Journal of Inequalities in Pure and
Applied Mathematics
AN ELEMENTARY PROOF OF THE PRESERVATION OF LIPSCHITZ
CONSTANTS BY THE MEYER-KÖNIG AND ZELLER OPERATORS
TIBERIU TRIF
UNIVERSITATEA BABEŞ-BOLYAI,
FACULTATEA DE MATEMATICĂ ŞI INFORMATICĂ,
STR. M. KOGĂLNICEANU, 1,
volume 4, issue 5, article 90,
2003.
Received 20 October, 2003;
accepted 02 November, 2003.
Communicated by: A. Lupaş
3400 CLUJ-NAPOCA, ROMANIA.
EMail: ttrif@math.ubbcluj.ro
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
149-02
Quit
Given the real numbers A ≥ 0 and 0 < α ≤ 1, we denote by LipA α the set
of all functions f : [0, 1] → R, satisfying
|f (x2 ) − f (x1 )| ≤ A|x2 − x1 |α
for all x1 , x2 ∈ [0, 1].
The main purpose of this note is to present an elementary proof of the following result:
Given the continuous function f : [0, 1] → R, it holds that
f ∈ LipA α
(1)
if and only if
(2)
Mn f ∈ LipA α
for all n ≥ 1,
where (Mn )n≥1 is the sequence of Meyer-König and Zeller operators.
It should be mentioned that similar proofs for other operators are to be found
in [2] and [3]. On the other hand, the equivalence (1) ⇔ (2) is a special case
of a much more general result [1, Theorem 1]. However, the proof presented in
[1] is completely different and does not have an elementary character.
Proof. Let f : [0, 1] → R be a continuous function and let n be a positive
integer. Recall that the nth Meyer-König and Zeller power series associated to
f is defined by (see [4])
Mn f (1) = f (1),
∞
X
k
mn,k (x),
Mn f (x) =
f
n
+
k
k=0
n+k k
mn,k (x) =
x (1 − x)n+1 ,
k
An Elementary Proof of the
Preservation of Lipschitz
Constants by the Meyer-König
and Zeller Operators
Tiberiu Trif
Title Page
Contents
JJ
J
II
I
Go Back
Close
x ∈ [0, 1[,
Quit
Page 2 of 6
k = 0, 1, 2, . . . .
J. Ineq. Pure and Appl. Math. 4(5) Art. 90, 2003
http://jipam.vu.edu.au
That (2) implies (1) follows from the fact that the sequence (Mn f )n≥1 converges
uniformly to f on [0, 1]. Thus it remains to prove that (1) implies (2). To this
end, let n be an arbitrary positive integer and let 0 ≤ x1 < x2 < 1 (since Mn f
is continuous at 1, it suffices to consider only the case x2 < 1). Then we have
Mn f (x2 )
∞
X
n+j j
j
=
f
x2 (1 − x2 )n+1
n+j
j
j=0
j
∞
X
n+j
x2 − x1 + x1 − x1 x2
j
n+1
=
(1 − x2 )
f
n
+
j
j
1 − x1
j=0
j ∞
X
j
n + j (1 − x2 )n+1 X j k
=
f
x1 (1 − x2 )k (x2 − x1 )j−k
j
n
+
j
j
(1
−
x
)
k
1
j=0
k=0
j
∞
XX
j
(n + j)!
xk (x2 − x1 )j−k (1 − x2 )n+k+1
=
f
· 1
n + j n!k!(j − k)!
(1 − x1 )j
j=0 k=0
∞ X
∞
X
j
(n + j)!
xk (x2 − x1 )j−k (1 − x2 )n+k+1
· 1
=
f
n + j n!k!(j − k)!
(1 − x1 )j
k=0 j=k
∞ X
∞
X
k+`
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
=
f
,
·
n+k+`
n!k!`!
(1 − x1 )k+`
k=0 `=0
where the change of index j − k = ` was used for the last equality. We have
An Elementary Proof of the
Preservation of Lipschitz
Constants by the Meyer-König
and Zeller Operators
Tiberiu Trif
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 6
J. Ineq. Pure and Appl. Math. 4(5) Art. 90, 2003
http://jipam.vu.edu.au
also
Mn f (x1 )
∞
X
k
n+k k
f
=
x1 (1 − x1 )n+1
k
n+k
k=0
∞
X
k
n + k k (1 − x2 )n+k+1
f
=
x1 ·
·
k
n
+
k
k
(1
−
x
1)
k=0
1
n+k+1
2 −x1
1 − x1−x
1
∞
X n + k + ` x2 − x1 `
`
1 − x1
`=0
∞
X
n + k xk1 (1 − x2 )n+k+1
k
=
f
n+k
k
(1 − x1 )k
k=0
∞ X
∞
X
k
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
=
f
·
.
n+k
n!k!`!
(1 − x1 )k+`
k=0 `=0
In particular, the above equalities show that
(3)
(4)
(5)
∞
X
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
·
= 1,
k+`
n!k!`!
(1
−
x
1)
k,`=0
∞
X
k+`
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
·
·
= x2 ,
n+k+`
n!k!`!
(1 − x1 )k+`
k,`=0
∞
X
k
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
·
·
= x1 .
n+k
n!k!`!
(1 − x1 )k+`
k,`=0
An Elementary Proof of the
Preservation of Lipschitz
Constants by the Meyer-König
and Zeller Operators
Tiberiu Trif
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 6
J. Ineq. Pure and Appl. Math. 4(5) Art. 90, 2003
http://jipam.vu.edu.au
Since f ∈ LipA α, we have
|Mn f (x2 ) − Mn f (x1 )|
∞
X
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
≤
·
n!k!`!
(1 − x1 )k+`
k,`=0
k
+
`
k
f
−
f
n+k+`
n+k α
∞
X
k+`
k
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
≤A
·
−
.
n!k!`!
(1 − x1 )k+`
n+k+` n+k
k,`=0
α
Taking into account (3) and the fact that the function t ∈ [0, ∞[ 7−→ t ∈ [0, ∞[
is concave, we deduce that
An Elementary Proof of the
Preservation of Lipschitz
Constants by the Meyer-König
and Zeller Operators
Tiberiu Trif
|Mn f (x2 ) − Mn f (x1 )|
" ∞
# α
X (n + k + `)! xk (x2 − x1 )` (1 − x2 )n+k+1 k + `
k
1
≤A
·
−
.
n!k!`!
(1 − x1 )k+`
n+k+` n+k
k,`=0
Using now (4) and (5) we get
|Mn f (x2 ) − Mn f (x1 )| ≤ A(x2 − x1 )α ,
Title Page
Contents
JJ
J
II
I
Go Back
i.e., Mn f ∈ LipA α. This completes the proof.
Close
Quit
Page 5 of 6
J. Ineq. Pure and Appl. Math. 4(5) Art. 90, 2003
http://jipam.vu.edu.au
References
[1] J.A. ADELL AND J. de la CAL, Using stochastic processes for studying
Bernstein-type operators, Proceedings of the Second International Conference in Functional Analysis and Approximation Theory (Acquafredda di
Maratea, 1992), Rend. Circ. Mat. Palermo, Suppl. 33(2) (1993), 125–141.
[2] B.M. BROWN, D. ELLIOTT AND D.F. PAGET, Lipschitz constants for
the Bernstein polynomials of a Lipschitz continuous function, J. Approx.
Theory, 49 (1987), 196–199.
[3] B. DELLA VECCHIA, On the preservation of Lipschitz constants for some
linear operators, Bol. Un. Mat. Ital. B, 3(7) (1989), 125–136.
An Elementary Proof of the
Preservation of Lipschitz
Constants by the Meyer-König
and Zeller Operators
Tiberiu Trif
[4] A. LUPAŞ AND M.W. MULLER, Approximation properties of the Mn operators, Aequationes Math., 5 (1970), 19–37.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 6
J. Ineq. Pure and Appl. Math. 4(5) Art. 90, 2003
http://jipam.vu.edu.au