I nternat. J. Math. & Math. Sci.
625
Vol. 5 No. 4 (I982) 625-673
RECENT DEVELOPMENTS IN THE THEORY OF WALSH SERIES
WILLIAM R. WADE
Department of Mathematics
University of Tennessee
Knoxville, Tennessee 37916 U.S.A.
(Received May 20, 1982)
ABSTRACT.
We survey research done on the theory of Walsh series during the decade
1971-1981.
Particular attention is given to convergence of Walsh-Fourier series,
gap Walsh series, growth of Walsh-Fourier coefficients, dyadic differentiation, and
uniqueness of Walsh series.
KEY WORDS AND PHRASES. Walsh-Paley system, dyadic group, maximal functns,
dyadic H p space, BMO, Vilenkin gups, weak type (p,p)
dyadic
differentiation.
980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 42C10, 43A75.
I.
INTRODUCTION.
This article surveys recent results on Walsh series.
material appearing in
Balaov and Rubintein [1970], a
trate on the decade 1971-1981.
To avoid duplication of
decision was made to concen-
References to earlier work will be made when neces-
sary to relate what is herein reported to that which preceeded it.
Discussion of
the relationships between this material and the general theory of orthogonal series
has been left to those more qualified for this task (e.g., Ul’janov [1972], Olevskii
[1975] and
Bokarev [1978b])o
In addition to this introductory section, there remain five sections:
II.
Walsh-Fourier Series, III.
Approximation by Walsh Series, IV.
Coefficients, V. Dyadic Differentiation, and VI.
Uniqueness.
Walsh-Fourier
These sections have
been further divided into consecutively numbered subsections, each dealing with a
626
W.R. WADE
particular facet of the subject and each carrying a descriptive title to help the
reader quickly find those parts which interest him most.
Section Vl is followed by a nearly complete listing of all articles on Walsh
series published during the decade 1971-1981.
ly and then chronologically.
This listing is ordered alphabetical-
In the course of our narrative these articles will be
cited, as above, by author and by year.
Let r O, rl,.., represent the Rademacher functions, i.e.,
rk(x) sgn(sin(2k+lx)),
k >_0, x E [0, I].
Let w O, w I,.., represent the Walsh functions, i.e., w 0
is a positive integer with n > n 2 >...> n > 0 then
and if k
2nl
+...+ 2ny
r n (x)...r n (x).
y
The idea of using products of Rademacher functions to define the Walsh system origi-
Wk(X)
nated with Paley [1932].
Paley
svstem.
Thus the system {w k} as defined above is called the Walsh-
It is interesting to note that this process of putting factors togeth-
er to make an orthonormal system only produces "Walsh-like" functions.
Waterman 969], [1982] proved that if
[0,I] which satisfies
lnl
<
defines an orthogonal system with
lkl
is a system of real functions on
0’ @I
a.e., n >0, if
Ck
@nl"’n
a.e., and if
large, then there exists a measure preserving map
Moreover, the system
Ik 2=
a.e. for k
the map
can be chosen to be a metric automorphism of [0,I].
{k
By a Walsh series we shall mean a series of type W
are real numbers.
for k
2 nl +...+
2ny
M for k sufficiently
of [0,I] onto itself such that
Wk k
1,2,.,.
Indeed,
is complete if and only if
k=O akw
th
The n
partial sums of W will be denoted by
k, where a O,
al,...
n-I
Wn k=OZ akw k ,n>_l
By a Walsh-Fourier series we shall mean a Walsh series of the form W[f]
where
f,
ao(f), al(f
.e.,
Z(R) ak(f)w k
k:O
represent the Walsh-Fourier coefficients of some integrable
62 7
THEORY OF WALSH S ERIES
I
ak(f)
Let L
p,
_< p
<
,
0
f(t)wk(t)dt
k
0
represent the collection of measurable functions whose
power is integrable over the interval [0,I].
pth
Let L represent the collection of
measurable functions whose essential supremum is finite on the interval [0,I], and
represent the collection of functions continuous on the interval [0,I].
let
well known that the Walsh functions form a complete orthonormal system in L
It is
2.
It is clear that the Walsh functions alternate between +I and -I on the interval
[0,I] and it is not difficult to see that the
kernel D
.=owk are always non-negative.
the very essence of the Walsh system.
orthonormal system whose functions
range {+I, -I}, and satisfy
fn(O)
fn
2nth
partial sums of the Dirichlet
These properties penetrate deeply into
Indeed, Levizov [1980] has shown that any
have exactly n sign changes on [0,I], have
O, for n
O, I,
is the Walsh system.
Price [1961] proved that among the orthonormal systems whose functions
signs on finer and finer partitions of [0,I], as n
one whose Dirichlet kernel has non-negative
2nth
/
,
fn
And,
alternate
the Walsh system is the only
partial sums.
Thus these two fea-
tures distinguish the study of Walsh series from that of other orthonormal systems.
Another distinguishing feature is that the Walsh functions can be identified
with the characters of a certain compact group, 2
r
_,
m.
is the cartesian product of countably many copies of the discrete group {0,I}
endowed with the product topology.
sequence (x I,
x2,...)
with each
xj
(x I,
identifies the dyadic group
with
Thus a typical element of the dyadic group is a
0 or I, j >_ I.
x2...)
the interval [0,I] much like the map e ix
represent the characters of the dyadic group, then
,
WkO
(
2
m.
The map
Zj:l xj2 -j
tifies the circle group T with the interval [0,2x].
each
This group, called the dyadic
is continuous on the group and satisfies
It turns out that if
k Wk
for k
/
x iden-
IO’ I
Thus
0,I,
WkO()WkO() WkO( + y-
for
This allows one to use theorems about Fourier analysis on the dyadic
group to solve problems about Walsh-Fourier analysis on the interval [O,l].
It also
allows one to pull the dyadic group structure back to the interval [O,l], defining a
628
W.R. WADE
[0,I] by
dyadic sum of two numbers x, y.
(x I,
where
x2,...)
k:l k Yk
come from the binary expansion of x and
(YI’ Y2
and
12-k’
Ix
(+)
x + y
y and the finite expansion is used when x or y is a dyadic rational.
property of
WkO
means that
In particular, one can almost view
and do most Walsh analysis there.
are w O, w
because
},
is not quite I-I.
holds for x, yE [0,I], k >0.
Wk(X)Wk(Y)
([0,I], ) as a
+ y)
Wk(X
-l(x)
Indeed
The character
compact group whose characters
The word "almost" is necessary
has two distinct images for each dyadic
rational x; one corresponds to the finite expansion and one corresponds to the infinite expansion, e.g.,
(0, I,
(I, O, 0
I/2.
Thus [0,I] is not a
(This
compact group under + unless each dyadic rational is split into two points.
point of view was introduced by Sneider [1940].)
Let f be defined on [0,I].
We shall represent the dyadic moduli of continuity
by
(a,
f)
If(x
sup
O<h<
0, I]
"
f(x)
h)
and
;p(a,
<
p
<
,
a
>
O.
O<h<a
If(x + h)
f(x)
f()IPdx} I/p,
0
The function f belongs to Lip(s, L p) for some 0 < s
exists a constant C such that
if
; h)-
If(x
f): sup
<
p(a,
s
f) <_ Ca for a
>
O.
<
Similarly f belongs to Lips
Ch s for h E [0,I].
A most useful distinguishing feature of the Walsh system is that the
sums of any Walsh series form a martingale.
gales contain information about
f E L
I,
if there
2nth
partial sums of Walsh series.
x]l
suplW2n[f,
n>O
x E [0, I]
its square function is
S(f)
partial
In particular, theorems about martin-
its (martingal )- maximal function is
f*(x)
2nth
(W2k[f ] W2k_ l[f])2
Thus, given any
THEORY OF WALSH SERIES
and it is well known (see Garsia [1973]) that
lls(f)
II L
<_
51{ f*ll l"
L
II f*IIL
II
<_
S(f)
Thus f belongs to
H
d.vadic H
--IIs(f)ll
ll
and
L
The function f is said to belong to dyadic H
l[fil
is finite.
629
if
L
if and only if f* is integrable.
Analogous to the classical case, the dual dyadic H
is the space of functions of
This is the space of functions f E L
dyadic bounded mean oscillation, dyadic BMO.
for which the norm
II flIBM0
is finite.
sup m(I)
I
,f(x)-
f
]dx"
is a dyadic interval
Included in this duality is the "HIder" inequality of Fefferman"
I
0
f(x)@(x)dxl <- /--l[ fll
BMO
H
This inequality only holds if the integral above is interpreted as
Iof(
n-lim IoW2n[f, x]W2n[@,
X) @( x> dx
Details and references can be found in Garsia [1973].
dyadic H
x]dx.
As in the classical case,
can be characterized by atomic decompositions (see Chao [1982b]) and dyadic
BMO enjoys the Carleson decomposition (see Chao [1982a]).
Both of these decomposi-
tions provide easy proofs of the duality between dyadic H
and dyadic. BMO.
The space dyadic H
nearly equivalent to it.
is clear that
H
is not only analogous to the classical Hardy space H
l,
but
Indeed, by the atomic theory of Coifman and Weiss [1977] it
is a
(proper) subset of classical H
I.
On the other hand, a
recent result of Davis [1980] proves that given f in classical H l, its translates
belong to dyadic H
for a.e.x.
Vilenkin [1947] showed that Walsh series and the group 2 m are a special case of
a broad theory of Fourier analysis on zero-dimensional groups.
Indeed, he proved
that if G is any compact, abelian, zero-dimensional group which satisfies the second
axiom of countability, then there exist primes
be identified with a sequence
(x I, x 2
PI’ P2
such that each
of integers where 0 <_
xj
....
group G can be identified with [0 I]
Indeed
for each integer k
>
<
(
G can
pj. Thus
set m k
the
IIj=ipj.k
630
W.R. WADE
:
Given
(x I, x 2
Z:ixj/mj.
let x
Then x E [0, I] and except for a counta-
ble set, the map / x is l-l and takes G onto [O,l].
ural ordering for the characters
of G.
0’ l
Vilenkin also introduced a nat-
In the case when all pj
group G is precisely the dyadic group and the characters
for k
WkO,
0’ l
2, the
are precisely
O, l,
We shall denote the Haar measure of a Vilenkin group G by m.
character series
k=Oakk by S and
Fourier series on G by S[f].
We shall denote
The context and the
square brackets will keep this from being confused with the square function S(f) defined above. We shall denote the L p spaces on G with respect to m by LP(G). The
moduli of continuity on G will be denoted by
k(f)
and
mP)(f)
If(x+h)-f(x)
sup
h EG k
sup
x EG
{I If(x+h)-f(x)IPdm(x)}I/P’
hEG k G
0 for j < k}, k
where G k =_ { E G" xj
l, 2,
Using the identificaton of G
with [O,l] we can also define G-moduli of continuity for functions f with domain
[O,l].
These will be denoted as above by re(a, f) and
rap(a,
f). A Vilenkin group G
is said to be of bounded type if limsup
<
Although the proofs usually require
n- Pn "
greater sophistication and often necessitate the introduction of new techniques, the
theory for Vilenkin groups of bounded type parallels that for Walsh series.
not the case for Vilenkin groups of unbounded type.
This is
In fact very little is known
about these groups.
An important example of Vilenkin groups is provided by the ring of integers of
any local field K.
field, the ring
is the maximal compact subring in K, and when K is the 2-series
is precisely the group 2
m.
This point of view provides a vehicle
for discussing dyadic analogues of Riesz transforms, Bessel potentials, Mellin and
Hankel transforms, and certain singular integral operators.
An elegant, clearly writ-
ten introduction to harmonic analysis on local fields is provided by Taibleson [1975].
For H p spaces in this setting see Chao [1975] [1982b] and Chao, Gilbert and Tomas
[1981].
For an analogue of the F and M Riesz theorem, see Chao and Taibleson [1979].
During the course of our narrative we shall have occasion to cite certain re-
631
THEORY OF WALSH SERIES
sults about Fourier analysis on Vilenkin groups or local fields.
In some cases the
cited result will generalize a Walsh series result discussed in that section.
other cases, the cited works will give new information about Walsh series.
In
In such
cases, we leave it to the reader to extract this new information.
Finally, we do not systematically discuss the many applications of Walsh series
to physical problems.
ences.
We suggest Maqusi [1981] and Harmuth [1972] for general refer-
Splettstsser [1978], [1980],
Concerning sampling theorems see Butzer and
and
Splettst6sser [1979].
II.
WALSH- FOURIER SERIES
I.
Pointwise convergence.
Billard [1967] showed that the Walsh-Fourier series of
L 2 functions converge a.e.
His proof was a dyadic group adaptation of the Carleson-
Hunt technique which reduces the problem of a.e. convergence to showing that the maximal function operator f-Mf
suplW n [f]l
is of type (2
2)
n>O
constant A such that
IIMfll 2 <All fll 2
holds for all f E
sition of this proof has been provided by Hunt [1970].
e
that there exists a
L2[O,I].
An excellent expo-
A different approach to a.e.
convergence of Walsh-Fourier series has been given by Gosselin [1979].
He avoids
the tedious estimates necessary to show that Mf is of type (2,2) by using Fefferman’s
L 2 method of breaking the partial sum operator into simpler pieces.
The best positive result of a.e. convergence of Walsh-Fourier series (and Fourier series as
well) belongs to Sjlin [1969]. He derived certain weak type inequal-
Llog+Llog+log+L,
ities involving Mf to show that W[f] converges a.e. when f E
i.e.,
when
If(x) o g+ If(x) flog+log+If(x)Jdx
0
It follows that the Walsh-Fourier series of any function in
verges a.e.
<
(R).
L(Log+L) l+E,
E
>
O, con-
It is an open question whether W[f] is a.e. convergent for all f E
Llog+Lo
In the negative direction, Ladhawala and Pankratz [1976] showed that there exists
an f in dyadic H whose Walsh-Fourier series diverges a.e.
function belongs to dyadic_H
if and only if it belongs to
Since a non-negative
Llog+L,
a modification of
their example could provide a negative answer to the question cited in the previous
paragraph.
For the larger spaces
Walsh-Fourier series can diverge
L(log+log+L)l-E,
evervwhere. It
E
>
O, Moon [1975] has shown that
is not yet known whether Walsh-Four-
631
THEORY OF WALSH SERIES
sults about Fourier analysis on Vilenkin groups or local fields.
In some cases the
cited result will generalize a Walsh series result discussed in that section.
other cases, the cited works will give new information about Walsh series.
In
In such
cases, we leave it to the reader to extract this new information.
Finally, we do not systematically discuss the many applications of Walsh series
to physical problems.
ences.
We suggest Maqusi [1981] and Harmuth [1972] for general refer-
Splettstsser [1978], [1980],
Concerning sampling theorems see Butzer and
and
Splettst6sser [1979].
II.
WALSH- FOURIER SERIES
I.
Pointwise convergence.
Billard [1967] showed that the Walsh-Fourier series of
L 2 functions converge a.e.
His proof was a dyadic group adaptation of the Carleson-
Hunt technique which reduces the problem of a.e. convergence to showing that the maximal function operator f-Mf
suplW n [f]l
is of type (2
2)
n>O
constant A such that
IIMfll 2 <All fll 2
holds for all f E
sition of this proof has been provided by Hunt [1970].
e
that there exists a
L2[O,I].
An excellent expo-
A different approach to a.e.
convergence of Walsh-Fourier series has been given by Gosselin [1979].
He avoids
the tedious estimates necessary to show that Mf is of type (2,2) by using Fefferman’s
L 2 method of breaking the partial sum operator into simpler pieces.
The best positive result of a.e. convergence of Walsh-Fourier series (and Fourier series as
well) belongs to Sjlin [1969]. He derived certain weak type inequal-
Llog+Llog+log+L,
ities involving Mf to show that W[f] converges a.e. when f E
i.e.,
when
If(x) o g+ If(x) flog+log+If(x)Jdx
0
It follows that the Walsh-Fourier series of any function in
verges a.e.
<
(R).
L(Log+L) l+E,
E
>
O, con-
It is an open question whether W[f] is a.e. convergent for all f E
Llog+Lo
In the negative direction, Ladhawala and Pankratz [1976] showed that there exists
an f in dyadic H whose Walsh-Fourier series diverges a.e.
function belongs to dyadic_H
if and only if it belongs to
Since a non-negative
Llog+L,
a modification of
their example could provide a negative answer to the question cited in the previous
paragraph.
For the larger spaces
Walsh-Fourier series can diverge
L(log+log+L)l-E,
evervwhere. It
E
>
O, Moon [1975] has shown that
is not yet known whether Walsh-Four-
THEORY OF WALSH SERIES
[1975] showed that if
condition, p
>
633
is a non-negative function on G which satisfies the
m
Ap
I, i.e., if
IE
(m(E)
(m--
dm)
IE
-I
holds for each coset E of the subgroups G
n
there exists a constant C <
such that
P
dm) p-I <B<
(p-l)
{x E G" x.
I IMflPdm -< CPl
G
0 for j
j
n}, n
<
>
O, then
fldm
G
It follows that S[f] converges a.e. when the right
holds for all measurable f on G.
hand side of this inequality is finite.
For the case when G is the dyadic group, Gundy and Wheeden [1974] proved that
< p <
under the A condition,
the weighted L p norms of f and of its square funcP
,
,
S(f) are equivalent. Thus they extended the pioneering work of Hirschman [1955]
tion
Ixl
who considered weights of the form re(x)
I.
also considered the limiting case when p
the A
for -l
<
<
p-l.
They showed that if
Gundy and Wheeden
m
>_0 satisfies
condition and if
m m (E)
then given
and B
>
>
!
JE
mdm
there exists a number y such that
am (f* > B, S(f)
<
y)
<
m (f*
>
),
>
O,
and such that a similar inequality holds with the roles of f* and S(f) reversed.
Concerning rearrangements of multiple Walsh-Fourier series, Kemhadze [1975]
proved that given f E
LP(G),
p
>
l, the multiple Walsh-Fourier series of f can be re-
arranged so that its spherical partial sums converge to f a.e.
2.
Sets of divergence.
Lukaenko
[1978] studied proper sets of divergence, i.e.,
sets of the form
E
for some integrable f.
.
such that (3) holds.
when f E
(3)
{x" W[f,x] diverges}
He proved that given any
However, there are
a
sets and
set E there is a function f E L
sets for which (3) never holds
Moreover, in [1980] he announced that there exists a continuous f such
that the set E defined by (3) is of Haar measure zero, is a
a
nor
an.
a
o
set but is neither
set.
A set E __c [0,I] is said to be a set of divergence for a class of integrable
W.R. WADE
634
functions if there exists an f
E such
that W[f] diverges on E.
Sj’lin’s result (see l), all sets of divergence for L p p
According to
are of measure zero
>
This result is best possible since any set of measure zero is a set of divergence
for L
p,
<
p
<
.
Indeed, for Vilenkin groups of bounded type, Heladze [1978] has
shown that every set E
<
p
<
.
_c G
of Haar measure zero is a set of divergence for
This problem is still open for p
and for
(G). An
LP(G),
n-dimensional
version of Heladze’s theorem was obtained by Sanadze and Heladze [1977].
.
For the trigonometric case, Kahane and Katznelson proved that any set of measure
zero is a set of divergence for
Us-
This question remains open for Walsh series.
ing the Banach-Steinhaus theorem, it is not difficult to prove that any countable
set is a set of divergence for
(2 m) were
(2m).
identified by Simon [1973].
The first uncountable sets of divergence for
His work is valid on any Vilenkin group G,
and his sets of divergence have uncountable intersection with each subgroup G n.
On-
neweer [1979a] proved that every set E of logarithmic Hausdorff measure zero is a set
of
(bounded) divergence for (2m). Harris and the author [1978] have constructed
certain perfect (uncountable) sets of divergence for
(2 m) but
it is not yet known
whether every perfect set of measure zero is a set of divergence for
(2m).
First, we point out
2
that although the Walsh-Fourier series of any L function converges a.e., it is not
3.
Pointwise converqence of rearranged Walsh-Fourier series.
necessarily a.e. convergent when rearranged.
For example, Heladze [1978] proved that
for every Vilenkin group G of bounded type there exists an f E
L2(G)
such that some
rearrangement of S[f] diverges unboundedly everywhere on G.
It is natural to ask whether a hypothesis stronger than "f E L 2" will guarantee
a,e. convergence of rearrangements of W[f].
For example, does there exist a monotone
increasing sequence {(k)} of positive numbers such that the condition
implies that a given Walsh series
ZakWk
.a k 2 (k)
is a.eo unconditionally convergent?
<
The
k s for > l, because
answer is yes if m(.k) increases fast enough, e.g., re(k)
2
<_ (..k-)(Zka k ). Tandori [1966] was first to obtain negative answers to
(ZIakl)2
this question"
no if re(k)
o(_loglogk). Nakata [1972], [1974], [1979] and Bokarev
In particular, Nakata
[1978a] have obtained increasingly stronger negative answers.
[1979] proved that given re(k) which satisfies
Z(I/(km(k)))
<
,
there exists a se-
THEORY OF WALSH SERIES
quence of real numbers
.akw k has
.ak2 re(k)
{a k} such that
an a.e. divergent rearrangement.
6 35
and such that the Walsh series
<
Thus it is difficult to find conditions
which guarantee that all rearrangements of a given Walsh series converge a.e.
However, certain rearrangements are more important than others. When dealing
with Walsh series, the Kaczmarcz rearrangement is an important rearrangement, for
historic reasons and for applications (see Balasov and Rubintein [1970], and Harmuth
[1972]).
It belongs to the general class of dyadic block rearrangements, i.e., re-
arrangements
o’ l
which satisfy
{k" 2n <- k
2 n+l}
<
{w
k. 2n<_ k
<
2 n+l}
O, l,...
for n
n-1
The Dirichlet kernel D
n
Wk in the Walsh-Kaczmarcz system behaves worse
than the Oirichlet kernel in the Walsh (-Paley) system in the sense that
Zk=O
limsup
n
Dn(X)
> C > O.
log n
Thus it is harder to prove a Walsh-Kaczmarcz-Fourier series converges, but easier
to find divergent Walsh-Kaczmarcz-Fourier series.
Balasov [1971] has shown that if a(n)+O, as n--, there exists an f E L
such that
IWn[f,
limsup
n-o
where
f.
Wn[f]
represents the
x]
+ a.e.,
oo(n) ogn
tnh partial
sum of the Walsh-Kaczmarcz-Fourier series of
In particular W[f] can diverge a.e. when f E
L(log+L)l-E
for E
On the other
O.
>
hand, Young [1974b] proved that if f E
L(log+L) 2
yet improved either of these results.
In particular, the problem of a.e. convergence
then W[f] converges a.e.
of Walsh-Kaczmarcz-Fourier series remains open for f E
L(log+L) p,
<
p
<
No one has
2.
Young [1974a] introduced a large class of dyadic block rearrangements and showed
that W[f] remains a.e. convergent under these rearrangements for f E L
2.
She also
proved that a smaller class of dyadic block rearrangements preserves a.e. convergence
of W[f] when f E
L(log+L)21og+log+L.
This program was carried out for Vilenkin
groups of bounded type by Gosselin and Young [1975].
The techniques are Carleson-
Hunt and involve proving certain maximal inequalities of Hardy-Littlewood type for
these rearrangements.
6B6
W.R. WADE
An easier program for studying a.e. convergence of rearranged Walsh-Fourier
series has been initiated by Schipp [1975].
j=O nj 2j
If n
integers written in binary notation, denote the integer
He calls a rearrangement
{k
and
Z=O Inj
mjl2
=0 mj
j
by n
2j
are
m.
of the Walsh functions linear if there exists a permu-
tation T of integers such that
T(n
T(n)
m)
T(m) and
k
for k >_0.
WTk
He
shows directly that the partial sums of Walsh-Fourier series in linear rearrange-
ments are closely related to partial sums in the usual (Paley) ordering.
can apply
Sjlin’s estimates
Thus he
without repeating the lengthy Carleson-Hunt argument.
In particular, he shows that linear rearrangements of W[f] converge a.e. when f E L p,
p
>
I.
The Kaczmarcz rearrangement is only piecewise linear.
Thus he modified his
procedure to show that piecewise linear rearrangements of W[f] converge a.e. when
f E L
2.
Bahsecjan[1979] has generalized this result from piecewise linear rear-
rangements to piecewise isomorphic rearrangements.
Skvorcov [1981b] proved a theorem about Walsh-Kaczmarcz-Fourier series which
contains the following corollaries.
If f is continuous on the group and if (a, f)
o(I/log(I/a)) as aO then W[f] is uniformly convergent.
test works for Walsh-Kaczmarcz series).
(Thus the Dini-Lipschitz
If f is continuous on the group and of
bounded variation over [O,l], then W[f] is uniformly convergent.
He also shows that
localization does not hold for convergence or (C,l) summability of Walsh-Kaczmarcz
Indeed, he constructed an f E L which equals 0 on [0, I/2] but whose
series.
Walsh-Kaczmarcz-Fourier series is not (C, l) summable at the point x
4.
Convergence in norm.
Wn[f]
f
Paley [1932] proved that if f E L
in L p norm, as n-=.
groups of bounded type.
p,
<
p
O.
<
,
then
Watari [1958] obtained this same result for Vilenkin
But it is now known to hold for any Vilenkin group. Indeed,
Young [1976a], Simon [1976], and Schipp [1976d] all showed that if G is any Vilenkin
group and f E
LP(G)
for p >
then
Sn[f]+f
in L p norm, as n-=.
Of the three meth-
ods, Schipp’s is most general and applies to certain arrangements of any product
system, but Young’s is simplest.
These results extend neither to p
that
Wn[f]/f
in L
nor to p
I.
Thus in order to conclude
norm one needs an assumption in addition to f E L
[1978a] obtained an L
analogue of the Dini-Lipschitz test,
Onneweer
for the Walsh-Paley
637
THEORY OF WALSH SERIES
ordering and Skvorcov [1981c] did the same for the Kaczmarcz ordering.
they showed if
I(,
f)
o(I/log(I/)), as 6-0, then
This test fails when "o" is replaced by "0".
in L
Wn[f]+f
Specifically,
norm, as n-=.
A Vilenkin group version of this test
was obtained by Simon [1979].
Simon [1978a] has introduced an analogue M of the Hardy-Littlewood maximal
function (see Stein and Weiss [1971]) on Vilenkin groups G, showing that M is of type
(p, p) for
<
ISn[f]ll p
tha/
p
<
and of weak type (I, I).
_ap II fll p
<
An application of his inequalities is
LP(G)
holds for f
<
p
n
<
>
where A
0
P
is an ab-
He also introduced an analogue T of the conju-
solute constant depending only on p.
gate function [1976] and has shown [1978b] that there is an absolute constant C such
that
l{x
holds for all f E
LI(G)
G" Mf(x) <y and
and all
,
y
>
ITf(x) l> Y}I
<
Ce-CY
O.
Results concerning uniform convergence of Walsh-Fourier series can be found in
the following two sections.
5.
Moduli o___f continuity and absolute convergence of Walsh-Fourier series.
A classical result of Fine [1949] is that if f E Lips,
absolutely.
>
I/2, then W[f] converges
In this section we report several extensions and refinements of this
result.
Yoneda [1973] weakened the condition "f E Lip,
>
I/2" to convergence of the
series
Z 2(I/n,
-,
(4)
f)/V-6
n:l
and obtained the Walsh analogue of a theorem of Zygmund"
if f is continuous and of
bounded variation and if
Z
V’(I/n, ’f)/ n
(5)
<
n--I
then W[f] is absolutely convergent.
He also introduced a summability method A and
localized Fine’s result in the following manner.
where
>
If f E Lips on some interval I,
I/2, then W[f] is absolutely summable A on I.
In connection with (5), Bokarev [1978b] has shown that if
continuity which satisfies
m is
any modulus of
W.R. WADE
638
n=l
then there exists an absolutely continuous f with (, f)
that
T. lak(f)
0(()), as
O, such
+.
Onneweer [1972] generalized Fine’s result to Vilenkin groups G of bounded type
by examining the role oscillation plays in determining absolute convergence of S[f].
Before stating his results we need some additional terminology.
G and let H be a subset of G.
The oscillation of f on H is defined by
osc(f, H)
>
O, let
Zq,
k (0
f(x2) l.
f(x
sup
x
For each integer k
Let f be defined on
x2E H
q < m k)
represent points of G for which
Zq,
k
+ G k exhaust the cosets of G in G. The function f is said to be of -generalized
k
bounded fluctuation if the terms of the series
mk-l
Zk=l q=OZ Iosc
Onneweer proved that if f E
are uniformly bounded in k.
converges then S[f] converges absolutely.
convergent when f E Lipe
GR)IP) I/p
(f z q,k +
>
(6)
LP(G),
p
2 and if (6)
He also showed that S[f] is absolutely
Onneweer [1974] proved that if f belongs to Lip (e, p) * Lip (B, q) for
2, q
l, 0
>
<
,
B
and (e + B) p
then S[f] converges absolutely.
bounded type" is redundant.
<
s
<
and
+ I/s
>
on some Vilenkin group G of bounded type,
Their method rests on a powerful factorization of Lip(,
that Lip (, r) Lip ( + I/s
r
>
p
Quek and Yap [1981] showed that the hypothesis "of
r) as Lip (e-B, r) * Lip ((B-l)/q
<
.
O, and is of p-generalized bounded fluctuation.
p) where I/p + I/q
and q
>
I/6.
I/r, s) holds on any Vilenkin group when 0
It follows
< e <
,
I/r.
Vilenkin and Rubinstein [1975] established Vilenkin group versions of results
Specifically, if G is of bounded
cited in the second paragraph of this section.
type then S[f] is absolutely convergent when either
k(2)(f) ,
L2(G)
<
f E
k=l
or when f is continuous and of bounded variation on G and
lmk+
RZ: v k
Cf)
<
(7)
(8)
THEORY OF WALSH S ERIES
holds.
In particular, if f E Lips,
then S[f] converges absolutely on G.
>
6 39
0 and if f is of bounded variation on G,
(The reason that (4), (7) and (5), (8) are not
exact analogues is that on the dyadic group,
corresponds to m(2 -k, f) not
k(f)
re(I/k, f)). McLaughlin [1973] has obtained exact Vilenkin group analogues of (4) and
(5) (see 12.) All these results hold for Vilenkin groups of unbounded type as
well (see Quek and Yap [1980], [1981]).
Combining results of Ladhawala [1976] and Butzer and Wagner [1973] one can show
LP),
that W[f] is absolutely convergent when f E Lip(s,
question ether this is true for
Rubintein
Finally,
6.
l, p
[1978] has shown that given any sequence
mn(P)(fp =mn for n >_0 and p
I.
It is an open
mn+ O, as n ,
fl E LI(G), f2 E L2(G),
l, 2, or
Uniform convergence of Walsh-Fourier series.
ogue of a theorem of Salem.
>
I.
and any Vilenkin group G there exist functions
such that
>
.
/
and
f(R)F (G)
Onneweer [1970] obtained an anal-
If f is continuous and periodic of period l, and if the
sequence
2_nl
Z
+
2k+l
(9)
k=l
converges to zero, as n
/-,
then W[f] converges uniformly on [O,l].
sult for Walsh-Kaczmarcz series is due to Skvorcov [1981b].
A similar re-
A version of this re-
sult also holds for Vilenkin groups G of bounded type (Onneweer and Waterman [1971]).
Thus, if
k(_f)
o(.I/k), as k
/
, and if f E (G)
then S[f] is uniformly convergent
on G (Vilenkin [1947]).
For Vilenkin groups of bounded type, Onneweer and Waterman [1971] proved that
if f is continuous and of l-generalized bounded fluctuation (defined in 5 above)
then S[f] is uniformly convergent,
tions in [1974].
Let A
{n
They strengthened this result in several direc-
In order to state their results we need additional terminology.
be a sequence of positive numbers which satisfy
=I I
+. A func-
tion f defined on G is said to be of A-bounded fluctuation if there exists an M
such that for every collection {I
n
of disjoint cosets in G it is the case that
.l osc(_f
n=1
In
< M,
<
W.R. WADE
640
The function f is said to be of harmonic bounded fluctuation if it is of {n}
ed fluctuation.
bound-
Onneweer and Waterman showed that there exist functions f of A-boundHow-
ed fluctuation whose Vilenkin-Fourier series S[f] diverge at at least one point.
ever, if f is of harmonic bounded fluctuation then S[f] converges at every point of
continuity and converges uniformly on any closed set of continuity for f.
Their
method centered on finding an analogue of Lebesgue’s test on G and showing that this
test is satisfied at points of continuity for functions of harmonic bounded fluctuatio
Their work also contains a very interesting condition sufficient to conclude that
Given an open set V c G,
a continuous function is of harmonic bounded fluctuation.
let N(V) represent the number (possibly infinite) of disjoint open "intervals" sep-
If f E ’(G) with range [0,I] and if
V.
arated by the elements of G
I
log N{x E G" f(x)
>
y}dy
<
o
Thus such functions have uniformly con-
then f is of harmonic bounded fluctuation.
This is a group analogue of the Garsia-Sawyer test
vergent Vilenkin-Fourier series.
(see Onneweer [1971b]).
Concerning the size of
IWn[f]
fl,
Tevzadze [1978] announced certain e3Limates
in terms of the modulus of continuity and variation of f which contain a group 2
version of the Garsia-Sawyer test. Specifically, he states that if u(n represets
the modulus of variation of f then there is an absolute constant C (independent of i)
such that
II f Wn[f]llL
is dominated by
n
C min
<m<n
{(I/n, f)
.v_(k)
+
k=m+l
k=l
(I0)
k2
It follows that if f belongs to the class H (’iV[u] for some modulus of continuity
and some modulus of variation u, then W[f] is uniformly convergent on [0.:
only if the sequence (I0) converges to zero, as n
.
if and
Hence for a given continuous
f E V[u], a necessary and sufficient condition that W[f] be uniformly convergent is
that
Zk =I (k)/k2
<
.
Gulicev [1980] (announced in [1979b]) has shown that the Walsh analogues of
theorems of Oskolkov and Busko fail to hold.
Specifically, if {L k} represent the
Walsh-Lebesgue constants (See Fine [1949]) then
641’
THEORY OF WALSH SERIES
liminf
k-x
I
f
I
W k[f]
0 for f E
J(I/k, f)L k
and
liminf
k-o
llWk[f]ll
0 for f E L
Lk
,
(ll)
.
(12)
However, if the limit infima in (II) and (12) are taken over certain subsequences of
integers then =0 is replaced by >0.
Thus for certain subsequences, analogy with the
trigonometric system is restored.
7.
Although there exist integrable functions
Summability of Walsh-Fourier series.
whose Walsh-Fourier series diverge everywhere, certain averages of these WalshFourier series still approximate the given function.
For example, Fine [1955] showed that if f ( L then W[f] is a.e. (C, )
O.
summable to f for all
[1976c].
A new proof of this result has been given by Schipp
He takes the Carleson-Hunt point of view, and shows that the maximal func-
) and of
tion associated with a large class of summability methods is of type (,
weak type (I, I).
type by
PI
This same program was carried out for Vilenkin groups of bounded
and Simon [1977a]. In particular, if
represents the n th partial
n(f)
Cesaro sum of S[f] and if *(f)
m{x"
Thus f E
LI(G) implies that
suplon(f)
n>O
l*(f, x)
>
then
y}
<
Cllflll/y,
*(f) is weakly integrable.
what conditions is *(f) actually integrable?
y
>
O.
It is natural to ask under
Nobuhiko [1979] used atomic
HI(G)
to prove that
l*(f)
Hence o*(f) E L
nk
I(G)
when f
HI(
dm
<
Cll f I
H I"
Baiarstanova [1979] has considered (C,I) summability of W [f] where
nk
2 k + 2 k-2 + 2 k-4 +...+ I. She showed that if /o(, f) (respectively,
l(a,
f))
O(I//log(I/)), as -), then W n [f] is uniformly (C,I) summable (respectively,
k
(C,I) summable in L norm). This result contrasts nicely with Schipp’s result
is
(see (I) and (2) in I).
Other methods of averaging can also be used to show that W[f] approximates f
in some sense.
proved that if f
One such example is the method of strong summability.
L
then
Schipp [1969a]
642
W.R. WADE
1.r
rn-1
mk !0 IW k
,
converges to zero a.e., as m
fl p} lip
[f]-
for any choice 0
<
(13)
p
<
.
An n-dimensional
version of this result was announced by Sarasenidze [1976] for functions in
L(log+L) n-l.
Apparently, the n-dimensional version of (13) fails to converge a.e.
(log+log+L)n-I
to zero for certain f E L
2.
for n
Several authors have considered weaker versions of (1.3).
Cybertowicz [1976] showed that if r
k
n
<
and
an_l,
k
0 for k
>
>
0 and if
For example,
(n-2+r-kr_l / (nl+r)r
an_l, k
for
I, then
n
n-I
-.
{k=O an’k Iwk [f] f12}I/2
a.e.
as n
for all f
L
Sarasenidze [1973] proved that given any positive,
regular method of summability
f there exist numbers r
n
/
[an,
.
Yano [1951b]
L p norms.
k=O
invest!gated
He proved that if
for 0
<
B
>
<
.
.
k+l
<an,
k and given any continuous
r
a n ,k
INk[f]- fl n} I/r n- 0
the growth of (C, B) sums of Walsh-Fourier series in
<
p <_,’0
I o n(f)
provided B
an,
]
k with
such that
n-I
uniformly, as n
0
<
and f E Lip(s, L p) then
<
O(n -), as n
fIIkp
Remarkably, Skvorcov [1981a] showed that this estimate holds even
Moreover, he obtained an order estimate for the limiting case
if f e Lip(l, L p) then
I o n(f)
In fact, he proved much more.
that if n
>
O, B
.
>
.
fllk
O(logn/n), as n
p
He showed that there is an absolute constant C such
0 and f ( L p then
I On(f)
where m is defined by 2 m
<
n
<
fllLp
2m+l
and
<
C
2-m
m
Z
k=O
an(f)
2kp( 2 -k
f),
represents the partial (C,B) sum of
W[f] in any piecewise linear rearrangement (see 3).
Thus the order estimates above
hold for Walsh-Kaczmarcz series as well as Walsh (-Paley) series.
f E L
p,
I"
In particular if
p <-, then the (C,B) sum of any piecewise linear rearrangement of W[f]
converges to f in L p norm. Skvorcov [1982] has generalized these results to Vilenkin
<
THEORY OF WALSH SERIES
643
groups of bounded type.
For results concerning (C,B) sums of Walsh-Fourier series when B
<
O, see
Tevzadze [1981].
Bicadze [1979] has obtained necessary and sufficient conditions for a multiple
Walsh series with monotone coefficients in the sense of Hardy to be (C, I) summable
to f in L p norm, 0
<
p
I.
<
Finally, Tateoka [1978] has studied localization on the unit square.
Recall
that if X is a collection of integrable functions then a summability method T has
the localization
propertv for X if given any f E X which vanishes on an open set V,
the double Walsh-Fourier series of f is uniformly summable (T) to zero on compact
subsets of V.
property for
,
Tateoka shows that square partial sums do not have the localization
p
that square Abel means have the localization property for L
but do not have it for L
p,
>
2
2, and that rectangular Abel means have the
but not for L p p >
<
localization property for
8.
p
p
<
Adjustment on sets of small measure to enhance converqence.
A celebrated result
of Menshov for trigonometric series is that given an a.e. finite-valued, measurable
f and an E
>
0 there exists a continuous function f which coincides with f off a
set of measure less than E such that the Fourier series of f.converges uniformly on
[0,2x].
Kotljar [1966] showed that this result is also true for Walsh-Fourierseries.
Price [1969] gave a new Walsh proof of Menshov’s theorem, utilizing the fact
that the Walsh functions are characters of the dyadic group.
This proof was adapted
to Vilenkin groups of bounded type by Onneweer [1971c].
Concerning whether adjustment can be made to improve the decay of Walsh-Fourier
coefficients,
f E L
Olevski"
[1978] showed that there is an f E
such that given any
which coincides with f on a set of positive measure, it is the case that
for all p < 2.
la n (f) p
n:O
Z
In particular, a continuous function cannot be adjusted so that its Walsh-Fourier
series is absolutely convergent.
(Guliev
[1979a] has also given a proof of this
corollary to Olevskii’s theorem.)
Adjustments can be made which result in a.e. and L
convergence.
Indeed,
Heladze [1977] has given an outline of a proof which establishes that given f E L
644
W.R. WADE
and given E
>
s(x) #
0 there is a measurable s with m{x:
Ll
converges a.e. and in
I} < E such that W[sf]
norm.
Kemhadze [1977] obtained an adjustment theorem for functions f continuous on the
He proved that given (
n-dimensional hypercube.
>
0 there is a g which coincides
with f off a set of measure less than ( such that the rectangular partial sum
multiple Walsh-Fourier series of f converges everywhere.
of the
He also shows that g can
be chosen so that its Walsh-Fourier series has certain prescribed gaps.
APPROXIMATION BY WALSH SERIES
]]I.
9.
Walsh series with gaps.
The study of gap Walsh series predates the study of
Walsh series since the Rademacher functions were introduced before the Walsh functions.
Coury [1974b] investigated differentiability of Rademacher series.
He showed
that a Rademacher series is either a.e. differentiable or almost nowhere differentiable.
Using an earlier result of
Balaov
[1965], who made several foundational
contributions to this problem, Coury proved that if a Rademacher series is differentiable at one point then it is of bounded
variation for all p
>
A deeper
I.
result he obtained is that any a.e. differentiable Rademacher series is of bounded
variation.
(Coury [1974a] has identified several conditions onthe coefficients of a
given Walsh series W sufficient to conclude that W is a.e. differentiable.)
For results about dyadic differentiability of Rademacher series see 15.
Zotikov [1976] has studied the problem of convergence of Rademacher series in
Vilenkin groups G.
He shows that the Rademacher system
is independent and
{Rn}n= 0
that if G is of bounded type then
Z anR n
(14)
n=O
is not a.e. summable by any T* method
(see Bary [1964]) when
a n2
When G is of
+
bounded type, he shows that (14) converges absolutely if and only if
n=O V’pn
in which case the limit F of (14) is bounded, is continuous on
.
{pn}-adic
irrationals,
and has Vilenkin-Fourier series (14).
A Walsh series
k=l
a
nkwnk
is called lacunar, if
nk+I/nk
>q
>
for k
I, 2
645
THEORY OF WALSH SERIES
Coury [1973] proved
Clearly every Rademacher series is a lacunary Walsh series.
that if E is a set of second category which has the property of Baire and if W is a
lacunary Walsh series which converges to zero (or is constant) on E, then W is actu-
A related result announced by Ebralidze [1976] is the fol-
ally a Walsh polynomial.
k=l, 2
Let {Rm k (m 1, 2
for k > and
as m
satisfies Rm k
be a sequence of real numbers which
lowing.
/
max
IRm,kl
2n<k<2n+
for m
>
I, n
>
.
<
min
c
O, where c is a fixed, finite number
um or the limit infimum, as m
Rm ,k
2n_l<k<2 n
If either the limit suprem-
I.
>
of the sums
Rm,
k akr k(x)
k=l
is finite for all x in a set E of the second category, then
Rm, k
when E is an interval and
.lakl
<
.
In the case
l, this result had been obtained for any lacunary
Walsh series by Morgenthaler [1957].
Concerning whether the sequence of coefficients of a Rademacher series belong
< p < 2, Rodin and Semjonov C1975] have examined analogues of
to any I p space,
Khinchin’s inequality for certain symmetric Banach spaces including those of Lorentz,
Marcinkiewicz, and Orlicz.
f
akr k
A corollary of their work is that a Rademacher series
has coefficients {a k}
I
EIP
for some
<
p
<
ly/(l + log2(I/xf(y)))l p-I dy
2 if and only if
<
.
0
Kolmogorov first noticed that convergence of a Rademacher series on a set of
positive measure is sufficient to conclude that its coefficients belong to Z
Morgenthaler [1957] showed this result also holds for lacunary Walsh series.
2.
Gapokin
[1971] weakened the lacunary condition considerably and Miheev [1979] pushed this
result even further.
He proved that if
n
/ as k, Z
measure, if n k
constant C such that
k=l
holds for I
>
ZakwnkZ,
with k
/
nk
>
and if for some p
k=IbkWnk
p
and any choice of real
is T* summable on a set of positive
bk’S,
then
2 <.
Zak
k=l
>
2 there is a
646
W.R. WADE
Morganthaler [1957] proved a central limit theorem for lacunary Walsh series.
The idea was that a lacunary Walsh series is close to being a Rademacheg series and
thus might behave like independent identically distributed random variables.
This
approach was developed further in a series of papers in which central limit theorems,
laws of iterated logarithm, and functional laws of the iterated logarithm were
obtained
[1979].)
(see FIdes [1972], [1975], Berkes [1974], Takahashi [1975], and Ohashi
akwnk
nk+l/n
For example, a Walsh series
is called weakly lacunary if there exist
+ ck -c for k
>
Suppose for
1, 2
k
the remainder of this paragraph that the coefficients of a weakly lacunary Walsh
c
>
0 and 0_
a <
1/2 such that
series satisfy
N
AN
Takahashi [1975] showed if a
a k 2 I/2 /+, as N
k=l
n
/.
I+E
O(AN/[N(Iog AN)
,
T]), as N
for some (
>
O,
then
N
limsup
N-
FIdes
(2A loglogA) -I/2 k=l akwnk
a.e.;
, then
N
y -t2/2
I,
e
<_--------I
dt,
y}
Z
Am k=l akwnk(x)
Z2
O(AN/Nm),
[1975] showed if a
N
as N
I
lim m{x"
N-=
Ohashi [1979] obtained a local version of
c
>
0 and 0
such that
a
N
O(AN/NC),
+ ck -m for k
>
as N
/
,,
>
.
(15)
He also showed that the
n 2 <... and coefficients a I, a 2
I, such that A N
as N +-, and such that
I/2 there exist integers n
nk+I/n k
<
Specifically, he proved that given any
condition is best possible.
<
y
(15) and used it to prove that there exist
weakly lacunary Walsh-Fourier series which diverge a.e.
FBldes growth
<
<
,
but such that (15) fails to hold.
As noted above, under certain conditions (e,g., convergence to zero on an inter-
val) the only lacunary Walsh series re Walsh polynomials. Such a condition was
obtained by Roider [1969] for gap Walsn-Fourier series.
positive integer with n
V(k)
y.
>
n2
> ...>
n
>
0
If k
2nl
+...+ 2 nY is a
the Vielfalt of k is defined to be
Thus a rather mild lacunary condition is given by
ak(f)
for some fixed integer v.
0 for V(k)
(16)
> v
Roider proved that if f E L
assumes oly finitely many,
or only integral, values, and if (16) is satisfied for some
.,
>_0, then f is actually
647
THEORY OF WALSH SERIES
a Walsh polynomial.
For related results see Gruber [1977/78].
For a local, condition sufficient to conclude that a gap series is a Walsh polyFor results concerning summability of gap series
nomial see [1982] by the author.
see ll.
lO.
The Walsh system as a basis.
Recall that a system
=lanfn which converges
some Banach space B if given f E B there is a unique series
in B norm to f.
The Walsh system is a basis for L
examined whether a subsystem
p,
is a basis in
{fl’ f2
<
p
.
<
Kazarjan [1978]
of the Walsh system can be multiplicatively
{wnk}k=
completed, i.e., whether there exists a measurable @ such that {@(x
basis for the spaces L
p,
<
p
<
-. He proved
)wnk (X)}k=
is a
that such a @ never exists no matter
how few and far between the gaps are.
Two systems
{fl’ f2 ’’’’}
and
{gl’ g2 ’’’’}
are e.quivalent bases in a Banach
space B if given coefficients a l, a 2
the series
n and Zangn are equiconvergent in B. Ciesielski and Kwapi6n [1979] proved that the Walsh system and the bounded
Zanf
polygonals are equivalent bases in L
p,
<
p
<
-. On
the other hand, Young [1976b]
showed that the trigonometric and Walsh systems are not equivalent bases in L
p,
p # 2.
Hence they will have different multipliers, and multipliers for the Walsh system
should be investigated.
Shirey [1973] examined the Walsh system as a quasi-basis, and proved that given
any set E of positive measure, the quasi-basis for
LP(E),
<
p
<
-,
obtained by
restricting the Walsh system to E, is a conditional quasi-basis.
II. Approximation by Walsh series. Skvorcov [1973b] has shown that unlike the
trigonometric case or the one-dimensional Walsh case, convergence of a double Walsh
series W on a set of positive measure is not sufficient to conclude that the coefficients of W converge to zero.
However, he proved that if the retangular partial
sums of a double Walsh series W converge on any dyadic irrational across (i.e., on a
set of the form {a} x (0, l) LI(O, l) x {b}, where a and b are dyadic irrationals)
.
/
then the coefficients of W satisfy a
n,m + O, as m + n
One of Menshov’s celebrated results is that given any measurable, a.e. finite-
valued function q there is a trigonometric series which converges to
techniques apply equally well to Walsh series.
a.e.
His
Thus Walsh series can be used to
648
W.R. WADE
approximate measurable functions which are finite a.e.
Talaljan [1960] has shown that given any L p basis, p
I, and any measurable
>
@ (finite or not) there is a series with respect to that basis which converges in
measure to @.
Thus to every measurable @ there corresponds at least one Walsh
series which converges to @ in measure.
is replaced by "a.e.".
This result does not hold if "in measure"
In fact, Talaljan and Arutunjan [1965] proved that there is
no Walsh series W which satisfies
+ on a set of positive measure, as m
W2m
.
/
Thus Walsh series cannot be used for a.e. approximation of general measurable
func-
ti ons.
It is natural to ask whether rearranged Walsh series or (C, I) partial sums of
Walsh series can be used for a.e. approximation of measurable functions.
The answer
to the second question is yes, and there is strong evidence that the first question
can also be answered in the affirmative.
Saginjan [1979] showed that given any
positive regular method T’ of summability there exist Walsh series W
with {a k} E Z 2+E for
E
>
0 and
Zk=l
n
exists a subseries of W which is a.e. T’ summable to @.
there
Thus given a measurable @,
there exist Walsh series which are a.e. (C, ) summable,
mable to @.
Zk=lakwnk
such that given any measurable
<
>
O, and a.e. Abel sum-
And, Ovsepjan [1973] proved that there exists a Walsh series which
has a rearrangement that diverges a.e. to +.
This question of divergence has been addressed for double Walsh series.
[1969] proved that if W is a double Walsh series then
on a set of positive measure, as n, m
/
.
Kemhadze
2 m cannot diverge to +
Thus even rectangular sums of double
W2n,
Walsh series cannot be used to a.e. approximate measurable functions unless the
functions are finite-valued a,e.
Pogosjan [1980] announced results concerning a.e. summability to + of the series
T
where
O, ’_
Yk
.k=O YkWk//kl
and w k is a dyadic block rearrangement on the Walsh function, i.e.
n
n+l
n
{w
<
{w
k" 2 <_ k < 2
k 2 <_ k
A consequence of his results is that if n < n 2 <
satisfies
2n+l
for n
imsup
k-
0,
nk+i/n k
then one can choose numbers
Yk
such that
Tnk
+
+ a e as k
/-
649
THEORY OF JALSH
On the other hand, if the ratios
are unifomly bounded, then
nk+i/n k
Tnk cannot
diverge to + on a set of positive measure.
Saginjan has obtained several conditions sufficient to conclude that
f(x)
(x) + liminf S (x)]
(17)
knk
nk
holds for a.e. x in some given measurable set E. For example, in [1974] he showed
that if S
Z akw k
I/2 [limsup S
is some dyadic block rearrangement of a Walsh series, if
converges to a finite measurable f on E, and if
(17) holds for a.e. x E E.
series with k
k2
<
liminfk_Snk(x
In [1981] he proved that if S
O(I/j) as j
kj/kj+
<
j
l2kj+
is a gap Walsh
and
kjl O(kn),
kj+ 2
for x E E then
>
ajwkj
n-2
j=l
S2n(x)
,
as n
if S is (C, l) summable a.e. on E to a measurable, a.e. finite-valued function f,
and if
liminfk_Snk(x)
E, then (17) holds a.e. on E.
for x
>
n-j m for fixed n, m
result includes gaps of the form k.
j
aginjan
ing the (C, l) analogue of (17),
rearrangement of a Walsh series, if
measurable f on E and if
f(x)
holds for a.e. x
E.
"Tauberian" theorem.
I/2[limsup
k-=
l, 2
l, j
>
Concern-
[1974] proved that if S is a dyadic block
o2n(x,
liminfk_Oonk(x
Notice that this
,
S) converges, as n
S)
onk(x
for x
>
S) + liminf
k-
to a finite-valued,
E, then
nk
(x, S)]
Contained in his proof is verification of the following
If
S)
limsuPn_ol2n(X
holds for all x E E, then
<
S2n(X)
converges a.e. on E, and
liminf
n-
holds for a.e. x
o2n(x,
S) <_ lim
n-
S2n(X
<
limsup
n
O2n(X,
E.
,.
Concerning structure of sets on which Walsh series diverge,
showed that given any
I f-
.-
(
akWkll p
L
p,
[1978]
Wn(X)
does not exist}.
n-x
He also proved that this result does not necessarily hold if
"- -set."
Lukaenko
set E there is a Walsh series W such that
E
For every f
S)
_< p
{x" lim
<
,
let
E p)(f)
"’-set"
is replaced by
denote the infimum of the expression
as {a k} take on all real values, for n
l, 2,... Golubov [1972]
650
N_:R, W_AI)E
has obtained many sharp estimates relating the "non-dyadic" moduli of continuity
m-(a, f)
I/p,i’’l
f.x.,Pdx}
]f(x + h)
sup
0
He proved that
O<h<a
to the sequences E p
)(f).
f) <_96 n -I/p
-p(I/n,
a <_ I,
EP)(f)
k I/p-I
k=l
holds for all f E L p and all n >_ I.
0 <_
He also obtained the following analogues of
trigonometric inequalities due to Ul’janov.
<_ p
If
<
q
<
there exists a constant
C depending only on p and q such that
q-2
!! f llq
<_. c{ I f
lip
Z kp
k=l
+ [
[EP)(f) ]q]I/q}
and
E
q)(f)
<_ C{E
P)(f)n I/p-I/q
Z kq/p-2[E p)(f)]q]I/q}
+ [
k:n+l
hold for n >I,. and f E L p.
(See Golubov [1970], p. 719, for ramifications).
Siddiqi [1971] proved that given a quasi-convex sequence
W
ZakWk
converges in L norm if and only if
IV.
WALSH-FOURIER COEFFICIENTS
12.
Growth of Walsh-Fourier coefficients.
aklogk
/
O, as k
{ak},
/
the Walsh series
(R).
Several authors have investigated
conditions under which the series
B
kY
(18)
k:l
converges for various choices of y and 1. A unified treatment of this problem (not
lakCf)
only in the Walsh case, but for Vilenkin groups of bounded type as well) has been
given by McLaughlin [1973].
His article contains an extensive bibliography and is
nearly a compendium of what is known about this problem.
most general of its kind, is that if
<_p
<
2, if 0
<
His main result, still the
B <_q and if I/p + I/q
I,
then (18) converges when
Z
k=l
kY-B/qlo--’n(-l/k, f)I
<
Ladhawala [1976] showed that (18) converges for
to dyadic H
-I and B
His proof contains the following interesting fact.
when f belongs
If {a k} is a se-
quence of real numbers which satisfies a k O(I/k), as k /(R), then a
k ak() for
some @ of bounded mean oscillation. Chao [1981] has obtained these results for
Vilenkin groups of bounded type.
In addition, Quek and Yap [1980] have shown that
THEORY OF WALSH SERIES
(18) converges for
651
0 when f E Lip (, p) on any Vilenkin group of bounded type
y
provided B > p/(p + p
l).
Moricz [1981] has studied integrability of a Walsh series with monotone coef-
ficients a >_a >_...>_a >_0. He proved that for r > l, B
k
0
series (18) converges if and only if
n
(sup
l) rdx <
r
k=O
n>_O
0
I1
In the case r=l, it turns out that
finite.,
r to be
Vilenkin and
y
Rubintein
Z akWk(X)
ak/k
<
r, and
.
is both necessary and sufficient for
[1975] have several estimates of the tails of
0 in the Vilenkin grouIsetting.
2, the
r
They proved that if f E
Z laj(f) 2}I/2 <_ l/---m2) (f),
L2(G)
(18) when
then
I, 2,
k
j =m
k
They obtained analogues of two theorems of Lorenz for Vilenkin groups of bounded
type"
Pk <- C
if
for k
I, 2,..., if f E Lip(G),
>
I/p
I/2 for some 0
p <2,
<
then
Z laj(f)I p}I/p <_ Cmkl/P--I/2.
j =m
.J=mklaj(f)
And, if
Horoko
<_
Cm
k
for some
>
0 then f E Lip(G).
[1972] considered the problem of determining exact estimates for the
growth of Walsh-Fourier coefficients of functions belonging to H v, the class of f
whose variation does not exceed the constant V > O. He proved that the maximum of
sup
fEH V
for 2 m
<
k
<
2m+
is V/
Rubintein [1980]
to p
2m+l
faR(f)
and that the minimum of (19) for
nmn(1)(f)
G
fdm
2m
< k <
2m+l
obtained an extension of Parseval’s identity from
for Vilenkin groups of bounded type.
then
(19)
k=Oak(f)ak(g ).
is V/2 m+2
<
p <
He announced that if
ran(g)/ O,
as n
+(R),
He indicated that this identity does not necessarily
hold if (20) is relaxed to
nmn(-l)(f)mn(g)
13.
(20)
> 0
liminf
n
Conditions on Walsh-Fourier coefficients sufficient to conclude that f is
W.R. WADE
652
constant.
Fine [1949] was among the first to notice that the Walsh-Fourier coeffi-
He showed that if f is absolute-
cients of a smooth function cannot decay too rapidly.
ly continuous and if
Bokarev
kak(f)
/
O, as k
then f is constant on [0,I].
[1970] sought to determine how rapidly Walsh-Fourier coefficients of
lak(f)
<d k, where d k + 0 and
and if f is continuous, then f is constant on [0,I]. Hence the Walsh-
continuous functions can decay.
Zd k
,
/
<
He proved that if
Fourier coefficients of a non-constant continuous f cannot satisfy
O(I/[k(logk)]),
lak(f)
for some
>
I. The condition
as k
cannot be relaxed.
>
/
,
Indeed,
Bokarev
constructed
a non-constant, continuous f whose Walsh-Fourier coefficients satisfy
lak(f)
One drawback to
(see Bokarev [1978b], p. 19).
O(I/[klogk]), as k
Bokarev’s
result is that it only applies to functions whose
Coury [1974a] worked to remove this
Walsh-Fourier series are absolutely convergent.
restriction.
He proved that if f is continuous and if
2P
where R
m
Z{lak(f) ak+l(f) I"
O, as p
R
m:p m
2m< k < 2m+l-l}, then f is constant.
(21)
It follows
that no non-constant, continuous f whose Walsh-Fourier coefficients are monotone
decreasing can satisfy
2na2n(f)
/
this result by weakening condition
O, as n
/
.
The author [1983] has generalized
(21).
Coury has proved (but not yet published) that there is no non-constant,
continuous f which satisfies
Z klak(f)
<
k=l
[1981]
generalized this result"
Powell and the author
Z=okak(f)wk
converges on (0,I) for some
>
if f is continuous, if
I, with the further assumption that
(j+l)2n-I
Z
exists when
j2na k
lim
n j=l k=j2 n
I, then f is constant.
(f)wk(x)
x E (0
I)
The author [1979b] has identified several conditions sufficient to conclude
that a function is constant on [0,I].
The most interesting one is that there is no
non-constant continuously differentiable function which satisfies
653
THEORY OF WALSH SERIES
k+l
sup n 2 -I
n>O k=O j=
E [O,l].
Z
for all dyadic rationals
V.
14.
p
.2k aj(f)wj(p)
<
DYADIC DIFFERENTIATION
The stronq dyadi c derivative.
Butzer and Wagner [1973] introduced the dyadic
derivative, which interacts with the Walsh-Paley functions in much the same way as
the classical derivative interacts with the exponential functions
{einX}.
Chief
among these interactions is the fact that Walsh functions are the eigenfunctions of
the induced differential operator, and
Dw k
(22)
O, l,
kw k, k
Butzer and Wagner [1972] also defined a derivative which yields (22) for the Walsh
functions in the Kaczmarcz ordering.
The Walsh-Paley definition is described below.
< p <
A given
Let X represent either the space or one of the spaces L p
.
f E X is said to be strongly dyadically differentiable in X (more briefly, X differ-
entiable) if the sequence of functions
n-l
Z 2J-l[f(x)
x)
dn(f,
2-J’l)]
fCx
j=O
converges in the norm of X, in which case the limit of
dyadic derivative of f and denoted by Df.
The dy_adi
F
c
(23)
(23) is called the strong
antiderivative
is defined by
Z k-lwk(t)
(24)
]dt
f(.x t) [I +
k:l
0
Butzer and Wagner [1973] proved that D is a closed linear operator, that if f E X
If (x)
then If is X differentiable with
D(If)
And I(Df)
(25)
f a.e,
f a.e. when f is X differentiable.
Thus the fundamental theorem of
dyadic calculus is true.
They also obtained the following characterization of strong
dyadic differentiation.
For any f E X the following three conditions are equivalentf is X differentiable and g
Df;
(27)
there is a g E X such that
ka
k(f)
a k(g) for k
(26)
l, 2,...;
and
there is a g E X such that f
Ig +
ao(f ).
(28)
W.R. WADE
654
This same characterization was extended to include X
dyadic H by Ladhawala [1976].
Butzer and Wagner [1975] also obtained a closed form for the "dyadic integral" operator (24).
They proved that if f E L
then
ao(f)
Z
ak(f)/k a.e.
k=l
Most results about the dyadic derivative have reinforced the idea that it is the
If
+
correct derivative to use in Walsh-Fourier analysis.
For example, Butzer and Wagner
[1973] confirmed the connection between differentiability and Lipschitz spaces by
showing that any f E Lip (, X) for
>
has a strong dyadic derivative Df E X, and,
moreover, any function f with a strong dyadic derivative Df E X necessarily belongs
to Lip (B, X) for 0
<
B
I.
<
when f belongs to dyadic H
l,
Ladhawala [1976] showed that If converges absolutely
i.e., if f is strongly dyadically differentiable in
dyadic H then W[f] is absolutely convergent. In particular, if Df exists in dyadic
H (or in any L p for p > l), then f is continuous on the group.
On the other hand, Penney [1976] has shown that not all strongly dyadically
differentiable functions are continuous on the group.
In fact, Ladhawala [1976]
then the best one can say is that f E BMO; f may not
proved that if Df exists in L
even be bounded.
Several authors have considered generalizations of the dyadic derivative to
objects other than functions defined on [O,l] with dyadic structure.
fined Df on the dyadic field, i.e., for functions f E
"Walsh transform"
exists then
J (.see
F(_Df)(y)
Fine
y(y)
Ll(o,
PI[1975]
) and showed that the
[1950]) interacts with D as it should: namely,
and
D(f)(x)
de-
if Df
(xf(x)) if xf(x) E L (0, ). In
[1977]
he constructed an indefinite integral for D and proved a fundamental theorem of
calculus in this setting.
Onneweer [1977] introduced a Vilenkin group analogue of the dyadic derivative,
showing that the characters of the Vilenkin group are the eigenfunctions of the in-
duced differential operator and argued that the Butzer-Wagner characterization (26),
(27), and (28) carries over without extra work to
this setting.
Pal and Simon [1977b]
proved a fundamental theorem of calculus for Onneweer’s derivative.
We shall call
this derivative "dyadic" below.
From the beginning, it has bothered some that the definition of the dyadic
THEORY OF WALSH SERIES
655
Onneweer [1978b] endeavored
derivative depends on the ordering of the characters.
to erase this deficiency, while at the same time coming up with a derivative whose
eigenvalues represented the "frequency" of the characters as in the classical case
with d/dx and
{einX}.
His efforts were concentrated on p-adic and p-series fields K.
(Recall that the group of integers of a 2-series field is precisely the dyadic group
2).
from those of the strong
where 2n < k < 2 n+l ,n=O,l
His definition yields different eigenvalues on 2
2nwk
dyadic derivative discussed above since Dw k
Nevertheless, the characters of K are again eigenfunctions of the associated differential operator, which turns out to be a closed, linear operator in
LI(K).
Onneweer
[1979b] introduced another "dyadic" derivative and compared all four derivatives
the Butzer-Wagner derivative, the one introduced by
Pl
on the dyadic field, his ear-
lier one on K, and this latest one defined on the dyadic group and the dyadic field.
This latest derivative behaves much as the others but has the twin virtues of simplicity and the hope of lending itself more easily to an interpretation with possible
applications in the physical sciences.
15.
Thee
pointwise dyadic derivative.
A function f defined on [0,I] is said to have
{dn(f, x)} (see (23))
f(x). This definition
a dyadic derivative at a point x if the sequence of real nun)ers
converges, as n
.
In this case, we denote this limit by
was introduced by Butzer and Wagner [1975], who identified a large class of functions
on which the pointwise dyadic derivative and the strong dyadic derivative agree a.e.
Skvorcov and the author [1979] observed that anytime
derivatives of f satisfy
D+f(x)
>
0
>
D f(x),
f(x)
exists, the classical Dini
Hence it is impossible for a non-
constant, (classically) continuous function to be dyadically differentiable at all
but countably many points in (0, I).
Since
Wk(X)
kWk(X)
for all x E [0,I], this
impossibility does not extend to functions continuous on the group.
Schipp [1974] proved a fundamental theorem for the pointwise dyadic derivative.
He showed that if f E L with
ao(f)
0 then
(If)
f a.e. (compare with (25)).
PI
and Simon [1977b] obtained this same result for the dyadic derivative on Vilenkin
groups of bounded type.
Schipp [1976b] extended his [1974] theorem to handle certain
Stieltjes measures associated with functions of bounded variation.
sults the method of proof is to show that the maximal operators
In all these re-
W.R. WADE
656
are of strong type (p, p),
sup dn(If, x), x E [0, I].
n>O
< p <_, and of weak type (l, l).
Onneweer [1977] showed that on a Vilenkin group G there exist continuous f
which are nowhere dyadically differentiable.
R
Zakr k
He also proved that a Rademacher series
is dyadically differentiable at x if and only if
Z
m a r (x)
k=O k k k
converges, in which case
R
Z mkakrk(x)
(29)
k=O
It follows that a Rademacher series on G is either a.e. or almost nowhere dyadically
differentiable.
Moreover, if the dyadic derivative of a Rademacher series is constant
on some open subset of G, then that series is actually a polynomial (see also [1982]
by the author).
16.
Thus the non-local property of the dyadic derivative is extreme.
Butzer and Wagner [1975] were first to
Term by term dyadi.c differentiation.
examine conditions under which a Walsh series
ZakWk
represents a dyadically differ-
entiable function f which satisfies
f(x)
Z kakWk(X).
(30)
k=l
This problem is completely solved for Rademacher series (see (29)) but is still
open even for lacunary Walsh series.
Several authors have identified conditions sufficient for (30) to hold.
Butzer and Wagner [1975] showed that (30) holds a.e. when the sequences {a k} and
.
{ka k} are quasi-convex and ka
They also showed that (30) holds everyk 0 as k
<
where when
i.e., when the derived series is absolutely convergent.
O, as
2 -n (n > O) when k K
Schipp [1976a] proved that (30) holds for all x
Zklakl ,
k
.
This result was generalized by Skvorcov and the author [1979], who showed
that if condition (21) holds, and if x # 2 -n for n
ficient condition for
f(x)
>
O, then a necessary and suf-
to exist is that the sequence
2n_l
Z
k=O
ka
kwk(x
converges in which case the limit of (31), as n
if a k 0, if
2ka2 k
/
O, as k
/
,
31
,
equals
f(x).
It follows that
and if the derived series converges then (30) holds
THEORY OF WALSH SERIES
for x # 2 -n (n
>
0). Also, if ka k
0 and if
2k
k
as k
/
,
j=k
f(x)
then
aj+ll
aj
exists if and only if
O,
(31) converges, as n
.
Powell and the author [1981] showed that (30) holds for any x which satisfies
Z kakWk (x)
<
k=l
> l, and that (30) holds for all x E [O,l] if a + O, as k
for some
and
k
<
They also found necessary and sufficient conditions for (30) to hold when
the derived series converges. In fact, if f(t) exists at t
x and t x + 2 -n-l
,
.
Zlakl
(n >_0) and if the right hand side of (30) converges, then a necessary and sufficient
condition for
f(x)
to exist is that
R(x)
im
n-
(j+l)2n-I
l
j=l
l
j
k=j2 n
2nakw k x
exist, in which case,
f(x)
Z kakWk(X)
R(x)
k=l
VI.
UNIQUENESS
17.
Uniqueness of Walsh series with monotone coefficients.
Balaov
[1971] proved
that if the coefficients of a Walsh-Kaczmarcz series S are convex and decrease
monotonically to zero, then S is the Walsh-Kaczmarcz-Fourier series of some function
in L
l.
Yano [1951a] had obtained this same result for Walsh-Paley series with "quasi-
convex" replacing "convex".
to this result.
Coury [1974a] showed that quasi-convexity is essential
Indeed he constructed a Walsh series with monotonically decreasing
coefficients which converges to a non-integrable function and is not a Walsh-Fourier
series.
In the trigonometric case, a series with convex coefficients which decrease
monotonically to zero converges to a non-negative function.
that this result does not hold for Walsh series.
monotone sequence {a k} for which W
Z akw k
Coury [1974a] showed
Indeed, he constructed a convex
assumes negative values.
He went on
k=O
to show that if the sequence of coefficients is completely monotone, then analogy
with the trigonometric case is restored and W > O.
Siddiqi [1971] proved that if a k + O, as k
,
and if
Z(ak ak+l )lgk
<
then
W.R. WADE
658
ZakWk
18.
is a Walsh-Fourier series.
A set E
Sets of uniqueness.
[0,I] or 2 m is said to be a set of uniqueness
__c
(or a U-set) if the zero series is the only Walsh series which converges to zero
off E.
Sneider [1949] showed that any countable subset of [0,I] is a L)-set and that
the Cantor set formed by removing middle halves is a
U-set has Lebesgue measure zero,
U-sets. Coury [1970],
LJ-set. He
also showed that any
but not all sets of Lebesgue measure zero are
[1975] found a class of sets of Lebesgue measure zero which
are dense in [0,I] but are not L3-sets.
positive, increasing h on [0,) with
Skvorcov [1977b] proved that given any
h(O)
0 there exists a perfect set E c [0,I]
L-set. Thus the problem
are L-sets is still open.
whose h-measure is zero such that E is not a
those perfect subsets of [0,I] which
Gevorkjan [1981] has shown that given any E
k
measure one such that W
akw k
with
W2n j
+
On the dyadic group, more is known.
of identifying
+ 0 there is a set E _c [0,I] of
0 off E and
akl
Ek implies W O.
The author [1979a] introduced a group
<
analogue of H-sets, and showed that every H-set in 2 m is a LO-set.
Building on this,
Yoneda [1982] proved that every closed subgroup of the dyadic group of Haar measure
zero is a
group are
L-set. It
U-sets.
follows that a large class of Cantor-like sets in the dyadic
Lippman and the author [1980] carried the Pyatetskii-Shapiro structure theorem
over to the group 2
m.
-set in the dyadic group is a
An elementary L-set E is one for which there
They showed that any closed
countable union of elementary L)-sets.
exists a sequence of functions
whose Walsh-Fourier series are absolutely
fl’ f2
convergent and vanish on E, which converges to
in the weak * topology (i.e.,
given any pseudo-function A, A(f
I) / O, as n / ). The author [1971] had
n
earlier shown that a countable union of closed L-sets is again a L]-set. It is not
known whether the condition "closed" can be relaxed in either of these results.
A set E c 2 m is said to be a L-set for a class
esis W E
,
with
Clearly, every
Borel set E is
W2n(x)
0 as n
L-set for
a LI-set for
lim
n-
is a
for x
L-set.
of Walsh series, if the hypoth-
E, implies that W is the zero series.
Crittenden and Shapiro [1965] proved that a
the class of Walsh series which satisfy
2-nw2n(x _+ O)
0 for all x E [0,I]
(32)
THEORY OF WALSH SERIES
if and only if E is countable.
The author [1975] introduced classes
19.
0
<
l,
(32), and showed that
which are smaller than the class of Walsh series which satisfy
a closed set E is a L]-set for
,
659
+ if and only if E is of s-capacity zero.
Uniqueness of approximatinq Walsh series.
Arutunjan and Talajan [1964] proved
that if W is a Walsh series whose coefficients a + 0 as k + ,if f is a finitek
valued integrable function, and if there exist integers n < n 2 <... such that
im
j
(x)
W2n j
f(x),
for all but countably many x E [0,I], then W is the Walsh-Fourier series of f.
Skvorcov [1968] showed that this result remains true if "integrable" is replaced by
"Perron integrable" and "Walsh-Fourier series" is replaced by "Perron-Walsh-Fourier
series".
In [1980b], however, he showed that it does not extend to functions which
A multiple Walsh-Fourier series version
are Denjoy integrable in the wide sense.
of the result of Arutunjan and Talaljan was obtained by Movsisjan [1974].
Skvorcov has also extensively examined the problem of obtaining uniqueness with
conditions weaker than
(33). On the positive side, he proved [1974], [1977a] that
if W is a Walsh series whose coefficients tend to zero, if n
of integers which satisfies-either
2j
<_nj
or
V(nj)
<A
<
<
2 j+l, j
<
n 2 <... is a sequence
(34)
O, I,...,
(see definition above display (16)), and if
limsuplWn.(X)
j
<M
<
J
for all but countably many x E [0,I], then W is the Walsh-Fourier series of the
function
f(x)
limsup
Wn(X),
as j
+
.
In [1980a] he proved that if W is a Walsh
series which satisfies (32) (a condition weaker than a k / 0), if f is a finite valued,
Perron integrable function, and if n < n 2 <... satisfies (34), then
l.im Wn(X
(35)
f(x)
for all but countably many x E [0,I] is sufficient to conclude that W is the PerronWalsh-Fourier series of f.
On the negative side, he showed [1975] that (35) is not
sufficient for uniqueness to hold if n
<
n 2 <... is arbitrary.
ed a non-zero Walsh series W and a sequence of integers
where, as j
/
.
{nj}
Indeed, he construct-
such that
Wnj
/
0 every-
W.R. WADE
660
Concerning uniqueness of non-convergent Walsh series, Crittenden and Shapiro’
[1965] showed that if W is a Walsh series which satisfies (32), if both
limsupIW2n(x)
n-
<
(36)
and
W2n(x)
> f(x)
liminf
n
x E [0,I], where f is an integrable function, then W
hold for all but countably many
is the Walsh-Fourier series of some g E L
I.
A portion of their argument which involve’.
a rather intricate application of the Baire category theorem has been shortened by
Lindahl [1971].
Ovsepjan [1973] showed that
(36) is crucial to uniqueness here.
In-
fact, he proved that given any continuous f and any permutation wO, w I,... of the
W > f, W n f a.e. as
Walsh functions, there is a Walsh series W
k such that n
n
,
akw
but W is not the Walsh-Fourier series of f.
Skvorcov [1973a] proved that if W is a Walsh series whose coefficients tend to
zero, if f is an integrable function, and if
(37)
< f(x) < limsup W2n(x)
liminf
nnfor all x E [O,I]E, where E is a closed L-set, then W is the Walsh-Fourier series
W2n(x)
of f.
The author [1977] proved that if (37) holds for x E [O,I]E, where E is a
countable set, and if (32) holds for x E E and all dyadic rational x, then W is
the Walsh-Fourier series of f.
(See [1980] by the author also).
For many years the problem of uniqueness for (C, I) summable Walsh series was
open.
Crittenden [1964] made some progress toward this problem, but never published
his results.
Skvorcov [1976b] solved the problem with an elegant proof that if W
(32) and if the Cesaro sums of W satisfy
is a Walsh series which satisfies
limsupl2n(x)
<
n-=
for all but countably many x E [0,I], and
lim
o2n(x)
f(x) a.e.,
for some Perron integrable f, then W is the Perron-Walsh-Fourier series of f.
20.
Null series.
null series.
A non-zero Walsh series which converges a.e. to zero is called a
Thus null series provide counter examples for certain conjectures
concerning uniqueness.
For example, Schipp [1969b] showed that non-negative partial
661
THEORY OF WALSH SERIES
sums is not a sufficient condition for uniqueness; he constructed a null series W
which satisfies W
n
>
0 for n
>
O.
Coury [1973] constructed a non-zero lacunary Walsh series which converges to
zero on a set of positive measure, but proved that there are no lacunary null series.
Skvorcov [1976a] proved that a null series can diverge to + on perfect sets.
Thus to insure uniqueness, conditions like (36) must hold for all but countablymany x.
Skvorcov [1977c] also examined how fast the coefficients of a null series can
By the Riesz-Fischer theorem they cannot converge to zero arbitrar-
converge to zero.
He proved that given M n
ily fast.
as n
/
,
there exists a null series W whose
coefficients satisfy
n
Z
He also obtained
21.
his
Closing comments.
a<Mn,
n >I
k=l
result for Vilenkin groups of bounded type [1979].
Up to this point we have mentioned some specific questions
which have not yet been answered.
In this final subsection we speculate about the
future in a more general way.
The initial
The study of pointwise dyadic differentiation is in its infancy.
kept track of zero crossings has not been exploited.
idea that
obvious problems, such as finding conditions under which
remain unexplored.
ator f
/
f
fn
/
Moreover, even
f implies
fn
/
f’
More information of this type could help decide whether the oper-
is really "differentiation" or merely a special multiplier.
Another frontier is provided by the spaces dyadi.c H
which need addressing include"
spectively, p
Which Walsh series results known for L
p,
Questions
p
>
(re-
) can be extended to dyadic H (respectively, d_zadic BMO)? Which
classical trigonometric results about H
if f E dyadic H
and d_adic BMO.
and BMO have Walsh analogues?
is of bounded variation, does W[f] converge absolutely?
For example,
In view of
the result of Davis [1980], cited in the introductory section above, d_dvadic H
tains information about classical H
I.
con-
Garnett and Jones [1982] have used this con-
nection to show how classical results can be obtained more easily by looking at the
dyadic case first.
Perhaps Walsh series will begin to provide theorems eventuating in
trigonometric analogues which solve long standing problems, or in trigonometric
proofs which are simpler than those now known.
Specific suggestions include
W.R. WADE
662
determining which perfect subsets of [0,i] are sets of uniqueness for Walsh series,
resolution of whether every Borel set of uniqueness for Walsh series is a set of first
p
category, and a new proof of a.e. convergence of L Walsh-Fourier series p > i.
Finally, work on multiple Walsh series has just begun.
since 2 m x 2 m is homeomorphic and isomorphic to 2
studying such Walsh series.
.
m,
Earlier it was held that
nothing new would be gained by
However, pulling back results from 2 m x 2 m to 2 via
such a homeomorphic isomorphism forces a non-standard enumeration on the characters
of 2
Thus multiple Walsh analysis provides a new way to study rearrangements of
Walsh series.
Since most rearrangements which arise in this way are rather wild (not
even dyadic block rearrangements), there is surely good reason to study Walsh series
in higher dimensions.
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