Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 223-244
223
CONCEPTS OF GENERALIZED BOUNDED VARIATION AND
THE THEORY OF FOURIER SERIES
M.
AVDiSPAHid
Univerzitet "Dzemal Bijedi"
Gradj evinski fakultet
A. Zuanida 14, 88000 Mostar
Yugoslavia
(Received April 26, 1985)
The aim of this article is to unify a large part of present knowledge about
ABSTRACT.
the behaviour of Fourier series of functions of generalized bounded variation.
The
connections between various concepts are discussed, particular attention being payed
to those due to Waterman and Chanturiya.
Exploiting the existing interactions and
utilizing the power of one or another approach to some typical questions of the Fourier
theory, a number of previously unnoticed results are obtained in the course of this
exposition.
KEY WORDS AND PHRASES:
abo
te
convergences
Fourier series, generaZized bounded variation,
Gibb
"
I0 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
uniform and
phenomenon.
42C10.
INTRODUCTION.
One of the features which distinguish Jordan’s class BV among the other standard
classes of functions in analysis is that the Fourier program can be carried out in it in
the
"most
classical
sense":
the Fourier series of any function of this class is every-
where pointwise convergent; in case of a continuous function the convergence is uniform;
continuity of a function may be characterized in terms of its Fourier coefficients, etc.
(As it is well known, the Fourier program in its "original sense" failed dramatically
already in 1876, when du Bois-Raymond constructed a continuous function whose Fourier
series diverges at a point.)
Another interesting occurrence is that already in order
to characterize continuity, Wiener came out of the class
ol variation of a higher order.
BV
and introduced the concept
Later investigations in Fourier analysis on the one
h,,d, and a mathematician’s always present desire for more elegance and/or more generality in treating a particular problem on the other hand, have lead to further interestig generalizations and new classes of functions.
Domains of validity of many
classical theorems were extended, sometimes to their "natural borders".
After the sec-
tion on notation, we survey some of these concepts and relationships between them in
sections 3
5.
In I0 sections which follow thereafter we try to picture the present
state of the Fourier theory of functions of generalized bounded variation, treating
AVIDISPAHI
M.
224
topics as: pointwise, uniform, and absolute convergence, summability, order of magnitude
of Fourier coefficients, continuity, determination of a jump, Gibb’s phenomenon, conju-
gate series, Parseval’s identities.
It is often indicated how to derive some recent results from each other and from
However what is being said in this
the older ones, usually enriching their substance.
direction is, as a rule, a sample rather than a detailed account of such possibilities.
OTATION AND CONVENTIONS.
2.
We consider real-valued
2.1.
a, b (a<b)
endpoints
M(f)
inf f,
2.2.
a
sup f
(b
an(f)
n
denotes, as usual, the n-th cosine (sine) Fourier coeffi-
bn(f))
n
f
cient of a function
(f)
and
O
(a + b2)i/2n
n
S (x)
(f)
and those
n
(6,f)
f(6)
f
n
If(x+h)
f(x)
{f e C:
6a,
(6)
If
g
C:
2.4.
o(a)}.
f(6)
(x).
m
defined on
[0,],
m(O)
0
f
C
g
Lip
is
sup
0x2
(,f)
sup
0
LP:
{f
lip (a,p)
2.5.
The norm of
(2
a ! I, Lip (a,p)
0
0}.
as
Integral moduli of continuity of
P
For
0((6))
f(6)
0 < a ! I, this class is denoted
llfll
P
if(x+h
LP:
{f
P
inf
L
f(x)
iPdx)I/p
are given by
0 <p <
O(a)}
P
o(6a)}.
(6,f)
0f2
p
f
m (6,f)
The best approximation of order
E (n,f)
if(x
n
to
T (x)
n
f
L
g
p
L
in
p
is
IPdx) I/p
where the infimum is taken over all trigonometric polynomials
n, n
2.6.
(C,a), a
[2]
p.76) and
2.7.
A
Tn
of degree not higher
1,2,
than
-i, denotes the
6a(f)
(a(f))
n
n
Cesro
summability method of order
the corresponding
Cesro
means of
a
([I], pp. 96-97;
S(f) ((f)).
is the class of functions with absolutely convergent Fourier series.
class of regulated functions
(Dieudonn [3], p. 139),
sided limits at each point.
For
W
n
subadditive, we set
H
{f
or
I.
Given an arbitrary nondecreasing continuous function
and
n(X,f)
and
Sn(f), Sn(x,f)
are denoted by
(f)
f
is
sup
h
0
is the Fourier series of
S(f)
analogously-
The modulus of continuity of
2.3.
S(f)
Partial sums of
is its conjugate series.
or simply
is an interval with the
For bounded functions, m(f)
f(a)
f(b)
f(1)
I
If
2-periodic functions.
we write
f
g
W
we
always suppose
is actually the universal class of this paper.
W
is the
i.e. unctions possessing the one-
f(x)
(f(x+O)
f(x-0))/2
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
225
Symbols for the classes of generalized bounded variation and related notions are
2.8.
to be found in Sections 3 and 5.
GENERALIZATIONS DUE TO N. WIENER, L.C. YOUNC, D. WATERMAN AND Z.A. CHANTURIYA.
3.
The classes of functions of bounded variation of higher orders were for the first
time introduced by Wiener in his paper
[4].
Various definitions of this notion appear
[5]. We choose
be of bounded p-variatlon on I0,2I, pit and
in the literature, but they all yield the same classes of functions,
A function
DEFINITION 3.1.
V
to belong to the class
V (f)
P
is said to
,
..If(li)IP}I/P
Z
sup
f
if
P
i
where the supremum is taken over all finite collections of
I
of [0,2]
V (f)
The quantity
nonoverlapping subintervals
is called the p-variation of
P
on [0,2].
f
of p-variation see also [6-12].
For a detailed study
Wiener’s concept has been genralized by Young [13] in the following way.
Let
DEFINITION 3.2.
creasing from
V(f)
0 to
{I.}i
over all systems
(u)
Clearly,
[0,2] is the supremum
on
of nonoverlapping subintervals of
u
gives the Jordan’s class
BV
It is customary to consider convex functions
O,
(x)/x
0 (x/O+)
nonnegative, continuous
sup {x:
@
f
-variation of a function
(If(l i )I)
i
P
(0)
The
and strictly in-
of the sums
Z
V
.
be a continuous function defined on [0, )
0} on.
(x)
and
(x)/x
V
(u)
up
and
={f:
V(f)<=}
gives Wiener’s
satisfying the conditions:
(x+)
Such a function
and strictly increasing from the point
We suppose that
complementary to
[0,2].
x
is nondegenerate, i.e.
in the sense of Young is defined by
@(y)
is necessarily
inf {x:
o
x
o
0.
max{xy
(x)> 0}=
The function
(x)}.
xZO
It is also convenient to suppose that
tive constants
x
and
o
(2x) ! d(x)
d (d
for
2)
0
satisfies the condition
V
There exist posi-
such that
x ! x
o
since this condition is necessary and sufficient for the space
The classes
42:
V
to be linear.
have been thoroughly studied in Musielak and Orlicz [14] and
Lesniewicz and Orlicz [15].
Another type of generalization, directly influenced by the convergence problems in
the theory of Fourier series, appeared in
DEFINITION 3.3.
Let
to infinity, such that
variation on
Z
A
{%
n
Z I/%
Waterman’s paper [16]
in 1972.
be a nondecreasing sequence of positive numbers tending
diverges.
n
[0,27] (or to belong
A function
f
is said to be of A-bounded
ABV) if
to the class
If(In)I/ n
for every choice of nonoverlapping intervals
is called the A-variation of
f
In case
I
A
[0,2].
n
{n}
The supremum of these sums
one speaks of harmonic bounded
AVDISPAHI6
M.
226
variation and the class HBV.
Am
Let
in
{An+m},
A function
0,1,2,
m
AcBV)
A-variation (to belong to
VAm(f)
if
f e ABV
O, as
is said to be continuous
m
([17]).
ABV, properties of A-variation function, A BV,
Properties of functions of the class
C
ABV
as Banach space, etc., have been investigated in
e.g., has shown in [18] that
ABV
BV
[18],[19], [20], [21].
W
is precisely the intersection and
Perlman,
the union of all
spaces and that no one of these results can be improved by taking countable inter-
sections or unions of
If
and
xy <
inequality
ABVs.
Young’s complementary functions,
are
(x) + (y)
(Young’s inequality) that
it follows directly from the
V# cHBV
I (I/n)
if
Actually, more can be said.
Z (I/n) <
THEOREM 3.1.
PROOF.
V HcBV.
implies
There exists a decreasing sequence of positive numbers
slowly that
@(__I
)n
n
Z
([22]).
lapping subintervals of [0,27].
Z
n
m+l
-<
If(In)
< I
n
n+m
nZ (If(In) l) +Zn
Let
f
V#
For every
If( I n )I
0,1,2
n
be
n
tending to 0 so
sequence of nonover-
one has
_<
(n+m)en+m
((n+m)len+m)
m
and {I
e
_<
V(f)
+ C.
0
(m
),
Hence
nZ
f
implying
-<
n+m
em+l(V(f)+C)
H BV.
C
Recently Schramm and Waterman [23] have combined concepts of
and
A
variation
into
DEFINITION 3.4.
A function f is said to be of A-bounded variation (f
for every system {I
of nonoverlapping subintervals of [0,27]
#ABV)
if
n
n
n
The supremum of these sums,
V#A(f),
is called the
#A-variation of
f
An intermediate step was Shiba’s [24] introduction of the classes
(#(u)
u
p
on [0,27].
ABV (p)
p > 1).
Finally, Chanturiya’s generalization [25], whose connections with that of Waterman
are going to play the central role in the course of our narrative, is described in
DEFINITION 3.5.
The modulus of variation of a bounded function
f
is the function
with domain the positive integers, given by
f(n)
where
Nn
{Ik:
n
p
n
k=Zl
k=l,...,n}
subintervals of [0,2hi.
is an arbitrary finite collection of
n
nonoverlapping
f
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
227
The modulus of variation of any function is nondecreasing and concave.
Vc V[n-l(I/n)]
{f:
W
and
f(n)
for which
f
the class of functions
4.
VIii
of an integral argument with such properties, then by
function
0((n)) (n
).
Given a
one denotes
We note that
o(n)} ([25]).
vf(n)
RELATIONSHIPS BETWEEN WATERMAN’S AND CHANTURIYA’S CONCEPTS.
Using a simple averaging argument the possibility of which lies in the heart of
{In
Definition 3.3 (Independent permutations of
and
{I
k}
pointed out in [26] the following inclusion relation between
are admissable!) we have
Waterman’s
and Chanturiya’s
classes.
ABV
THEOREM 4.1.
n
V
n
i11/x.
Analogously we obtain for the new classes given by Definition 3.4.
i
Vine-l(
CABV
THEOREM 4.2.
)]
n
Z 1/I.
l
i=l
Let {I
PROOF.
f
and
i}
n overlapping intervals
I
i
[0,2]
By Jensen’s inequality we have
CABV
6
be an arbitrary collection of
n
(i--E1
If(Ii) I/I
n
k=El i/k k
(If(Ii) l)/l.
n
i--El
n
Z
k=l
n
Z i/l k
k=l
i/l k
This yields
n
i--El
If(li)
I.
-< C-I(
n
)"
kZ=l
kl /k
n
1/1
k
and in the next step, proceeding as in Theorem 4.1.,
n
i If(li)
!
Cn-l(
n
k
I/I
k
The result in the opposite direction we shall need in a slightly more precise form
than it was stated in [26].
THEOREM 4.3.
If
kl A(I/Xk)f(k)
<
’
f
then
f e A BV.
c
PROOF.
n If(Ik)
kl Ik+m
CA.
Ak+m
-<
as
m
-
By Abel’s partial sunation and the facts that
one obtains
I
k
Ak-t illf(li)
(A.
)9 (k) +
f
kZl Ak+m )f(k+m)
f
+
(n)
k--Zn+m AI
k
k__Zn+m
(A
+
n-I-
<
)f(k)
A
is continuous in
im+
vf
kl
(A
variation, i.e.,
is nondecreasing and
If(Ik)
.)f(i)
<
o(I)
A
n
/
,
AVIDSPAHI
M.
228
COROLLARY I.
i)
v(k)/k
--El
k
ii)
k__El
(k)/k 2 <
i)
If
II0
COROLLARY 2.
ii)
dx
I
If
/I
i)
log--7--Cx]
dx
0
0 < a < I, then
dx <
-
l
n=l
yields
{n a} BV
V.
c
V#c HcBV
then
l-a(x
I(I)
<
Now the assertion follows
V[n-l(I/n)].
Vc
from Corollary I. i), since
Follows in a similar way from Corollary I. ii).
ii)
COROLLARY 3.
i)
C
f
If
k=El A(l/k)kmf(I/k)
and
{n
Lip a
ii)
PROOF.
,
<
V[v]c {n a} C BV;
HcBV
V[]c
implies
l-a(x
0
PROOF.
l+a <
(R), 0 < a < I, implies
C
BV
ii)
AcBV;
f
o(kf(I/k))
f(k)
Follows from Theorem 4.3 and the fact that
i)
then
0 < a <
6 > l-e,
for
,
<
([25],Th. 4).
Immediately from i).
An especially interesting case is given by
6 <
0 < a
For
THEOREM 4.4.
{na}BV
V[n
Vl/(l_a
Let
i)
f
E
{na}BV
m
d
k
kl= a
k
K
d
<
f(Ik)
k
Vrna(f)t
{I
and let
lapping subintervals of [0,2w].
numbers
{nB}c BV
In particular, Lip
All inclusions are strict.
PROOF.
one has
a
k
From
k
Hence
m
V
f
dk
k(l-a)(p-l)d
k
k
I
Therefore
a
d
m
l-a
a
m
is independent of the choice of intervals
dp
l-a
{I
d
m
it follows
< K
k}
< K p-I I
for every
adk
<
m
p =I/(l-a)
For
__k
Cml-adm
KP-IV{na
(f)
{n }BY.
m
k
and the constant
one has
what implies
Lip(l-a)
The inclusion
In the interval [0,]
Vl/(l_a
Let us show that
is obvious.
we shall take the points
x
o
O,
x
n
k=l
klog
s
I/(l-a) (k+l)
f (0)
f(Xn
f()
lim s
n/
n
The function
O,
(/s)
(/s)
l-a
l-a
n
k=El
k__Zt
(-I) k+l
(-1)
kl-alog(k+l
k+l
kl-alog(k+l
f
Lip(l-s)#
s /s,
n
n
n
m
< d
P
ii)
S
-
be an arbitrary finite collection of nonover-
We may suppose that they are denumerated so that the
descend.
for every
k}
(l-a) {na}BV
is defined by
where
229
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
I
and to be linear on each
(/s)l-c
If(Ik)
f
1,2
k
is extended to
[0,2] by
Now
0 < x _<
for
f(-x)
f(x+)
[Xk_l,Xk],
k
llkll-
kl-log (k+l)
Figure
k
(/s)
ka
I
interval
kl
f(x") ({f(x)
I
x
on
I
k
k
I
k
m)
(n
klog(k+l)
one can always find
k
Lip(l-a)
l-a
x’, x"
Really, for any two points
Lip(I-a).
f
n
[f(Ik)
n
Hence
so that
{na}BV.
f
[0,2], not contained
II k
such that
O
x"l
! x’
and
However,
in the same
f(x’),
O
See Figure I.)
(Geometrically it is obvious.
Since
O
(with the Lipschitz constant independent of
k), the assertion
follows.
iii)
V[ na]
Vl/(l_a
I/2
i-I
and
iv)
f(2)
f(O)
[1,2],
where
V[nU]{n } C BY
since the values of
5.
0
B
and
linear on the intervals
f
f(I/2 i-l)
f:
example that the inclusion is strict is provided by the function
f(3/2 i+l)
An
follows straightforwardly from the Hider inequality.
[I/2 i 3/2 i+l]
I/i
[3/2
l-a
i+l
i=1,2
follows from Theorem 4.3.
This inclusion is obviously strict,
lie in the open interval.
BANACH INDICATRIX.
In 1925 Banach [27] introduced, for a continuous function
[m(f),M(f)]
which
f(x)
[0, m]
setting
if and only if
/M(f)
BV
Nf
the Banach indicatrix of
[0,2]
m(f)
Nf(y)
and a continuous function
If
f
F
on
[29]; rediscovered in Tsereteli [30]).
theorem is also valid for
f
[31].
Nf(y)
otherwise.
Nf:
for
He proved that
W
there exist an increasing function
[(0),(2)]
Defining
Nf
such that
N
F
f
F-
on
(Sierpiski
one obtains that the Banach
since that variation of a function does not vary for mono-
tone transformations of the argument.
Lozinski
and
[0,2]
x
(see also Natanson [28], p. 253 ff., who called
N (y)dy <
f
f).
the function
equal to the number of values of
y, if this number is finite,
f
f
This generalization of Banach’s result is due to
The approach presented here is from [32].
M.
230
be an increasing concave function on [0,), (0), (x)
Now let
(x+).
(x)/x +0
AVDISPAHI
(x/),
[32] have recently proved the following
Asatiani and Chanturiya
theorem.
THEOREM 5.1.
If
W
f
and
I M(f) (N f (y))dy <
re(f)
(5 I)
f
then the modulus of continuity of
satisfies
Z [2(n) -(n-l) -(n+l)]
.
(5.2)
for which (5.2) is valid but (5.1) is not.
C
f
There exists a
<
f(n)
n=l
Theorem 5.1 and Theorem 4.3 imply
{f
THEOREM 5 2
PROOF.
/M(f) (Nf(y))dy
W:
< }
Now
I/K
(n)-(n-l)
would have
K
If there were a constant
is concave.
{
COROLLARY
If
COROLLARY 2.
(f
COROLLARY 3.
{f
p
k
W:
K
n
Hence
IM(f) NfI/p (y)dy
}
<
n
N
follows from the fact that
n
for every
[(k)-(k-l)] n/K
(n)/n /0 (n/).
trary to the assumption that
>
n+l
(n)-(n-l)
n
so defined, possesses all the
n
such that
n
(n)
and
where
c
It suffices to check that the sequence
properties required by Definition 3.3.
then we
n
for every
n, con-
n
n
Finally
{n l-I/p
m(f)
c
k I/k=(n)/
BV.
(Zerekidze [33]; Garsia and Sawyer [34], p.591)
I M(f)
m(f)
W:
{f
W:
NfI/p(y)dy
IM(f)
m(f)
6.
A BV
m(f)
log
<
}
Nf(y)dy
V
P
< }
p > I.
H BV
c
POINTWISE CONVERGENCE OF FOURIER SERIES.
In the question of convergence of the Fourier series of functions belonging to the
classes
V
and
V[], Salem and Chanturiya, respectively, have concentrated their
attention on continuous functions of the class and uniform convergence of their series.
The concept of A-variation again (and harmonic variation, in particular) has originated
Waterman’s
from Goffman’s and
investigation [35] of the conditions under which a Fourier
series converges everywhere for every change variable.
(According to their own words,
they were inspired by the possibilities hidden in Salem’s method of [22].)
The next
result, obtained by Waterman ([16],[36]), is the farthest reaching result on the conver-
gence of functions with bounded generalized variation.
THEOREM 6.1.
bounded.
If
f
HBV, then the partial sums of its Fourier series are uniformly
The series converges everywhere and converges uniformly on closed intervals of
points of continuity.
If
ABV
HBV, then there is a continuous
f
ABV
whose
Fourier series diverges at a point.
Using Theorem 4.2 and 4.3 we state three corollaries, the first two of them in
parelal form.
COROLLARY I. evzadze [37]), All funcitons of the class
everywhere convergent Fourier series.
V[]
with Z
(n) <
n2
have
231
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
COROLLARY 2.
If
f e
If
E
V
and
I
-I
()
<
=, then the
Fourier series of
f
converges
eve rywhere.
COROLLARY 3.
-I
<
n
,
the partial sums of the Fourier series of
i__sii/ i
f c
ABV
are uniformly bounded, the series converges everywhere and the convergence is
uniform on the closed intervals of points of continuity.
In virtue of Chanturiya’s result that
Z
condition
@() ,
<
I
-I
<
is equivalent to
Salem’s
Corollary 2 is the known result, obtained in an interesting man-
ner (Koethe space setting) by Goffman
[38]. However his method gives no information
regarding uniform convergence (what is the initial Salem’s result and the omitted part
of Corollary 2.) By a result of Sevast’yanov (see Section 7 for more details) Coroand 2 are in fact equivalent.
llaries
Corollary 3 is new.
Unfortunately, the substance of Shiba’s paper [39] on uniform
ABV(P) is unknown to us, but we expect that the
convergence of funcitons of the class
corresponding result is in accordance with Corollary 3, i.e.
If
f
ABV (p) n C
and
Z
n(
i:i
I/
i
I/p
<
=,
then the Fourier series of
f
converges uniformly.
In view of Theorem 4.4, everywhere convergence of the Fourier series of a function
{na}BV
< I, follows already from Hardy-Littlewood’s result concerning the
0 <
classes Lip(I/p,p) ([40]) and uniform convergence, if f is continuous, from Yano’s
f
observation [41]. (For other proofs in case of Wiener class V c Lip(I/p,p) see
P
Young [9] and Marcinkiewicz [8].) There is an interesting estimate, due to Bojanid and
Waterman [42] of the rate of convergence in this case.
THEOREM 6.2.
If
f e
{na}BV,
V()
For
denotes
=0 (i.e.
{n}-variation
BY)
f
I, then
n
V(E)
kE__
(n+l)l-
general result (classes ABV
7.
<
< (2-)(i+2/)
iSn(x)_f(x)l
where
0 <
of
gx(t)
f(x+t) + f(x-t)
2f(x).
this is an earlier result of Bojanid [43].
which can be closer to HBV) see Waterman
For a more
[44].
UNIFORM CONVERGENCE.
As an application of his convergence criterion, Salem [22] has proved
ready noted in Section 6) that all continuous functions of the class
convergent Fourier series if the function
tion
r. @(I/n)
@, complementary to
(7.1)
V@
as Salem’s classes.)
was an open question whether the condition (7.1) is definitive.
log
-d
<
For a long time it
It was answered in
positive independently by Baerstein [45] and Oskolkov [46] in 1972.
that (7.1) is equivalent to
0
it was al-
@, satisfies the condi-
<
(Later on, we shall refer to such classes
i
V@
s
have uniformly
Oskolkov has shown
(7.2)
AVIDISPAHI
M.
232
and this is further, according to the result of Chanturiya [47], equivalent to
-I
Z
(7.3)
<
For the deviation of
Vn
f
from the partial sums of its Fourier series Oskolkov
C
gave the estimate
ls n(f)-fll
<
v
c(/">
0
is an increasing function on [0,), with
Here he supposes merely that
In case
not necessarily convex.
(f)
og:d
=
#()
#(0)
0,
one obtains the earlier result of Stechkin
[48]:
If
f
BVoC
E
then
llf-Sn(f)ll
! C(/n) log(V(f)(/n)).
([2],
Using a nonincreasing rearrangement
f
V
Nf
@
(()
he constructed a convex function
f
a function
p.30)
of the Banach indicatrix of
i
0
(iM(f)
and showed that the criterion of Garsia and Sawyer
[34], Theorem I) is a consequence of Salem’s result.
du
N
re(f)
log
such that
Nf(y)
dy <
(Till then, the cases of the
See, e.g., Goffman [38],
Garsia-Sawyer class and Salem classes were treated separately.
Waterman [16].)
In view of the fact that
O(n
vf(n)
-I
(I/n))
for
V
f
and equivalence of
(7.1) and (7.3), Salem’s theorem follows from Chanturiya’s result [47]:
CV[v]
Fourier series of all functions of the class
converge uniformly if (and
only if)
Z v(k__) <
(7.4)
k2
However, (7.4) is equivalent to (7.1).
such that
f
6
V#
one can construct
Sevast’yanov [49]).
and (7.1) is satisfied.
Since all the conditions above on
E v(k)/k 2 <
(If
f
f
yield
part statements are corollaries of Theorem 6.1.
HBV
corresponding sufficient
What is new here, are the estimates,
the most general one being
THEOREM 7.1.
(Chanturiya, [47])
n(f) ll
! C
If
min
l<_m !
[]
f
E
C[0,2], then
{m (I/n)
f
m
k
I/k +
[(n-l)/2]
2
k=+l f(k)/k
n>3
It implies the estimates of Lebesque and Oskolkov.
8.
SUMMABILITY OF FOURIER SERIES.
The usual task of summability in Fourier analysis is to recover a function from its
Fourie series by some regular method, since, in general, the series fails to converge.
However, in the class
ferent:
HBV
this is not the case and our question here is quite dif-
Can one state some assertion, stronger than convergence, about the behaviour of
the Fourier series of functions belonging to certain subclasses of
what extent stronger?
ation" of
f
and, if so, to
HBV
It is natural to expect that this depends on the "order of vari-
The methods which we consider are Cesro methods (C,),
-I
<
< O.
The central result in this direction is the following theorem due to Waterman [17].
233
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
8.1.
THEOREM
(C,a-l)
everywhere
S(f)
The Fourier series
of onlinuity of
of a function
bounded and is uniformly (C,a-l)
If
f
{n a} BV,
f
S(f)
then
c
{na}BV
f
a < I, is
0
bounded on each closed interval
(C,a-l)
is everywhere
summable to
and the summability is uniform on each closed interval of continuity.
f
boundedness and convergence, by a well known convexity theorem ([I],
(C,a-l)
From
p. 127), (C,B)
B
summability for
a-I
follows.
V {na}cBV
Analogous to Theorem 3.1 one realizes that
0 < a < i.
I (I/ka) <
if
Therefore, as the first corollary of Theorem 8.1 we can state a theorem of
Akhobadze [50], proved independently and at the same time as Theorem 8.1.
COROLLARY I.
If
is (C,-)
V
f
Z (i/k
and
l-a
O< a
)<
i, then the Fourier series of
I @(I/k
Akhobadze establishes also that the condition
f e
sufficient in order that the assertion holds for every
f e C n ABV
Waterman proves that there exists
(Asatiani, [51] If f C n v[J
uniformly (C,-) summable to f.
COROLLARY 2.
is
l-a)
<
V
is continuous.
is necessary and
and, on the other hand,
whose Fourier series is not
(C,a-l)
{na}BV.
ABV
bounded at some point, if
S(f)
f
summable everywhere and the summability is uniform if
Z
and
(k 2-a
<=,
0< a
i, then
According to Theorem 8.1, if in Corollary 2 we let out the assumption of continuity
of
the we have
f
-
(C,-a)
everywhere summability in conclusion.
Corollary 2, so en-
riched, and Theorem 4.2 imply
COROLLARY 3.
If
then
()n
i=Zll/%i
n
S(f)
and
I
-i
i
n=l
ABV
f
(C,-a)
is everywhere
summable to
and the summability is uniform on
f
each closed interval of continuity.
In a similar way, using the results of of Section 5, one can state corresponding
(C,8)
assertions for the classes defined by means of the Banach indicatrix.
bility for
4.4,
V
f
B
P
-l/p,
>
Young [9].
was proved by
summa-
which is an immediate consequence of Theorems 8.1 and
It follows also from a more general result of Hardy and
Littlewood [40] on the classes Lip(i/p,p) (see also
[2], p. 66).
In the end we mention Asatiani’s extension of Chanturiya’s theorems on uniform con-
vergence.
THEOREM 8.2.
If
f
C
and
m
lim
min
n- l_<m<n-I
series of
f
{f(I/n) k I
is uniformly
i/k
I- + n-I
1
(C,-a)
The condition above (with
m
km+ f(k)/k2-}
summable to
and
O, 0 -<
f
fixed) is also necessary for uniform
summability of all Fourier series of functions of the class
9.
.
i, then the Fourier
H
(C,-a)
n VIii.
ABSOLUTE CONVERGENCE.
Bernstein has proved that the Fourier series of
n
-/2
f(i/n)
(see [2],
p. 241).
f
is
absolutely convergent if
The condition is best possible in the sense
AVDISPAHI
M.
234
,
-i/2m (l/n)
Z n
that if
f e H
then there exists
1/2
and there exists
BY, the condition
Now if
posed on its modulus of continuity may be significantly weakened.
of Zygmund ([2]
(Stechkin [52];
A
f
such that
see [53], p. 14 f.). In particular Lip a
A for a
f e Lip 1/2 \A (This was already proved by Bernstein.)
Namely, by a theorem
p. 241) in this case
i/n/2(l/n)
Z
(Hence BVc Lip a
S(f).
is sufficient for absolute convergence of
A
0.)
for all
That this condition is also definitive (and for every uniformly bounded complete orthonormal system!) was proved by Bochkarev [54] ([54a]) in 1973.
(It is understandable
that his method has become fundamental for necessity type results of this kind.)
The question arises:
I/ n
Hence, the fact that
V
2
ABV
f
HBVoH
’
is of generali?ed
and by HIder inequality
Z I/
with
convergence of the Fourier series of
absol,te
for
What is an apprepriate condition if
Trivially, Lip I/2
bounded variation?
H
f
a
V
ABV
2
if
does not contribute to
n
(Both in [20] and[16] theorems
have been stated!)
A
Siasz has noted that what is actually needed in Bernstein’s theorem is the conver-
gence of the series
I’ n
-112
I02 (]/n,f)
Stechkin [55] gave a criterion in terms of the best approximations to
Sasz’s theorem from
and deduced
E
it
f
in
L2-metric
by means of the Jackson type relation
O[m 2(I/k,f)].
2(k,f)
(9.1)
In [56] McLaughlin has generalized Stechkin’s result to
{’k
Let
THEOREM 9.1.
sequence of natural numbers and
Z
k-/2[E2(m(k)
{akt
where
Assume that {m(k)} is an increasing
be an orthonormal system.
f)]B
<
0
-< 2.
then
Z
If
kla m(k) ]B
<
denotes the sequence of Fourier coefficients of
f
for
{k
The theorem and the relation (9.1) stress the importance of the estimate for
2(1/k,f).
The essential result for our purposes here is due to Chanturiya [57]:
(n)+n
2 (l/k
where
W
f
Cn-1/2(k=Z(n)
f)
max {m:
(n)
and
(k)/k
2) i/2
f
(9 2)
f(m)/m_> f(I/n)}.
Now from Theorem 9.1, (9.1) and (9.2) one obtains immediately Chanturiya’s theorem
announced in
[58].
THEOREM 9.2.
Let
Z
n=l
For
n
f
6-3 .n+/-(n)
(kL (n)
6
THEOREM 9.3.
0,
B =I
([57])
H
V[v],
2
(k)/ k2 3/2 <
(n)
o(n).
then
Z
n=l
B
0
If
6
n 0
B
n
<
< 2
.
this yields the sufficiency part of
For a Fourier series of the class
H
m
be absolutely convergent it is necessary and sufficient that
l-2a
n=l
]2(l-a)
![()
n
and
V[n
a
0 < a < I/2, to
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
nong
235
a number of corollaries which one can state for particular classes, we point
to
Lip a n V[n
COROLLARY I. ([59]) Cor. I.)
A
B
0 <
for
I/2, a > O.
A (a,B as above) and furBy Theorem 4.4 this is equivalent to Lip a n {n }BV
n
<
to Hirschman ([60],
result
due
was
an
earlier
a
what
A
V
2)
(p
ther to Lip
P
Lemma 2d).
Similarly
([59]
COROLLARY 2
I/2
V[n
f
If
Z O
k
then
for
in [6] (a special case of Th. 6
P
(Corollary 2 may be obtained from Theorem 9.1 via (9.1) and 10(3).)
2/(3-26),
there).
Theorem 3)
Golubov’s result on
is equivalent to
V
Using Theorem 4.2 and 4.4 it is also possible to deduce some informations (in
ABV.
various forms) about the classes
i0.
ORDER OK MAGNITUDE OF FOURIER COEFFICIENTS.
ml(,f)
A standard argument (see [61], p. 80 f.) leads to the relation
O(6f([I/]))
f2
0
[I/
if(x+h)_f(x)id X
Let
We repeat it.
f
for bounded
2
kl
[l/h]
0 < h !
Then
If(x+kh)-f(x+(k-l)h)l dx
2
! [I/h]
vf([I/h]
+|)
(10.1)
Since
vf(n)/n
f(n)/n ! f(n+l)In
is nonincreasing, we have
Further
[I/6]
[I/h]
(l+lln)f(n+l)l(n+l) ! (l+lln)f(n)In.
implies vf([I/h])/[I/h] ! vf([I/6])/[i/6].
!
Hence
ml (6’f)
<spi
0
6
g2
if(x+h)_f(x)id x
Since the absolute value
dominated by
0
ml(/n,f)/(//),
V[]
of the classes
0(6f([1/6])).
of the n-th Fourier coefficient of
n
f
is always
it follows that the Fourier coefficients of functions
(n)/n (see also Chanturiya [59])
are of the order
From this
and the results of Section 4, follow all known estimates for Fourier coefficients of
functions of generalized bounded variation.
Pn
0(1/
ABV, e.g., one obtains
In case of
n
i=Z1 I/i).
This was independently proved by Wang
[62] in 1982.
[23] and this author
Littlewood [40]) and hence
O
n(f)
For
f
O(n
-I/p)
Lip(a, p)
if
[20], Schramm and Waterman
0n(f)
f e V
O(I/n a) (Hardy and
Lip(I/p,p) (Marcinkiewicz
P
[8]).
Waterman’s result
o(na-l),
if
iM(f) llp(y)dy
Nf
m(f)
{n
f
a
<
Wang [20] has proved
on summability (Theorem
}c BV
or if
0n(f)
0
I
a <
I.
f(n)/n
o(I/
n
i__E1
8.1) implies ([2],
In particular,
2-1/p <
I/ i)
or if
for
f
0 (f)
n
f
V
o(n
p.78) that
On(f)
-I/p)
I)
with
(P
if
I @(I/n I-I/p) <.
A BV, in general.
c
Hence func-
tions of Salem classes and of the class of Garsia and Sawyer have Fourier coefficients
of order
o(I/log n).
(See Theorem 3.1 and Corollary 3 in Section 5).
M.
236
REMARK.
AVDISPAHI
In case of Fourier coefficients one can use the presence of the factor cos nx
(resp. sin nx) in a relation analogous to (I0. I) to eliminate
(a modification due to
[63]).
[zumi
on the right hand side
The group theoretic nature of the argument was
([64],3.22) in 1947. Vilenkin’s observation escaped notice
of Taibleson [65], Edwards ([66], p.35), Benedetto ([67], p. 120).
noted already by Vilenkin
Another information on the magnitude of Fourier coefficients is contained in the
estimate of the rate of approximation of a function by trigonometric polynomials in
L2-norm
(kn 0) I/2
0(2(I/n,f)).
(10.2)
(See section 9, relation (9.1).)
Clearly,
V
for all
a
kn
n
P! 2
P
Lip(I/2,2)
f
if
0(|)
Ok
Hence one has the latter estimate
{na}BV,
and then, via Theorem 4.4, for all
a
! I/2 (and
This is significantly better than Shiba’s result in [68].
i/2).
tion of the estimate for
l(i/n,f)
and the fact that
is bounded
tinuous) does not lead to satisfactory results in (10.2).
Vine],
A simple applica-
(eventually con-
We have already seen the
advantages of Chanturiya’s approach in matters of absolute convergence. Sacrificing
(n),
2(I/n,f)
we illustrate the principle on which his estimate of
(9.2) in Section 9) relies.
If(x+k/r)- f(x+(k-l)/n)l,
dk,n(X)
/20
If(x+/n)-f(x)IZdx
dk, n (x)
since
(k)/k
f
I/n
k
O[ (
n
(k)/k2
k=l,2,...,n.
/20 (kl dk,n2
if we denumerate
that these numbers decrease when
2(,f)
(see relation
Let us denote
k
I/2
{d
(x))dx =0(
k ,n
increases.
Then
nI
(x)} nk=l
k
(k)/k2)’
for each particular
x
so
Therefore
]’
(I0.3)
and, in general,
2(6
0[(6
f)
[I/6]
k1
f(k)/k2) I/2 ].
2
((9.3) follows, of course, from (9.2), for
For
n
{na}BV,
2-2a
2
k=n
Ok
a
9.(k)/k
decreases.)
i/2, this gives us, for example, (via Theorem 4.4 again)
0(I)
and this is also better than the corresponding result of Shiba.
II.
CHARACTERIZATION OF CONTINUITY.
Of the following five conditions
n
k
2
0 sin
k
o(i)
(11.1)
n
k k2 p
n
k!n kPk
k Ok
o(n)
o(n)
o(Zog n)
(11.3)
(11.4)
(11.5)
237
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
the first three are equivalent (Tanovid-Miller [69]).
They imply (11.4) and this one
By a well known theorem of Lukcs ([2],
implies further (11.5).
f
cient to insure continuity of
p. 60) (11.5) is suffi-
is supposed merely to be regulated.
f
if
Wiener
BY.
[4] has shown that (Ii.I) and (11.3) are also necessary for continuity of f
Sidon [70] has indep@ndently established the same for (11.4), which is, in case BY,
trivially equivalent to (11.3), because of
Hence (11.5) is also necessary
0(I).
ko k
in this case, what is known as Lozinski’s theorem [71].
necessity of (Ii.I) if
V
Co
f
lim
h/O
+
(i Of 2
which holds for
This is contained in a more general result
if(x+h)_f(x) lqdx )I/q
V
f
If(Xk+O)-f(Xk-O)I q)I/q
(Z
k
(Golubov [7]).
< q
! p
P
(see [4], p.77 the bottom, and p.78,
< 2
! p
P
relation (21); see also Golubov [72]).
Actually, Wiener has proved the
In view of (10.2) condition (11.2) is obviously fulfilled if
particular, by (9.2), (II.I)
{f
tinuity of functions of the class
ABV
also for the classes
Lip(I/2,2).
f
n
with
2
k 9(k)/k < }
[# I(I/
i I/li)]2
(Chanturiya [73]).
Therefore
e.g. (See Theorem 4.2)
f e V
On the other hand, there is no necessary condition for the continuity of
in terms of the absolute value
kl
kI
sin kx
BV
k
k(x+logk)k
sin
of its Fourier coefficients.
Ok
(The function
with a jump at O, and Hardy-Littlewood function
V2
In
(ii.5) are necessary and sufficient conditions for con-
Lip I/2
fl(x)
f2(x)
I/k .)
0k(f 2)
have
V2
2
Let us now prove
THEOREM 11.1.
Each of the conditions (II.I)
the continuity of
PROOF.
{n
f
I/2}
BV.
C
For the necessity part it is enough to prove
Sufficiency is clear.
n
k If(x+k/n)-f(x+(k-l)n/n)l 2
4n
In
k 0sin2
Then
k
Given
I/2
g
d
k ,n
0,
(x)
o(I)
nk /20 (kl
and then (II.I) will hold.
0(I)
n
n
maX{no,nl }"
2
_<
no (I/n)
k=Zl
+ C
x
<
V(no)I/2
For n > n
d
2
(x)
k,n
n
k__Zn O +I
as
x
n
since
n
n
{dk, n (x) }k=l
let
(See Theorem 4.4
be as in Section 10
part i) of the proof
x) such that
(f;[O 4])
{k
Further there exists
n
k=Zl dk’n (x)
and
(independent of
O
kno +i dk, (x)/kI/2
Let
x
uniformly in
n
(See Definition 3.3)
uniformly in
If(x+k/n)-f(x+(k-l)/n)12)dx
For every
there exists
(11.5) is necessary and sufficient for
+
n
such that
f(I/n)
for
we have then
2
n
k=Zn
+I
O
(x)
n
kl /2dk ,n (x) dk,I/2
-<
K
dk,n(X)/kl/2 no2 + C,
uniformly in
x.
n
n
AVDISPAHI
M.
238
COROLLARY ([32], Theorem 6)
fM,f,t NfI/2 (y)dy
{fe W:
In the class
<
}
conditions
m(f)
(11.5) characterize the continuous functions.
(11.1)
Still easier than Theorem 11.1 one proves the theorem due to Cohen [74], saying
that such characteriz&tion is possible in the class
U2/(U)
_<
0
O(I) (u/O+).
For
Uo
In fact, given
C n
f
V_
take
no
x
and
if
$
e > O, there exists
u
eV(f)
for every
f(I/n)_
so that
n
kl dK,n(X)
_>
n
V$
no
satisfies the condition
such that
O
for
Uo
_>
n
u2
no.
e(u),
Then
and the assertion follows immediately.
DETERMINATION OF A JUMP.
12.
By the theorem of Lukcs the jump of an integrable function at a point
x
of its
o
discontinuity of the first kind may be determinated from its Fourier series using the
first logarithmic mean
In case
BV
f
([2],
{nb cos nx
p. 106, f.) of the sequence
n
na sin nx }.
n
o
o
the logarithmic mean may be replaced by (C,I) method (essentially
Fejr [75]; see Czillag [76]) and indeed by any (C,)
> O.
In some sense we have
completed this picture proving
([77], Theorems
THEOREM 12.1.
{kbk(f)cos
kx
ABV
HBV
If
kak(f)sin
and 2)
kx]
If
(C,)
is
HBV, then the sequence
f
summable to (f(x+O)-f(x-O))/
then there exists a continuous function
f
for every
x.
n
ABV
k kbk(f)
such that
O(n).
([78], Theorem 3)
THEOREM 12.2.
{kbkCOS kXo
then the sequence
>
for every
If
l-I/p (
B)
V
and every
{nB}BV
I, (f
p
P
kaksin kXo
{kbkCOS kXo
(C,1)-limit of
f
(C,)
is
f e V[n ], 0<8<I)
or
summable to (f(x+O)-f(x-O))/
x.
is, of course, equal to
kakstn kXo
S’(
n Xo)
lim
n+l
(C,1) result is established, it turns out that it is possible to
Therefore, once the
take any of expressions:
lim
n
s(2r-1)
(xo)
n
n
lim
n
2r-I
g(2r) (xo)
n
n
x
for the determination of the jump at
implied by
[77].
(C I)
r=l 2
2r
(s(k)(x)
n
O
since the summability methods here involved are
is the k-th derivative of
[79].
GIBB’S PHENOMENON.
x
converging at
O
classes is due
P
It is a corollary of our Theorem 12.2.
A sequence of functions
x
V
(C,I) in the case of
An earlier result on the effectivness of
13.
A similar defini-
n
cases.)
ton applies to other
to Golubov
S (x)
to limit
s
{f
(but not necessarily
O
if to every
.Ifn(X)-Sl
<
for
x
defined in the neighbourhood of a point
n
0
IX-Xol
for
x
# xO
(e)
there is a
<
and
is said to converge
n > n
.o
and a
n
O
and
uniformly at
n (e)
O
o
such that
239
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
p.58).
If
lim f
f
n+
for each sequence
fn(Xn)/S
An equivalent definiton is that
is defined in the neighbourhood of
n
the absence of the Gibb’s phenomenon at the point
{f
vergence of
If
THEOREM 13.1.
x
at
n
(ibid., p. 61).
o
f
lim S (x,f)
n+ n
f(x)
f(Xo),
(Waterman’s
formly at
g(x)
An inspection of Waterman’s
by Theorem 6.1.
x
f
is continuous at
is large enough and
n
is arbi-
close enough to
x
S(f)
Hence
k
is continuous at
k
Xo
converges uni-
converges uniformly at
x
is known to show Gibb’s phenomenon.
at
S (x,f)
n
The behaviour of
o
(0)
The func-
Xo.
HBV.
and belongs to
Xo
d
the same as the behaviour of the partial sums of the series
14.
Sn(x,f)
Xo
f(Xo+0)-f(Xo-0)=d
has a jump
f
k(x-x o)
sin
d
(x)
S(g)
Hence
shows Gibb’s phenomenon
S(f)
and Gibb’s phenomenon is absent.
o
Let us suppose now that
tion
o
idea is repeated in the proof of Theorem 14.1.)
x
o
and only there.
for every
provided
x
and continuous at
o
We have
proof of this fact in [36] shows that if
trarily close to
(Zygmund [2],
is equivalent to the uniform con-
is of harmonic bounded variation,
f
at every point of discontinuity of
PROOF.
x
x
Xn/Xo
near
x
sin k(x-x
o)
k
o
is then
which
k
S(f).
Hence so does
CONJUGATE SERIES.
Another among the classical theorems about functions of bounded variation which ad-
HBV
mit to the whole class
Young’s
is
theorem on the convergence of the conjugate
series.
THEOREM 14.1.
(f)
HBV, a necessary and sufficient condition for the convergence of
f
If
is the existence of the conjugate function
x
at
(x)
lim
g/O
f
e
f(x+t)-f(x-t) dt
2tan(t/2)
(f)
which represents then the sum of
PROOF.
x(t)
x(+O)=O, for
at
x.
(x
f(x+t)-f(x-t)
Let us denote
We may assume
x, i.e. of the limit
at
f
h)
h
x (t)(t/2)
2tan
dt"
otherwise both the series and the integral are known to
diverge.
n
0
n
Given
(x,n/n) =-!n fn
(x)
/In
<
0
0 < 6 < n
/n f
+
n
<
Ix(t)l
In
2g
cos nt
+ o(1) +
!
n x(t)
/n
2tan(t/2) dt
1-cos nt
2tan(t/2) dt + o(1)
< e
for
sin2nt/2
dt
0
x(t)2-a2) dt
by Riemann-Lebesgue.
dt
n 0
nt
ll-cos
2tat/2) Idt
6
3
2tan(t/2)
such that
n
Further
II
1-cos nt
f/n x(t)
cos nt
fn @x(t) -r2)
dt
n/n
e > 0, we can chose
i21
x(t)
and
f...
Itl
! n/n
n 2 f,ln
-< 2
0
o(i)
11
+
12
+ o(1)
Then
tdt
< g.
(n/) for every
AVIDSPAHl
M.
240
/
The method of estimating
Waterman’s proof
is essentially that of
/n
of
Theorem 6.1 in [361.
6
/n
[n/].
E+l
where
may
(k+l)/n
f
k/n
kZ--I
I’
Clearly
necessary, we
way,
N
cos nt
x(t)2tan(k/2)dt
N
(t)
cos nt
f(N+i i/n I[
+
2ta t/2) dt
+
i,
in the same
and removing the last term in
o(I)
,ssume
x
After an obvious change of variable
to be even.
one obtains
II
N
_I
/
0
t+k
-)
x-n
kE--I
n
(-l)k cos
nt
dt
t+k
tan---n]
The ab,olute value of the integrand her, is dominated by
N-I
l__
--)cot
ik__Zi [x (t+k
n
2n
t+(k+l )
x-- )ct’t+
t+k
2n
n
(k+l
2n
.!
Let us write the general term of the sum
indicates summation over odd indices.
where
in the form
x(t+(k+t))
n[*x (t+k____!)
n
cot
n
+
in
n
x (,t+(k+l) )tcot--n
t+k
cot
n
t+(k+l)
]"
2n
By the mean value theorem
t+k
2-I co t-n
N
Choosing
cot
t+(k+l
2n
such that
o
sin
I/k
kN o +I
t+k
[co t
n
k-Z-I lx( t+(k+l))
4n 2
Sk
,
2
N-}
2n
2
co
<
r
4 .t+k 2
-----n
N-I
2-z iki
if
6
sup
0<t<
/(n+l)
h
Ix(t+(k+l))ln kZ--I
I/k2
o
o(I) as
n
since
@
-t+k
t+(k+l)n
.t+kn
! Nx()]cnln
n
kl lX) -x(I t+(k+l))[
< V
H(Ox ;[0’6]) <
[mx (t+kn.
T
n
k2
Further
(Waterman [21]).
is sufficiently small
Hence
<
N
tt+(k+l
)--}l
2n
The first summand after the inequality sign is clearly
O.
4k2
we have
l<k<N
tinuous at
<
4n2
(x)
(x,/n)
/n
the proof is complete."
o(1).
The earlier extension of
l(x,h)
Since
Young’s
(x,/n)
theorem to the classes
V
O(I/n)
P
for
-
+
e"
is con-
<
p>[, is due to
Marcinkiewicz [8].
THEOREM ]4.2.
f
If
HBV
g
and
g
W, then both
S(f)
and
(f)
are uniformly con-
vergent.
PROOF.
Already
f,
f(xo+O)-f(xo-O)
> 0
W
imply that
what contradicts the fact that
for
([61], p. 592).
and
are continuous.
Really, if, e.g.,
n(Xo,f)/ -" Hence n(Xo,f)
((Xo+O)-(Xo-O))/2. The
n(X o,)
(Xo,f)
is completely analogous.
by Theorem 6.1.
f
then, by Lukc’s theorem
But if
f,
Now
C
and
f
HBVO C
S(f)
proof
implies uniform convergence of
converges uniformly, then so does
S(f)
(f)
241
GENERALIZED BOUNDED VARIATION AND FOURIER SERIES
PARSEVAL IDENTITIES.
15.
Zygmund ([2],
K
p. 157) calls two classes of functions,
if for any pair
K
e
complementary
the Parseval formula
K
g
K
and
/2f(x)g(x)d x a(f)a(g)
0
+
+
k=Zl{ak(f)ak(g)
bk(f)bk(g)}
(15.1)
lolds, in the sense that the series on the right is summable by some method of summation.
require that this series converges, as we shall do, examples of complementary
[!- we
(L,BV), (L p,L q)
classes are
L
15.1.
]IEOREM
f
for [/p+I/q =I
I, and
p
(Llog
L,L)([2],
pp.159 and
Waterman [36] has generalized the first result to
27).
ABV
L, g
HBV
and
ABV
if
are complementary classes,
dBV
then there exist
such that the series in (15.1) diverges.
We would like to point out another, less known, type of "Parseval identity" due to
9 ].
Young
TItEOREM 15.2.
f V
If
V
g
P
l/p+l/q Z 1, and
q
discontinuity, then the Stieltjes integral
!
S
0
2
fdg
n =l
have no common points of
exists in the Riemann sense and
(15.2)
we obtain the interesting
V
If
g
n[an(f)bn(g)-an(g)b n (f)].
Taking the pair
(’ROLLARY.
f fdg
and
f
V
P
1/p+l/q
q
l, then
np
Z
n=l
(15 3)
It is a classical theorem due to Hardy and Littlewood (see [2],
conditions
BY
f
imply
f
(Clearly, if
A
f
A
and
ko k
p.286 f.)
that
the
0(I), (15.3) is
satisfied.)
Generalizations of Theorem 15.2 to the classes
esniewicz and Orlicz
ACKNOWLEDGEMENT.
have been investigated by
[15].
would like to thank the referee for several valuable suggestions.
Tanks are also extended to Professor Z. A.
([ISSR)
V
Chanturiya of the University of Tbilissi
who read the manuscript and called my attention to Tevzadze’s paper [37] as a
reference for Corollary
of Theorem 6.1.
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