n,j
x(x). Thus, for anyn ≥en and x∈[c, d], we haveBKn(M)(f)(x) =BKn(M)(g)(x). The proof is complete now.
As in the case of the Bernstein max-product operator, we can present a local direct approximation result as an immediate consequence of the localization result in Theorem 5.1.
Corollary 5.2. Letf : [0,1]→[0,∞)be bounded on[0,1]with the lower bound strictly positive and 0< a < b <1 be such that f|[a,b] ∈Lip[a, b] with Lipschitz constantC.
Then, for anyc, d∈[0,1]satisfying a < c < d < b, we have
BKn(M)(f)(x)−f(x) ≤C
n for alln∈Nandx∈[c, d], where the constant C depends only onf anda, b, c, d.
Proof. Let us define the functionF : [0,1]→R, F(x) =
f(a) if x∈[0, a], f(x) if x∈[a, b], f(b) if x∈[b,1].
The hypothesis immediately imply thatF is a strictly positive Lipschitz function on [0,1]. Then, according to Theorem 4.1 and noting that the minimum of F is above the minimum off,mf, it results that
BKn(M)(F)(x)−F(x) ≤2C
C mf
+ 5
· 1
n,for allx∈[0,1], n∈N.
Now, let us choose arbitraryc, d∈[a, b] such thata < c < d < b.Then, by Theorem 5.1 it results the existence ofne ∈ N which depends only ona, b, c, d, f, F such that BKn(M)(F)(x) = BKn(M)(f)(x) for all x ∈ [c, d]. But since actually the function F depends on the function f, by simple reasonings we get that in factne depends only ona, b, c, dandf.
Therefore, for arbitraryx∈[c, d] andn∈Nwithn≥newe obtain
BKn(M)(f)(x)−f(x) =
BKn(M)(F)(x)−F(x) ≤2C
C mf + 5
· 1 n, whereC1 andendepend only ona, b, c, d andf.
Now, denoting
C2= max
1≤n<en{n· kBKn(M)(f)−fk[c,d]}, we finally obtain
|BKn(M)(f)(x)−f(x)| ≤ C
n,for alln∈N, x∈[c, d], withC= max{2C
C mf + 5
, C2}depending only on a, b, c, dandf. In a previous section we proved thatBKn(M) preserves monotonicity and more generally quasi-convexity. By the localization result in Theorem 5.1 and then applying a very similar reasoning to the one used in the proof of Corollary 5.2, we obtain local versions for these shape preserving properties. Indeed, in all cases it will suffice to consider the sameF as in the proof of Corollary 5.2 as this function will be monotone or quasi-convex/quasi-concave, respectively, whenever f will be monotone or quasi- convex/quasi-concave, respectively. For this reason we omit the proofs of the following corollaries (see also the corresponding local shape preserving properties proved for the operatorB(Mn )in [6]).
Corollary 5.3. Let f : [0,1]→ [0,∞)be bounded on [0,1]with strictly positive lower bound and suppose that there exists a, b∈[0,1],0< a < b <1, such thatf is nonde- creasing (nonincreasing) on[a, b]. Then for any c, d∈[a, b] witha < c < d < b,there exists en∈ N depending only ona, b, c, d and f, such thatB(Mn )(f) is nondecreasing (nonincreasing) on [c, d]for alln∈Nwithn≥en.
Corollary 5.4. Letf : [0,1]→[0,∞)be a continuous and strictly positive function and suppose that there exists a, b∈[0,1],0 < a < b <1, such that f is quasi-convex on [a, b]. Then for any c, d∈[a, b]with a < c < d < b, there existsen∈Ndepending only ona, b, c, dandfsuch thatBn(M)(f)is quasi-convex on[c, d]for alln∈Nwithn≥en.
Corollary 5.5. Letf : [0,1]→[0,∞)be a continuous and strictly positive function and suppose that there exists a, b ∈ [0,1], 0 < a < b < 1, such that f is quasi-concave on [a, b]. Then for any c, d∈[a, b]with a < c < d < b, there exists en∈N depending only ona, b, c, dandf, such thatBn(M)(f)is quasi-concave on[c, d]for alln∈Nwith n≥en.
Remark 5.6. As in the cases of Bernstein-type max-product operators studied in the research monograph [2], for the the max-product Kantorovich type operators we can find natural interpretation as possibilistic operators, which can be deduced from the Feller scheme written in terms of the possibilistic integral. These approaches also offer new proofs for the uniform convergence, based on a Chebyshev type inequality in the theory of possibility.
Remark 5.7. In the recently submitted paper [3], we have introduced the more gen- eral Kantorovich max-product operators based on a generalized (ϕ, ψ)-kernel, by the formula
Kn(M)(f;ϕ, ψ)(x) =1 b ·
Wn k=0
ϕ(nx−kb) ψ(nx−kb)·h
(n+ 1)R(k+1)b/(n+1)
kb/(n+1) f(v)dvi Wn
k=0
ϕ(nx−kb) ψ(nx−kb)
, (5.3) where b > 0, f : [0, b] → R+, f ∈ Lp[0, b], 1 ≤ p ≤ ∞ and ϕ and ψ satisfy some properties specific to max-product operators and proved pointwise, uniform or Lp convergence quantitative approximation results. For particular choices of (ϕ, ψ), we have obtained approximation results for many other max-product Kantorovich oper- ators, including for example the sampling operators based on sinc-type kernels.
Remark 5.8. In another recently in preparation paper [4], we have generalized the max-product Kantorovich operators from the above Remark 2), by replacing the clas- sical linear integralR
dv, by the nonlinear Choquet integral (C)R
dµ(v) with respect to a monotone and submodular set functionµobtaining and studying the max-product Kantorovich-Choquet operators given by the formula
Kn(M)(f;ϕ, ψ)(x)
=1 b ·
Wn k=0
ϕ(nx−kb) ψ(nx−kb)·h
(C)R(k+1)b/(n+1)
kb/(n+1) f(v)dµ(v)/µh
kb
n+1,(k+1)bn+1 ii Wn
k=0
ϕ(nx−kb) ψ(nx−kb)
, (5.4) It is worth noting that the max-product Kantorovich-Choquet operators aredoubly nonlinear operators: firstly due to max and secondly, due to the Choquet integral.
Acknowledgement.The work of both authors was supported by a grant of the Roma- nian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-III-P1-1.1-PD-2016-1416.
References
[1] Bede, B., Coroianu, L., Gal, S.G., Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern. J. Math. Math. Sci.,2009, Art. ID 590589, 2009.
[2] Bede, B., Coroianu, L., Gal, S.G., Approximation by Max-Product Type Operators, Springer, 2016, xv+458 pp.
[3] Coroianu, L., Gal, S.G., Approximation by truncated max-product operators of Kantorovich-type based on generalized (ϕ, ψ)-kernels, Math. Meth. Appl. Sci., online access, 2018: 1-14, DOI: 10.1002/mma.5262.
[4] Coroianu, L., Gal, S.G., Approximation by max-product operators of Kantorovich- Choquet type based on generalized(ϕ, ψ)-kernels, in preparation.
[5] Coroianu, L., Gal, S.G.,Classes of functions with improved estimates in approximation by the max-product Bernstein operator,Anal. Appl.,9(2011), 249–274.
[6] Coroianu, L., Gal, S.G., Localization results for the Bernstein max-product operator, Appl. Math. Comp.,231(2014), 73–78.
[7] Gal, S.G.,Shape-Preserving Approximation by Real and Complex Polynomials, XIV+352 pp., Birkh¨auser, Boston, Basel, Berlin, 2008.
Lucian Coroianu University of Oradea
Department of Mathematics and Computer Sciences Universit˘at¸ii Street, No. 1
410087 Oradea, Romania e-mail:[email protected] Sorin G. Gal
University of Oradea
Department of Mathematics and Computer Sciences Universit˘at¸ii Street, No. 1
410087 Oradea, Romania e-mail:[email protected]
