Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
21 (2005), 177–183
www.emis.de/journals
ISSN 1786-0091
ON THE POINTWISE ESTIMATION OF CESARO KERNEL OF
NEGATIVE ORDER WITH RESPECT TO WALSH-PALEY
SYSTEM
V. TEVZADZE
Abstract. Some pointwise properties of (C, α) kernel (−1 < α < 0) with
respect to the Walsh–Paley system are established.
1. Introduction
Let r0 (x) be the function defined by
(
£
¢
1 if x ∈ 0, 12 ,
£
¢
r0 (x) =
−1 if x ∈ 12 , 1 ,
r0 (x + 1) = r0 (x).
The Rademacher system is defined by
rn (x) = r0 (2n x),
n ≥ 1, and x ∈ [0, 1).
Let ψ0 (x), ψ1 (x), ψ2 (x), . . . represent the Walsh functions, i.e. ψ0 (x) = 1, and if
k = 2n1 + 2n2 + · · · + 2ns is a positive integer with n1 > n2 > · · · > ns , then
ψk (x) = rn1 (x) · rn2 (x) · · · rns (x).
The idea of using the products of the Rademacher functions is to define the Walsh
system originated by Paley [4].
Denote by Knα (t) the kernel of the method (C, α) and call it the (C, α) kernel,
or the Cesaro kernel:
n
1 X α
Knα (t) = α
A
ψν (t),
An ν=0 n−ν
(α + 1)(α + 2) · · · (α + k)
k!
It is well-known ([8], Ch. 3) that
n
X
Aα−1
(1)
Aα
=
n
n−k ;
Aα
k =
(α 6= −k).
k=0
(2)
(3)
α
α−1
Aα
;
n − An−1 = An
α
α
An ∼ n .
2000 Mathematics Subject Classification. 42C10.
Key words and phrases. Walsh functions, Dirichlet kernel, Cesaro means.
177
178
V. TEVZADZE
Some properties of the (C, α) kernel (α > 0) have been established by Fine [1],
Yano [7], Gát [2], [3] and [5]. Using these properties they studied the problems of
pointwise and uniform (C, α) summability of Walsh–Fourier series.
In the present paper we study some pointwise properties of (C, α) kernel
(−1 < α < 0) with respect to the Walsh–Paley system. The results of this paper have been published without proof in [6].
2. Main Results
The main results of the paper are presented in the form of the following propositions.
Theorem 1. The estimation
Kn−α (t) ≤ c(α) ·
1
1
· 1−α ,
t
A−α
n
t ∈ (0, 1),
0 < α < 1,
holds.
Theorem 2. For any α ∈ (0, 1) and p ≥ 2m the equality
m
¶
µ 2X
−1
−α
Ap−ν ψν (t) = Sgn ψ2m −1 (t), t ∈ [0, 1),
Sgn
ν=0
is valid.
3. Auxiliary Results
We shall need the following
Lemma 1. For any α > 0 and p > 2m − 1 + α the sum
m
2X
−1
A−α
p−ν ψν (t)
ν=0
is representable in the form
m
2X
−1
(1)
A−α
p−ν ψν (t)
ν=0
=
m
2X
−1
`ν A−α−i
p−qν ,
ν=0
where `ν , qν , i are nonnegative integers depending only on the point t ∈ [0, 1] and
m ∈ N (and not depending on p and α); moreover, i + qν ≤ 2m − 1.
Proof. Using the method of mathematical induction, we can verify that the lemma
is valid for m = 1. We have
1
2X
−1
−α
−α
A−α
p−ν ψν (t) = Ap ψ0 (t) + Ap−1 ψ1 (t).
ν=0
Since on the interval 0 ≤ t <
1
2
−α
−α
−α
A−α
p ψ0 (t) + Ap−1 ψ1 (t) = Ap + Ap−1 ,
and on the interval
1
2
≤t<1
−α
−α
−α
−α−1
A−α
,
p ψ0 (t) + Ap−1 ψ1 (t) = Ap − Ap−1 = Ap
ON THE POINTWISE ESTIMATION OF CESARO KERNEL. . .
179
our lemma for m = 1 is valid. Let the lemma be valid for m − 1 ∈ N, and we prove
that the lemma is valid for m ∈ N. Indeed, we have
m
2X
−1
A−α
p−ν ψν (t)
2m−1
X−1
=
ν=0
A−α
p−ν ψν (t)
+ ψ2m−1 (t)
2m−1
X−1
ν=0
A−α
p−2m−1 −ν ψν (t)
ν=0
(α > 0,
p > 2m − 1 + α).
We consider two cases: (1) ψ2m−1 (t) = 1; (2) ψ2m−1 (t) = −1.
Let ψ2m−1 (t) = 1. By the assumption
p − 2m−1 > 2m − 1 + α − 2m−1 > 2m−1 − 1 + α,
2m−1
X−1
A−α
p−ν ψν (t)
2m−1
X−1
=
ν=0
and
2m−1
X−1
cν A−α−i
p−mν
ν=0
A−α
p−2m−1 −ν ψν (t)
=
2m−1
X−1
ν=0
cν A−α−i
p−2m−1 −mν .
ν=0
Hence we have
m
2X
−1
A−α
p−ν ψν (t) =
2m−1
X−1
ν=0
cν A−α−i
p−mν +
ν=0
2X
−1
2
A−α
p−ν ψν (t) =
ν=0
(2)
cν A−α−i
p−2m−1 −mν =
ν=0
Let now ψ2m−1 = −1. Since
m
2m−1
X−1
Aα
n
−
Aα
n−1
=
=
we have
A−α
p−ν ψν (t)+ψ2m−1 (t)
ν=0
2m−1
X−1 ³
`ν A−α−i
p−qν .
ν=0
Aα−1
,
n
m−1
X−1
m
2X
−1
2m−1
X−1
A−α
p−2m−1 −ν ψν (t)
ν=0
´
−α
A−α
p−ν − Ap−2m−1 −ν ψν (t)
ν=0
2m−1
X−1 ³
=
´
−α−1
−α−1
A−α−1
+
A
+
·
·
·
+
A
p−ν
p−ν−1
p−2m−1 −ν−1 ψν (t).
ν=0
By the assumption,
2m−1
X−1
A−α−1
p−j−ν
=
2m−1
X−1
ν=0
cν A−α−1−i
p−j−mν ,
j = 0, 1, 2, . . . , 2m−1 − 1,
ν=0
hence from (2) we have
2m−1
X−1
A−α−1
p−ν ψν (t)
ν=0
=
m−1
2m−1
X−1 2 X−1
j=0
cν A−α−1−i
p−j−mν
ν=0
=
m
2X
−1
`ν A−α−1−i
,
p−qν
ν=0
i.e. in both cases equation (1) holds. It is evident that in these cases
i + qν ≤ 2m − 1.
Thus the lemma is proved.
¤
Lemma 2. For any α > 0 and p > 2m + α the equality
m
m
µ 2X
¶
µ 2X
¶
−1
−1
−α
−α−1
Sgn
Ap−ν ψν (t) = − Sgn
Ap−ν ψν (t) ,
ν=0
is valid.
ν=0
t ∈ [0, 1),
180
V. TEVZADZE
Lemma 2 follows directly from Lemma 1 if we take into account that in the
−α−1
conditions of Lemma 2 Sgn A−α
p−ν = − Sgn Ap−ν .
4. Proof of Main Results
Proof of Theorem 1. Let t ∈ (0, 1) and m ∈ N (N is a set of natural numbers),
such that 2−m ≤ t < 2−m+1 . We write n ≥ 1 in the form n = p · 2m + q, where
0 ≤ q < 2m . We have1
m
Kn−α (t)
p·2
n
X−1
1 X −α
1
= −α
An−ν ψν (t) = −α
A−α
n−ν ψν (t)
An ν=0
An
ν=0
+
1
A−α
n
n
X
A−α
n−ν ψν (t)
ν=p·2m
m
p−1 2 −1
1 X X −α
= −α
A
ψr·2m +ν (t)
m
An r=0 ν=0 n−r·2 −ν
(3)
+
q
1 X −α
An−p·2m −ν ψp·2m +ν (t)
A−α
n ν=0
m
p−2
2X
−1
1 X
= −α
ψr·2m
A−α
n−r·2m −ν ψν (t)
An r=0
ν=0
m
2X
−1
1
+ −α ψ(p−1)·2m (t)
A−α
n−r·2m −ν ψν (t)
An
ν=0
+
q
X
1
m
A−α
ψ
(t)
p·2
q−ν ψν (t) = A1 + A2 + A3 .
A−α
n
ν=0
Estimate A1 . Using Abelian transformation, we have
m
¯ p−2
¯
2X
−1
¯
1 ¯¯ X
−α
A1 = −α ¯
ψr·2m (t)
An−r2m −ν ψν (t)¯¯
An r=0
ν=0
m
¯X
p−2
2X
−2
1 ¯¯
ψr·2m (t)
A−α−1
= −α ¯
n−r·2m −ν Dν (t)
An r=0
ν=0
¯
p−2
X
¯
¯,
m
ψr·2m A−α
D
(t)
+
n−(r+1)2m +1 2
¯
r=0
where
Dk (t) =
k−1
X
ψk (t).
i=0
Since (see [8])
(4)
α
c1 (α)nα ≤ Aα
n ≤ c2 (α)n ,
α > −2,
1Here the use is made of the equality ψ
r+s (t) = ψr (t)ψs (t), if in the binary expansion r, s ∈ N
the same terms are omitted.
ON THE POINTWISE ESTIMATION OF CESARO KERNEL. . .
181
and |Dk (t) ≤ k, t ∈ [0, 1], we obtain
m
p−2 2X
−1
X
1
|A1 | ≤ −α · c(α) · 2m
(n − r · 2m − ν)−α−1
An
r=0 ν=0
(5)
1
1
m
m −α
≤ −α · c(α)2m(1−α)
−α · c(α) · 2 (n − (p − 1) · 2 )
An
An
1
1
≤ c(α) · −α · 1−α .
t
An
≤
For A2 we have
m
¯
¯
2X
−1
¯
1 ¯¯
−α
|A2 | = −α ¯ψ(p−1)2m (t)
An−(p−1)2m −ν ψν (t)¯¯
An
ν=0
m
2 −1
1 X
≤ c(α) · −α
(n − (p − 1)2m − ν)−α
An ν=0
m
2 −1
1 X m
≤ c(α) · −α
(2 + q − ν)−α
An ν=0
(6)
m
2 −1
1 X m
1
≤ c(α) · −α
(2 − ν)−α ≤ c(α) · −α 2m(1−α)
An ν=0
An
≤ c(α) ·
1
1
−α · 1−α .
t
An
Estimate now A3 ,
|A3 | =
(7)
¯ q
¯
µ
¶
q
X
¯
1 ¯¯ X −α
1
−al
¯
1
+
(q
−
ν)
A
ψ
(t)
≤
c(α)
q−ν ν
¯
¯
A−α
A−α
n
n
ν=0
ν=0
≤ c(α)
1 1−α
1
1
1
q
≤ c(α) −α 2m(1−α) ≤ c(α) −α 1−α .
t
A−α
A
A
n
n
n
Taking into account (5), (6) and (7), from (3) we get
|Kn−α | ≤ c(α)
1
1
,
1−α
t
A−α
n
which was to be proved.
¤
Proof of Theorem 2. We use the method of mathematical induction. For m = 1
the lemma is valid. Indeed,
1
2X
−1
−α
−α
A−α
p−ν ψν (t) = Ap ψ0 (t) + Ap−1 ψ1 (t);
ν=0
on the interval 0 ≤ t <
1
2
−α
−α
−α
A−α
p ψ0 (t) + Ap−1 ψ1 (t) = Ap + Ap−1 > 0,
and on the interval
1
2
≤t<1
−α
−α
−α
−α−1
A−α
< 0.
p ψ0 (t) + Ap−1 ψ1 (t) = Ap − Ap−1 = Ap
182
V. TEVZADZE
Since ψ1 (t) = 1 if 0 ≤ t < 12 , and ψ1 (t) = −1 if
1
µ 2X
−1
Sgn
1
2
≤ t < 1, therefore
¶
A−α
p−ν ψν (t)
= Sgn ψ1 (t).
ν=0
Let the lemma be valid for m − 1 ∈ N and let us prove that the lemma is valid for
m ∈ N. We have
(8)
m
2X
−1
A−α
p−ν ψν (t)
2m−1
X−1
=
ν=0
A−α
p−ν ψν (t)
2m−1
X−1
+ ψ2m−1 (t)
ν=0
ν=0
m−1
Let ψ2m−1 = 1. Since p − 2
µ 2 X−1
m−1
Sgn
A−α
p−2m−1 −ν ψν (t).
m−1
≥2
, by virtue of the assumption,
¶
A−α
ψ
(t)
= Sgn ψ2m−1 −1 (t),
p−ν ν
ν=0
Sgn
µ 2m−1
X−1
¶
A−α
p−2m−1 −ν ψν (t)
= Sgn ψ2m−1 −1 (t).
ν=0
Hence from (8) it follows that
m
µ 2X
¶
−1
−α
Sgn
Ap−ν ψν (t) = Sgn ψ2m−1 −1 (t)
ν=0
= Sgn ψ2m−1 −1 (t) · Sgn ψ2m−1 (t)
= Sgn (ψ2m−1 −1 (t)ψ2m−1 (t)) = Sgn ψ2m −1 (t).
If, however, ψ2m−1 (t) = −1, equality (8) yields
m
2X
−1
(9)
A−α
p−ν ψν (t)
=
2m−1
X−1 ³
ν=0
´
−α
A−α
p−ν − Ap−2m−1 −ν ψν (t)
ν=0
=
2m−1
X−1 ³
´
−α−1
−α−1
A−α−1
+
A
+
·
·
·
+
A
m−1
p−ν
p−ν−1
p−2
−ν+1 ψν (t).
ν=0
Taking into account Lemma 1, for all j = 1, 2, . . . , 2m−1 − 1 we obtain
Sgn
m
µ 2X
−1
¶
A−α
p−ν ψν (t)
= Sgn
µ 2m−1
X−1
ν=0
¶
A−α−1
ψ
(t)
,
p−j−ν ν
ν=0
and consequently, using Lemma 2, from (9) it follows that
Sgn
m
µ 2X
−1
¶
A−α
p−ν ψν (t)
ν=0
= − Sgn
m
µ 2X
−1
= Sgn
µ 2m−1
X−1
¶
A−α−1
p−ν ψν (t)
ν=0
¶
−α
Ap−ν ψν (t) = − Sgn ψ2m−1 −1 (t)
ν=0
= Sgn ψ2m−1 (t) Sgn ψ2m−1 −1 (t) = Sgn ψ2m −1 (t).
Thus the theorem is proved.
¤
ON THE POINTWISE ESTIMATION OF CESARO KERNEL. . .
183
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Received April 15, 2005.
Department of Mechanics and Mathematics,
Tbilisi State University,
Chavchavadze str. 1, Tbilisi 0128,
Georgia