I nternat. J. Math. Math. Sci.
(1978)47-56
Vol.
47
ON L I-CONVERGENCE OF WALSH-FOURIER SERIES
C. W. ONNEWEER
Department of Mathematics
University of New Mexico
Albuquerque, New Mexico 87131
(Received February 14, 1977 and in revised form October 25, 1977)
ABSTRACT.
Let G denote the dyadic group, which has as its dual group the
Walsh(-Paley) functions.
LI(G)
Ll(G)-norm. As
In this paper we formulate a condition for
functions in
which implies that their Walsh-Fourler series converges
in
a corollary we obtain a Dini-Lipschltz-type theorem for
L I (G) convergence and we prove that the assumption on the
LI(G)
modulus
Similar results also
of continuity in this theorem cannot be weakened.
hold for functions on the circle group T and their (trigonometric) Fourier
series.
Let G be the direct product of countably many groups of order 2.
Thus G-- {x
x,y
E
x
G the sum x
(xl) 0
+ y
with x i
{0,i} for each
i
>_ 0},
and for
is obtained by adding the i-th coordinates of
C.W. ONNEWEER
48
>_ 0.
x and y modulo 2 for each i
The topology of G can be described
by means of the (non-archimedean) norm
if x 0
0 and x
k
Xk_ I
G and for k
{x e G
llxll
<_
2 -k}
lxll
on G, where
II011
i, and
the subgroups Gk of G by G O
Gk
II’II
0.
Also, if we define
0},
0 --...=Xk_ I
For k
then the Gk form a basis for the neighborhoods of 0 in G.
k,
-k
>_ I
{x e G; x
we define the cosets l(n,k), 0 < n < 2
2
>_
0
k
If 0 < n < 2
of Gk as follows
then n can be represented uniquely as
n
with b i
e{0,1}
in G and let
b02k-I + bl 2k-2 + ...+ bk_ I,
Let e(n,k)
for each i.
e(n,k)
l(n,k)
Gk.
(bo,bl,...,bk_l,0,0,...)
So, in particular, l(0,k)
Gk.
Furthermore, in order to simplify the notation, we shall denote e(l,k)
by e (k).
Next, let
Its elements are the
denote the dual group of G.
Walsh functions and Paley defined the following enumeration for them.
For each k
>_
0 and x
#k(X)
e G define
(xi) 0
by
k(X)
exp(ixk).
If n > 0 is represented as
n
with a i
e{O,l}
a0
a12+ ...+
+
a
k
2k
for all i, then the n-th Walsh function
defined by
a
Xn(X)
0
0
is
a
(x)-..."
kk(x).
Let dx denote normalized Haar measure on G.
.
Xn
For f e
LI(G)
we define
its Walsh-Fourier series by
f(x)
k--O
}(k) Xk(X),
where
(k)
g
f(t)Xk(t)dt.
CONVERGENCE OF WALSH-FOURIER SERIES
For the partial sums of this series we have
n-1
7.
Sn(f;x)
f(k)
f(x-t)Dn(t)dt--
Xk(X)
k--0
where
,
Dn(X)
G
n-i
k=O
k (t)
Dn (t)
f
49
is called the Dirlchlet kernel of order n.
The following properties hold.
(D1) If n > 0 is expressed as n
2
k
+ n’
with 0 <
n’
< 2k
then
D2k(X) + k(X)Dn’ (x).
Dn(x)
(D2) For each k > 0 we have
k,
2
D2k(X)
(D3) If f e
LI(G)
1 <
x e l(m,k)
IS2k(f)-f
Ii
.I
O.
n.
< 2 k and 0 < n < 2
m
Gk,
if x e G k G
k
[0
0 we have D (0)
n
(D4) for each n >
(DS) If k > 0,
lk_=
then
if x e:
k,
then for each
e(m,k) + Gk we have
ISn(x)
[Dn (e(m k))l
<
m-12k+l
A proof of these properties and additional information on Walsh-Fourier
[2].
series can be found in
y e G the function f
Theorem I.
as n
/
Proof.
0 <
n’
.
Y
Finally, if f is a function on G and if
is defined by f
Let f be a function in
Then
ISn(f)-f Ii
Y
(x)
LI(G)
o(I) as n
f(x-y).
for which n
/
.
Let n > 0 be given and assume that n
< 2
k. Then
lSn(f)-fll I
<_
2
k
+ n’
f-f
e(n)
with
lSn(f)-S2k(f)ll I + lS2k(f)-f II I.
o()
C.W. ONNEWEER
50
Thus, according to (DI) and (D3) we have
IISn(f) IIl
Ik Dn’
<
fl
*
I
+ o(i)
A + o(i), as n
/
.
In order to find the appropriate estimate for A we continue as follows.
2k_l
I (p,k)
G p=O
k (t)Dn’ (t) f (x-t) dt
Clearly,
Dn,(t)
l(p,k)
l(2p,k+l)l(2p+l,k+l)
l(2p+l,k+l) then
whereas if t
[
A
<
[
D
p--0
G
l(p,k)G.
is constant on each set
n’ (e(p k))
IDn’(e(0’k))I
2k_l
and if t E l(2p,k+l) then
k(t)
k(t)
(x-t)d
(x-t)dt
G
I (2p+l, k+l)
If(x-t)-f(x-t-e(l,k+l))Idt
dx
I
G
Gk+ I
-f(x-t-e(2p+l,k+l))Idt
=B+C.
According to (D4) we have
IGk+ Ilf(x-t)-f(x-t+e(k+l))Idx
B <n’
I
II
n’
f-f
dt
G
e (k+l)
II I
at
o(i) as n
/
Gk+l
Finally, if we use (D5) and apply Fubini’s Theorem we obtain
2k_l
C
<_
p=l
e(k+l)
p
G
I
dt
i,
Therefore
-i.
I (2p,k+l)
I IGk+
p--i
Also
dx
CONVERGENCE OF WALSH-FOURIER SERIES
51
2k_l
I p-I 2k+l 2-(k+l) ilf_f e (k+l)
i
p=l
<
cI
log
2kl
f-f
e (k+l)
o(i) as n
i
/
according to the assumption of the Theorem.
II
ISn(f)-f
Thus
o(i) as
n/
Before stating a corollary to Theorem i we first introduce some
additional terminology.
LI(G)
For f
Definition i.
> 0 the integral modulus of continuity
and
is defined by
sup{l
ml(; f)
2-(n+l)
lle(n) ll
si==
Ify-fl Ii; lYll
for each n
>__
<_ 6}.
0 we see immediately that the
following holds.
Corollary i.
If f e
ISn(f)-f Ii
then
Remark i.
LI(G)
and if
o(i) as n
o(llog
ml(;f)
61-1
as
/
0,
/
Corollary i can be considered as the L I analogue of the Dini-
Lipschitz test for uniform convergence of Walsh-Fourier series, see
[2, Theorem XIII].
We first show that Corollary i is weaker than Theorem i by giving an
example of a function f e
as
/
by
fk(x)
i
>_
Xk;
such that (i)
If-f e(n) II I
0 and (ii) n
For each k
LI(G)
0 and x
then
f(x)
fk
(xi) 0
e L
I)
o(i) as n
ml(6;f)
+ o(llog 61-1
/
in G define the function
and
lfk
(k+i)-3/2 fk(x)"
k=O
i
I
"
fk
Next let
C.W. ONNEWEER
52
Clearly, the sum defining f(x) converges for each x e G and applying
the Monotone Convergence Theorem to its partial sums we see that
I(G).
f e L
Fix r
>_
Then for all k
0.
>__
if k
k(X),
0 and x e G we have
+ r,
fk(x-e(r))
-fk(x),
if k
r
Thus,
[
f(x-e(r))
k--O
(k+l)-3/2fk(x) + (r+l)-3/2(l-2f r (x)).
Hence,
f(x)
and since
f(x-e(r))
2fr(X)
1
(r+l)
_r(X),
-3/2
(2f r (x)-l)
we obtain
II f-f e(r) II i (r+I)-3/21 l*rl 11
Next, let d(r)
k--r
e(k).
o(r
Then for each k
I
k(x)’
I
fk(x),
fk(X_d(r))
-i
>_
0 and x e G we have
if k <
if k
>_
Thus
f(x)
(k+l)-3/2(2fk(X)-l)
[
f(x-d(r))
rl
r
k--r
[ (k+i)-3/2 k(X)"
k--r
From a well-known inequality for Rademacher functions, see [4, Chapter V,
Theorem (8.4) ], we obtain
llf-fd(r) ll I >_AI(
for some positive constants A
I
[
(k+i)-3)i/2>_
k;r
and A
2.
A2r-i
Therefore, since
lld(r) ll
2
-(r+l)
CONVERGENCE OF WALSH-FOURIER SERIES
i 2-r ;f)
we see that
+ o(r-l),
+ o(IXog
ml(6;f)
that is,
53
as 6+0.
We shall now show that Corollary I is the best possible in the
following sense.
Theorem 2.
There exists a function f in L
(i)
converge in L
I(G)
Proof.
61-1
0(flog
l(;f)
properties:
I(G)
with the following two
and (ii) {S (f)} does not
n
In [2, p. 386] it was shown that if
n
2
n
I
+
n
2n2 +
+ 2
with n
I
> n
2
> ...> n
then
-n
-i
llDnll
[
n
P(
p=l
22s + 22(s-l)
Thus, if n
2
[
2
r).
r=p+l
+
22 + 20
for some s > 0 then
s
p-i
p=l
r=0
+
[ 2- 2p [
--(s+l)-
2
mr)
S
>(s+)
[ >
p=l _f
2
Also, it follows immediately from (D2) that for each k > 0 we have
llD2klll
Next
i.
if k is odd and a
k
for each n > 0, let
I if k is even.
n
[
n-i
k= 0
n
D
2n+l
(x) and
2
k
with a
Furthermore, for each n
let
P (x)
ak
Qn(X)
D
2n+ n
(x)
k
>_ 0,
0
54
C.W. ONNEWEER
{Ienlll
Then
polynomial
1 and if n is even then
Qn
"part" of
is
{]QnlII
the polynomial P
>
n/4, and the (Walsh)
In order to simplify
n
the notation we shall from here on write k’ for 2
k
and k" for (k’)’.
Consider the function
f(x)
k=l
2-k (k+l)
(x) Pk, (x)
Clearly, the series defining f(x) converges for all x
0 and f
L
I(G).
For k > i we have
Qk’l IS(k+l),,+
(f)-S
(k+l)’
Thus, the sequence
{Sn (f)
with 0 < g < 1 and let
Then
lYll
all k
_<
<_
does not converge in L
<__
(x)
X(s+2)
.
implies that
n < 2
k:l
2-(1+1)<
Ps
(x)
hence
n(X+y)
is of degree
_> ’.
< 2
$
-1.
The last condition
that (s+2)’ > k-l, so that 2
Xn(X)
-s
Consequently, we have
G
2-k IX (k+l)" (x-y) D (k+l)" (x-y)-x (k+l)" (x) D (k+l)" (x)
G
+
2
-k
k=s+l
+
k:s+l
G
2-k
(k+l),, (x-y)D (k+l),, (x_y)
il(k+l) ’’(x)D(k+l)
G
0(.
-1
for all x in G and all n such
Ilf(x-Y)-f(x)
lfY-fl Ii
<
Next, take any
G. Choose the natural number s so that for
X(k+l),,(X)Pk, (x) is of degree < ’, whereas
implies that 2(s+2)" > 1’
that 0
I(G).
i
implies y e
$
the polynomial
G
l2-kx (k+l)" Qk’ II i >-- 2- k/4
ll i
be the natural number for which
s the polynomial
Also y
(k+l)" (f)
’’(x)
).
.
CONVERGENCE OF WALSH-FOURIER SERIES
0
+ 2
2
-k
0(2 -s)
55
0(-i).
k--s+l
Therefore, if
that is,
lYll
<_
6 then
I-i).
0(flog
l(;f)
lfy-f Ii
0(flog
2
-(+1)
This completes the proof of Theorem 2.
As was observed in the abstract, except for some minor
Remark 2.
modifications the theorems presented thus far also hold for functions
defined on the circle group T and their (trigonometric) Fourier series.
In this context we have
Theorem i’
LI(T)
If f e
ISn(f)-f ll
o(1) as n
and if log n
/
.
If-f /n II i
o(I) as n
/
then
Theorem l’ can be proved by modifying the proof of a test for uniform
[i, Chapter 4, 5]. Also,
convergence of Fourier series due to Salem, see
in order to see more clearly the similarity between Theorem i and Theorem i’
If-f e(n) II i
le(n) ll)-ll If-f e(n) II 1
we mention that the condition n
is equivalent to
Theorem 2’.
as
/
log(
There exists an f e L
0 and (ii)
{Sn (f)
I(T)
o(i) as n +
o(I) as n
such that (i)
does not converge in
LI(T).
In order to prove Theorem 2’ we use a result of F.
Riesz, who showed
that for each n > I there exists a trigonometric polynomial P
2n such that
"part"
of
en
llenlll
i, and a polynomial
and such that
....llQnll I
in Theorem I
Qn
n
of degree
of degree n, which is
> C log n, see
[i, Chapter VIII, 22]
or [3].
REFERENCES
i.
Bary N. K., A Treatise on Trigonometric Series, Vol. II, MacMillan,
New York, 1964.
2.
Fine, N. J., On the Walsh functions, Trans. Amer. Math. Soc. 65(1949),
372-414
56
C.W. ONNEER
3.
Zygmund, A., A Remark on Conjugate Series, Proc. London Math. Soc.,
34(1932)
4.
392-400.
Zygmund, A., Trigonometric Series.
Vol. I, 2nd. ed. Cambridge
University Press, New York, 1959.
KEY WORDS AND PHRASES.
Walsh-Fourier Series, Convergence in norm,
Dini-Lipschltz-type test.
AMS(MOS)
SUBJECT CLASSIFICATION
(1970). 42A20, 42A56, 43A75.