Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 5, Article 90, 2003
AN ELEMENTARY PROOF OF THE PRESERVATION OF LIPSCHITZ
CONSTANTS BY THE MEYER-KÖNIG AND ZELLER OPERATORS
TIBERIU TRIF
U NIVERSITATEA BABE Ş -B OLYAI ,
FACULTATEA DE M ATEMATIC Ă ŞI I NFORMATIC Ă ,
S TR . M. KOG ĂLNICEANU , 1,
3400 C LUJ -NAPOCA , ROMANIA .
ttrif@math.ubbcluj.ro
Received 20 October, 2003; accepted 02 November, 2003
Communicated by A. Lupaş
A BSTRACT. An elementary proof of the preservation of Lipschitz constants by the Meyer-König
and Zeller operators is presented.
Key words and phrases: Meyer-König and Zeller operators, Preservation of Lipschitz constants.
2000 Mathematics Subject Classification. 41A36, 26A16.
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.
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
149-03
2
T IBERIU T RIF
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 f (x) =
f
mn,k (x),
n
+
k
k=0
n+k k
mn,k (x) =
x (1 − x)n+1 ,
k
x ∈ [0, 1[,
k = 0, 1, 2, . . . .
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
∞
X
n+j j
j
Mn f (x2 ) =
f
x2 (1 − x2 )n+1
n
+
j
j
j=0
j
∞
X
j
n+j
x2 − x1 + x1 − x1 x2
n+1
=
f
(1 − x2 )
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
(n + j)!
xk (x2 − x1 )j−k (1 − x2 )n+k+1
j
· 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
·
,
k+`
n
+
k
+
`
n!k!`!
(1
−
x
1)
k=0 `=0
where the change of index j − k = ` was used for the last equality. We have also
∞
X
k
n+k k
Mn f (x1 ) =
f
x1 (1 − x1 )n+1
n+k
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
k
n + k xk1 (1 − x2 )n+k+1
=
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
J. Inequal. Pure and Appl. Math., 4(5) Art. 90, 2003
http://jipam.vu.edu.au/
P RESERVATION OF L IPSCHITZ C ONSTANTS
3
In particular, the above equalities show that
∞
X
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
(3)
·
= 1,
k+`
n!k!`!
(1
−
x
1)
k,`=0
(4)
(5)
∞
X
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
k+`
·
·
= x2 ,
k+`
n
+
k
+
`
n!k!`!
(1
−
x
1)
k,`=0
∞
X
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
k
·
·
= x1 .
n+k
n!k!`!
(1 − x1 )k+`
k,`=0
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
(n + k + `)! xk1 (x2 − x1 )` (1 − x2 )n+k+1
k+`
k
·
−
.
≤A
k+`
n!k!`!
(1
−
x
n
+
k
+
`
n
+
k
1)
k,`=0
Taking into account (3) and the fact that the function t ∈ [0, ∞[ 7−→ tα ∈ [0, ∞[ is concave, we
deduce that
|Mn f (x2 ) − Mn f (x1 )|
" ∞
# α
X (n + k + `)! xk (x2 − x1 )` (1 − x2 )n+k+1 k + `
k
≤A
· 1
−
.
k+`
n!k!`!
(1
−
x
n
+
k
+
`
n
+
k
1)
k,`=0
Using now (4) and (5) we get
|Mn f (x2 ) − Mn f (x1 )| ≤ A(x2 − x1 )α ,
i.e., Mn f ∈ LipA α. This completes the proof.
R EFERENCES
[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.
[4] A. LUPAŞ AND M.W. MULLER, Approximation properties of the Mn -operators, Aequationes
Math., 5 (1970), 19–37.
J. Inequal. Pure and Appl. Math., 4(5) Art. 90, 2003
http://jipam.vu.edu.au/