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An extremal property for a class of positive linear operators
Journal of Approximation Theory 162 (2010) 6–9 www.elsevier.com/locate/jat
An extremal property for a class of positive linear operators Ioan Gavrea, Mircea Ivan ∗ Department of Mathematics, Technical University of Cluj-Napoca, Str. C. Daicoviciu 15, 400020 Cluj-Napoca, Romania Received 24 February 2008; accepted 4 February 2009 Available online 21 February 2009 Communicated by Kirill Kopotun
Abstract We generalize a recent result of de la Cal and C´arcamo concerning an extremal property of Bernstein operators. c 2009 Elsevier Inc. All rights reserved.
Keywords: Positive linear operators; Finitely defined operators; Bernstein operators
1. Introduction and main results Let n be a natural number greater than 1. Consider a set of linear independent functions {ϕ0 , . . . , ϕn } ⊂ C[0, 1] such that ϕk (x) > 0, for x ∈ (0, 1), and k = 0, . . . , n. Let xk ∈ [0, 1], k = 0, . . . , n, be distinct points. Denote by δx , x ∈ [0, 1], the evaluation functional δx : R[0,1] → R, δx ( f ) := f (x). P Definition 1. A linear operator U : C[0, 1] → C[0, 1] of the form U = nk=0 ϕk δxk is said to be of discrete type. Denote by V the set of all linear P operators V : C[0, 1] → C[0, 1] preserving the affine functions and having the form V = nk=0 ϕk λk , where λk are linear and positive functionals defined on C[0, 1]. ∗ Corresponding author.
E-mail addresses: [email protected] (I. Gavrea), [email protected] (M. Ivan). c 2009 Elsevier Inc. All rights reserved. 0021-9045/$ - see front matter doi:10.1016/j.jat.2009.02.006
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
7
Theorem 2. Let U be an operator of discrete type preserving the affine functions. Then, for all V ∈ V, the following inequality is satisfied f ≤ U f ≤ V f,
for any convex function f ∈ C[0, 1].
Moreover, if there exist a strict convex function g ∈ C[0, 1] and a point x ∈ (0, 1) such that U g(x) = V g(x), then V = U. Suppose that the points xk satisfy 0 = x0 < · · · < xn = 1. Theorem 3. Let U : C[0, 1] → C[0, 1] be an operator of discrete type preserving the affine functions and V : C[0, 1] → R[0,1] a linear positive operator. Then, the following propositions are equivalent: (a) V ∈ V. (b) There exists a linear positive operator L: C[0, 1] → C[0, 1] preserving linear functions such that V = U ◦ L. (c) There exists a linear positive operator L: C[0, 1] → R[0,1] preserving the affine functions such that V = U ◦ L. Remark 4. Theorems 2 and 3 are respective extensions of Theorems 2 and 3 in [3], where k/n are replaced by arbitrary knots xk and the Bernstein basis is replaced by the functions ϕk . See also [2, Th. 1]. 2. Auxiliary results The divided difference [x1 , x2 , x3 ; f ] of a function f ∈ R[0,1] on the distinct knots x1 , x2 , x3 ∈ [0, 1] is defined by [x1 , x2 , x3 ; f ] :=
f (x1 ) f (x2 ) f (x3 ) + + . (x1 − x2 )(x1 − x3 ) (x2 − x3 )(x2 − x1 ) (x3 − x1 )(x3 − x2 )
A function f ∈ R[0,1] is said to be convex (strictly convex) if [x1 , x2 , x3 ; f ] ≥ 0 (> 0),
∀ distinct knots x1 , x2 , x3 ∈ [0, 1].
As usual, denote by ei the monomials ei (t) := t i , i = 0, 1, 2. For t ∈ [a, b], consider the positive part function, ψt : [a, b] → R, ψt (x) = (x −t)+ := (x −t +|x −t|)/2. We note that the functions ψt are convex. ˇ skin [5], and Lupas¸ [4]. Throughout this note we use some results of Popoviciu [6], Saˇ In the case of the Tchebyshev system {e0 , e1 , e2 }, Theorem 1 of [5] (see also [1, p. 117, Example 3; p.116, Theorem 2.5.4]) becomes: ˇ skin). Let x0 ∈ [a, b]. If F: C[a, b] → R is a linear positive functional such that Theorem 5 (Saˇ F(ei ) = ei (x0 ),
i = 0, 1, 2,
F( f ) = f (x0 ),
∀ f ∈ C[a, b].
then
In the case of n = 2, Popoviciu’s Theorem [6, Th´eor`eme 12] becomes: Theorem 6 (Popoviciu). If F: C[a, b] → R is a linear functional such that: (i) F(e0 ) = F(e1 ) = 0, F(e2 ) > 0, (ii) F(ψt ) ≥ 0, ∀t ∈ [a, b],
(1)
8
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
then, for any f ∈ C[a, b], there exists a set of distinct points θ1 , θ2 , θ3 ∈ [a, b], depending on f , such that F( f ) = F(e2 )[θ1 , θ2 , θ3 ; f ]. Theorem 7 (Lupas¸ [4, Th. 1]). Let A: C[a, b] → R be a linear positive functional with A(e0 ) = 1. Then, for any f ∈ C[a, b], there exist distinct points θ1 , θ2 , θ3 ∈ [a, b], depending on f , such that A( f ) = f (A(e1 )) + (A(e2 ) − (A(e1 ))2 )[θ1 , θ2 , θ3 ; f ].
(2)
3. Proofs The proof for Theorem 3 and the first part of Theorem 2 are essentially the same (with obvious changes) as those given in [3]. Proof of Theorem 2 (Second Part). From U e0 = e0 = V e0
and U e1 = e1 = V e1 ,
using the linear independence of the set {ϕ0 , . . . , ϕn }, we obtain: λk (e0 ) = 1 = e0 (xk ) and
λk (e1 ) = xk = e1 (xk ),
k = 0, . . . , n.
(3)
Let g ∈ C[0, 1] be a strictly convex function and x ∈ (0, 1) such that V g(x) = U g(x), i.e., n X (λk (g) − g(xk ))ϕk (x) = 0.
(4)
k=0
Using the Jensen inequality for the functionals λk , we obtain λk (g) ≥ g(λk (e1 )) = g(xk ),
k = 0, . . . , n.
(5)
By using the positivity of ϕk , Eqs. (4) and (5) imply λk (g) = g(xk ),
k = 0, . . . , n.
(6)
From Theorem 7, we deduce that there exist distinct points θ1 , θ2 , θ3 ∈ [0, 1] such that λk (g) = g(xk ) + (λk (e2 ) − (λk (e1 ))2 )[θ1 , θ2 , θ3 ; g],
k = 0, . . . , n.
(7)
Since g is strictly convex, Eqs. (6) and (7) imply λk (e2 ) = (λk (e1 ))2 = xk2 = e2 (xk ),
k = 0, . . . , n.
(8)
Eqs. (3) and (8), and Theorem 5 give λk ( f ) = f (xk ), k = 0, . . . , n. This concludes the proof. ˇ skin’s Theorem. For readers’ convenience, we insert below an original simple proof of Saˇ Proof of Theorem 5. Since the class of all Lipschitz functions f : [a, b] → R is dense in C[a, b], it is sufficient to prove the theorem for Lipschitz functions only. Let f ∈ Lip M [a, b]. The following inequalities are satisfied: | f (x) − f (x0 )| ≤ M |x − x0 |,
∀x ∈ [a, b],
F(| f − f (x0 )|) ≤ M F(|e1 − x0 |).
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
By using (1), the Schwarz inequality implies F(|e1 − x0 |) ≤
p
9
F((e1 − x0 )2 ) = 0, hence
F(| f − f (x0 )|) = 0. Since |F(g)| ≤ F(|g|), ∀g ∈ C[a, b], Eq. (9) yields F( f ) = f (x0 ).
(9)
We present below a simple original proof of Lupas¸’s Theorem 7. Proof of Theorem 7. By Schwarz’s inequality we deduce that A(e2 ) − (A(e1 ))2 ≥ 0. Consider first the case A(e2 ) − (A(e1 ))2 > 0. Define the auxiliary linear functional B: C[a, b] → R, B( f ) = A( f ) − f (A(e1 )) (see A(e1 ) ∈ [a, b]). We have: B(e0 ) = A(e0 ) − e0 (A(e1 )) = 1 − 1 = 0, B(e1 ) = A(e1 ) − e1 (A(e1 )) = A(e1 ) − A(e1 ) = 0, B(e2 ) = A(e2 ) − e2 (A(e1 )) = A(e2 ) − (A(e1 ))2 > 0. Using the Jensen inequality for the linear positive functional A we deduce that B(ψt ) ≥ 0,
for any t ∈ [a, b].
By Popoviciu’s Theorem 6 we deduce that there exist distinct points θ1 , θ2 , θ3 ∈ [a, b], such that B( f ) = B(e2 )[θ1 , θ2 , θ3 ; f ]. This concludes the proof of the first case. Consider now the case A(e2 ) = (A(e1 ))2 . We have: A(e0 ) = 1 = e0 (A(e1 )), A(e1 ) = e1 (A(e1 )), A(e2 ) = (A(e1 ))2 = e2 (A(e1 )). ˇ skin’s Theorem 5, we deduce that A( f ) = f (A(e1 )) for all f ∈ C[a, b], and Eq. (2) is By Saˇ satisfied for all distinct points θ1 , θ2 , θ3 ∈ [a, b]. Acknowledgments The authors thank the editors and the referees for valuable comments and useful suggestions that improved this note. References [1] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, in: de Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter & Co, Berlin, ISBN: 3-11-014178-7, 1994, p. xii+627. MR1292247 (95g:41001). [2] J. Bustamante, J.M. Quesada, On an extremal relation of Bernstein operators, J. Approx. Theory 141 (2) (2006) 214–215. MR2252100. [3] J. de la Cal, J. C´arcamo, An extremal property of Bernstein operators, J. Approx. Theory 146 (1) (2007) 87–90. MR2327474. [4] A. Lupas¸, Mean value theorems for positive linear transformations, Rev. Anal. Num´er. Th´eor. Aprox. 3 (2) (1974) 121–140 (1975) (in Romanian) MR0390601 (52 #11426). ˇ skin, Korovkin systems in spaces of continuous functions, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962) [5] Ju.A. Saˇ 495–512 (in Russian) MR0147905 (26 #5418). [6] Popoviciu Tiberiu, Notes sur les fonctions convexes d’ordre sup´erieur. IX. In´egalit´es lin´eaires et bilin´eaires entre les fonctions convexes. Quelques g´en´eralisations d’une in´egalit´e de Tchebycheff, Bull. Math. Soc. Roumaine Sci. 43 (1941) 85–141 (in French) MR0013170(7,116d).
An extremal property for a class of positive linear operators Ioan Gavrea, Mircea Ivan ∗ Department of Mathematics, Technical University of Cluj-Napoca, Str. C. Daicoviciu 15, 400020 Cluj-Napoca, Romania Received 24 February 2008; accepted 4 February 2009 Available online 21 February 2009 Communicated by Kirill Kopotun
Abstract We generalize a recent result of de la Cal and C´arcamo concerning an extremal property of Bernstein operators. c 2009 Elsevier Inc. All rights reserved.
Keywords: Positive linear operators; Finitely defined operators; Bernstein operators
1. Introduction and main results Let n be a natural number greater than 1. Consider a set of linear independent functions {ϕ0 , . . . , ϕn } ⊂ C[0, 1] such that ϕk (x) > 0, for x ∈ (0, 1), and k = 0, . . . , n. Let xk ∈ [0, 1], k = 0, . . . , n, be distinct points. Denote by δx , x ∈ [0, 1], the evaluation functional δx : R[0,1] → R, δx ( f ) := f (x). P Definition 1. A linear operator U : C[0, 1] → C[0, 1] of the form U = nk=0 ϕk δxk is said to be of discrete type. Denote by V the set of all linear P operators V : C[0, 1] → C[0, 1] preserving the affine functions and having the form V = nk=0 ϕk λk , where λk are linear and positive functionals defined on C[0, 1]. ∗ Corresponding author.
E-mail addresses: [email protected] (I. Gavrea), [email protected] (M. Ivan). c 2009 Elsevier Inc. All rights reserved. 0021-9045/$ - see front matter doi:10.1016/j.jat.2009.02.006
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
7
Theorem 2. Let U be an operator of discrete type preserving the affine functions. Then, for all V ∈ V, the following inequality is satisfied f ≤ U f ≤ V f,
for any convex function f ∈ C[0, 1].
Moreover, if there exist a strict convex function g ∈ C[0, 1] and a point x ∈ (0, 1) such that U g(x) = V g(x), then V = U. Suppose that the points xk satisfy 0 = x0 < · · · < xn = 1. Theorem 3. Let U : C[0, 1] → C[0, 1] be an operator of discrete type preserving the affine functions and V : C[0, 1] → R[0,1] a linear positive operator. Then, the following propositions are equivalent: (a) V ∈ V. (b) There exists a linear positive operator L: C[0, 1] → C[0, 1] preserving linear functions such that V = U ◦ L. (c) There exists a linear positive operator L: C[0, 1] → R[0,1] preserving the affine functions such that V = U ◦ L. Remark 4. Theorems 2 and 3 are respective extensions of Theorems 2 and 3 in [3], where k/n are replaced by arbitrary knots xk and the Bernstein basis is replaced by the functions ϕk . See also [2, Th. 1]. 2. Auxiliary results The divided difference [x1 , x2 , x3 ; f ] of a function f ∈ R[0,1] on the distinct knots x1 , x2 , x3 ∈ [0, 1] is defined by [x1 , x2 , x3 ; f ] :=
f (x1 ) f (x2 ) f (x3 ) + + . (x1 − x2 )(x1 − x3 ) (x2 − x3 )(x2 − x1 ) (x3 − x1 )(x3 − x2 )
A function f ∈ R[0,1] is said to be convex (strictly convex) if [x1 , x2 , x3 ; f ] ≥ 0 (> 0),
∀ distinct knots x1 , x2 , x3 ∈ [0, 1].
As usual, denote by ei the monomials ei (t) := t i , i = 0, 1, 2. For t ∈ [a, b], consider the positive part function, ψt : [a, b] → R, ψt (x) = (x −t)+ := (x −t +|x −t|)/2. We note that the functions ψt are convex. ˇ skin [5], and Lupas¸ [4]. Throughout this note we use some results of Popoviciu [6], Saˇ In the case of the Tchebyshev system {e0 , e1 , e2 }, Theorem 1 of [5] (see also [1, p. 117, Example 3; p.116, Theorem 2.5.4]) becomes: ˇ skin). Let x0 ∈ [a, b]. If F: C[a, b] → R is a linear positive functional such that Theorem 5 (Saˇ F(ei ) = ei (x0 ),
i = 0, 1, 2,
F( f ) = f (x0 ),
∀ f ∈ C[a, b].
then
In the case of n = 2, Popoviciu’s Theorem [6, Th´eor`eme 12] becomes: Theorem 6 (Popoviciu). If F: C[a, b] → R is a linear functional such that: (i) F(e0 ) = F(e1 ) = 0, F(e2 ) > 0, (ii) F(ψt ) ≥ 0, ∀t ∈ [a, b],
(1)
8
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
then, for any f ∈ C[a, b], there exists a set of distinct points θ1 , θ2 , θ3 ∈ [a, b], depending on f , such that F( f ) = F(e2 )[θ1 , θ2 , θ3 ; f ]. Theorem 7 (Lupas¸ [4, Th. 1]). Let A: C[a, b] → R be a linear positive functional with A(e0 ) = 1. Then, for any f ∈ C[a, b], there exist distinct points θ1 , θ2 , θ3 ∈ [a, b], depending on f , such that A( f ) = f (A(e1 )) + (A(e2 ) − (A(e1 ))2 )[θ1 , θ2 , θ3 ; f ].
(2)
3. Proofs The proof for Theorem 3 and the first part of Theorem 2 are essentially the same (with obvious changes) as those given in [3]. Proof of Theorem 2 (Second Part). From U e0 = e0 = V e0
and U e1 = e1 = V e1 ,
using the linear independence of the set {ϕ0 , . . . , ϕn }, we obtain: λk (e0 ) = 1 = e0 (xk ) and
λk (e1 ) = xk = e1 (xk ),
k = 0, . . . , n.
(3)
Let g ∈ C[0, 1] be a strictly convex function and x ∈ (0, 1) such that V g(x) = U g(x), i.e., n X (λk (g) − g(xk ))ϕk (x) = 0.
(4)
k=0
Using the Jensen inequality for the functionals λk , we obtain λk (g) ≥ g(λk (e1 )) = g(xk ),
k = 0, . . . , n.
(5)
By using the positivity of ϕk , Eqs. (4) and (5) imply λk (g) = g(xk ),
k = 0, . . . , n.
(6)
From Theorem 7, we deduce that there exist distinct points θ1 , θ2 , θ3 ∈ [0, 1] such that λk (g) = g(xk ) + (λk (e2 ) − (λk (e1 ))2 )[θ1 , θ2 , θ3 ; g],
k = 0, . . . , n.
(7)
Since g is strictly convex, Eqs. (6) and (7) imply λk (e2 ) = (λk (e1 ))2 = xk2 = e2 (xk ),
k = 0, . . . , n.
(8)
Eqs. (3) and (8), and Theorem 5 give λk ( f ) = f (xk ), k = 0, . . . , n. This concludes the proof. ˇ skin’s Theorem. For readers’ convenience, we insert below an original simple proof of Saˇ Proof of Theorem 5. Since the class of all Lipschitz functions f : [a, b] → R is dense in C[a, b], it is sufficient to prove the theorem for Lipschitz functions only. Let f ∈ Lip M [a, b]. The following inequalities are satisfied: | f (x) − f (x0 )| ≤ M |x − x0 |,
∀x ∈ [a, b],
F(| f − f (x0 )|) ≤ M F(|e1 − x0 |).
I. Gavrea, M. Ivan / Journal of Approximation Theory 162 (2010) 6–9
By using (1), the Schwarz inequality implies F(|e1 − x0 |) ≤
p
9
F((e1 − x0 )2 ) = 0, hence
F(| f − f (x0 )|) = 0. Since |F(g)| ≤ F(|g|), ∀g ∈ C[a, b], Eq. (9) yields F( f ) = f (x0 ).
(9)
We present below a simple original proof of Lupas¸’s Theorem 7. Proof of Theorem 7. By Schwarz’s inequality we deduce that A(e2 ) − (A(e1 ))2 ≥ 0. Consider first the case A(e2 ) − (A(e1 ))2 > 0. Define the auxiliary linear functional B: C[a, b] → R, B( f ) = A( f ) − f (A(e1 )) (see A(e1 ) ∈ [a, b]). We have: B(e0 ) = A(e0 ) − e0 (A(e1 )) = 1 − 1 = 0, B(e1 ) = A(e1 ) − e1 (A(e1 )) = A(e1 ) − A(e1 ) = 0, B(e2 ) = A(e2 ) − e2 (A(e1 )) = A(e2 ) − (A(e1 ))2 > 0. Using the Jensen inequality for the linear positive functional A we deduce that B(ψt ) ≥ 0,
for any t ∈ [a, b].
By Popoviciu’s Theorem 6 we deduce that there exist distinct points θ1 , θ2 , θ3 ∈ [a, b], such that B( f ) = B(e2 )[θ1 , θ2 , θ3 ; f ]. This concludes the proof of the first case. Consider now the case A(e2 ) = (A(e1 ))2 . We have: A(e0 ) = 1 = e0 (A(e1 )), A(e1 ) = e1 (A(e1 )), A(e2 ) = (A(e1 ))2 = e2 (A(e1 )). ˇ skin’s Theorem 5, we deduce that A( f ) = f (A(e1 )) for all f ∈ C[a, b], and Eq. (2) is By Saˇ satisfied for all distinct points θ1 , θ2 , θ3 ∈ [a, b]. Acknowledgments The authors thank the editors and the referees for valuable comments and useful suggestions that improved this note. References [1] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, in: de Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter & Co, Berlin, ISBN: 3-11-014178-7, 1994, p. xii+627. MR1292247 (95g:41001). [2] J. Bustamante, J.M. Quesada, On an extremal relation of Bernstein operators, J. Approx. Theory 141 (2) (2006) 214–215. MR2252100. [3] J. de la Cal, J. C´arcamo, An extremal property of Bernstein operators, J. Approx. Theory 146 (1) (2007) 87–90. MR2327474. [4] A. Lupas¸, Mean value theorems for positive linear transformations, Rev. Anal. Num´er. Th´eor. Aprox. 3 (2) (1974) 121–140 (1975) (in Romanian) MR0390601 (52 #11426). ˇ skin, Korovkin systems in spaces of continuous functions, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962) [5] Ju.A. Saˇ 495–512 (in Russian) MR0147905 (26 #5418). [6] Popoviciu Tiberiu, Notes sur les fonctions convexes d’ordre sup´erieur. IX. In´egalit´es lin´eaires et bilin´eaires entre les fonctions convexes. Quelques g´en´eralisations d’une in´egalit´e de Tchebycheff, Bull. Math. Soc. Roumaine Sci. 43 (1941) 85–141 (in French) MR0013170(7,116d).