Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
27 (2011), 223–231
www.emis.de/journals
ISSN 1786-0091
CONVERGENCE ALMOST EVERYWHERE OF FOURIER
SERIES OF THE FUNCTIONS OF BOUNDED VARIATION
LARRY GOGOLADZE AND VAKHTANG TSAGAREISHVILI
Abstract. Let (ϕn (x)) be an orthonormal system on [0, 1] (ONS). If f (x)
is a function of bounded variation (f (x) ∈ V ), then the convergence of
Fourier series of f (x) generally does not occur (see [2]).
In the present paper we prove that if ONS (ϕn (x)) satisfies some conditions, then lim Sn (f, x) exists a.e. on [0, 1], where f (x) ∈ V is an arbitrary
n→∞
function.
1. Some definitions and theorems
Let V be a class of a functions of bounded variation and
Z 1
kf kV =
|f 0 (x)| dx + sup |f (x)|
x∈[0,1]
0
be a norm. The sum
∞
X
ϕ
bn (f )ϕn (x)
n=1
is the Fourier series of the function f (x) ∈ L(0, 1).
n
1 X α−1
α
A Sk (f ; x) (α > 0)
σn (f, x) = α
An k=1 n−k
is the Césaro mean of the sum Sn (f ; x). Let
(1.1)
Pn (x) =
2n+1
X−1
ck ϕk (x),
k=2n
where (ck ) are arbitrary numbers, and let
Z
Z i/2n
(1.2)
A(Pn ) = maxn Pn (x) dx = 1≤i<2
0
0
in /2n
Pn (x) dx.
2010 Mathematics Subject Classification. 42A16.
Key words and phrases. Orthonormal system, function of bounded variation, Fourier
series.
223
224
LARRY GOGOLADZE AND VAKHTANG TSAGAREISHVILI
The following theorems are valid (see [1, pp. 87, 132]).
Theorem A. If (ϕn (x)) is ONS on [0, 1] and
∞
X
c2n log2 n < +∞,
n=1
then
∞
X
(1.3)
cn ϕn (x)
n=1
converges a.e. on [0, 1].
Theorem B. If (ϕn (x)) is ONS on [0, 1] and
∞
X
c2n (log log n)2 < +∞,
n=1
then lim σnα (f, x) exists almost everywhere on [0, 1] for any α > 0.
n→∞
Lemma 1.1. If Φ(x) ∈ L2 (0, 1) and f (x) ∈ L2 (0, 1) is a function finite in
every point of [0, 1], then
Z
(1.4)
0
1
Z i/n
n−1 X
i
i+1
f (x)Φ(x) dx =
f
−f
Φ(x) dx
n
n
0
i=1
Z 1
n Z i X
n
i
Φ(x) dx + f (1)
+
f (x) − f
Φ(x) dx.
i−1
n
0
i=1
n
Proof. It is evident that
Z
(1.5)
1
f (x)Φ(x) dx =
0
n Z
X
i=1
i
n
f (x)Φ(x) dx.
i−1
n
Applying Abelian transformation we have
Z i
Z i/n
n
n−1 X
X
n
i
i
i+1
(1.6)
f
Φ(x) dx =
f
−f
Φ(x) dx
i−1
n
n
n
0
i=1
i=1
n
Z 1
+ f (1)
Φ(x) dx.
0
Combining (1.5) and (1.6) we get (1.4).
CONVERGENCE ALMOST EVERYWHERE OF FOURIER SERIES . . .
225
2. The main results
Theorem 2.1. Let (ϕn (x)) be ONS on [0, 1] satisfying the following conditions:
∞
R1
P
2
2
1)
ϕ
bn (1) log n < +∞ ϕ
bn (1) = 0 ϕn (x) dx ;
n=1
2) for every sequence of polynomials (Pn (x)) (see (1.1) and (1.2))
2
∞ X
A(Pn )
n
< +∞.
kPn kL2
n=1
(Here and below if A(Pn ) = kPn kL2 = 0, then we assume that
0.)
A(Pn )
kPn kL2
Then for any function f (x) ∈ V
∞
X
ϕ
b2n (f ) log2 n < +∞.
n=1
Proof. Suppose that
Pn (x) =
2n+1
X−1
(ϕ
bk (f ) log2 k)ϕk (x),
k=2n
then
2n+1
X−1
(2.1)
Z
1
ϕ
b2k (f ) log2 k =
f (x)
0
k=2n
Z
=
2n+1
X−1
ϕ
bk (f ) log2 kϕk (x) dx
k=2n
1
f (x)Pn (x) dx.
0
Considering equality (1.4) with f (x) ∈ V and Φ(x) = Pn (x), we have
Z
(2.2)
0
1
Z in
2
i+1
i
f (x)Pn (x) dx =
f
−f
Pn (x) dx
n
n
2
2
0
i=1
Z 1
2n Z in X
2
i
+
Pn (x) dx + f (1)
Pn (x) dx.
f (x) − f
n
i−1
2
0
n
i=1
2
n −1 2X
If f (x) ∈ V , then
n
2X
Z in
−1
2
i
i+1
(2.3)
f
−f
Pn (x) dx ≤
n
n
2
2
0
i=1
Z in
2n −1 2
X
i
i
+
1
< kf kV · A(Pn )
f
≤ maxn −
f
Pn (x) dx
n
n
1≤i<2
2
2
0
i=1
=
226
LARRY GOGOLADZE AND VAKHTANG TSAGAREISHVILI
A(Pn )
= kf kV ·
kPn kL2
nA(Pn )
≤ 2kf kV
kPn kL2
On the other hand,
n Z
X
2
(2.4)
i=1
2n+1
X−1
i−1
2n
2
X
ϕ
b2k (f ) log2
f (x) − f
sup
i−1 i
i=1 x∈[ 2n , 2n ]
Z
k
12
k
i
2n
Pn (x) dx ≤
Z in
2
f (x) − f i |Pn (x)| dx
2n i−1
12 X
2n
1
2
Pn (x) dx
V
≤
.
k=2n
n
≤
12
k=2n
2n+1
X−1
i
2n
ϕ
b2k (f ) log4
0
2n
2i−n
(i−1)2−n (f )
i=1
−n
2
≤ kf kV · 2
2n+1
X−1
ϕ
b2k (f ) log4
≤ 2kf kV n2
2n+1
X−1
n
21
k
k=2n
−n
2
· 2− 2
ϕ
b2k (f ) log2
21
k
k=2n
and
Z
0
1
2n+1 −1
Z
X
2
Pn (x) dx = ϕ
bk (f ) log k
0
k=2n
≤
2n+1
X−1
k=2n
ϕ
b2k (f ) log2
k
1
ϕk (x) dx
21 2n+1
X−1
ϕ
b2k (1) log2
21
k
.
k=2n
Using (2.3) and (2.4) from (2.2) we get
Z 1
(2.5)
f (x)Pn (x) dx ≤
0
21 2n+1
X−1
1
A(Pn )
2
−n
2
2
+2 +
≤ 2nkf kV
ϕ
b (1) log k
×
kPn kL2
n k=2n k
2n+1
12
X−1
2
2
×
ϕ
bk (f ) log k .
k=2n
In such a way (2.1) implies
21 2n+1
2n+1
X−1
X−1
A(Pn )
1
2
2
−n
2
2
+2 2 +
ϕ
bk (f ) log k ≤ 2kf kV n
ϕ
bk (1) log k
×
kP
n
n kL2
n
n
k=2
k=2
CONVERGENCE ALMOST EVERYWHERE OF FOURIER SERIES . . .
227
2n+1
21
X−1
2
2
×
ϕ
bk (f ) log k .
k=2n
Hence,
2n+1
X−1
ϕ
b2k (f ) log2
k≤
4kf k2V n2
k=2n
n
A(Pn )
1
+ 2− 2 +
kPn kL2
n
2n+1
X−1
ϕ
b2k (1) log2
21 2
k
.
k=2n
Since
∞
X
2 −n
n2
2
∞ X
nA(Pn )
< +∞,
n=1
n=1
kPn kL2
< +∞ and
∞
X
ϕ
b2n (1) log2 n < +∞,
n=1
we have
∞
X
ϕ
b2n (f ) log2 n < +∞.
n=1
Theorem 2.1 is proved.
Theorem 2.2. Let (ϕn (x)) be ONS on [0, 1] satisfying the conditions of Theorem 2.1, then the Fourier series of any function f (x) ∈ A converges almost
everywhere on [0, 1].
The validity of Theorem 2.2 follows directly from Theorems A and 2.1.
Theorem 2.3. Let (ϕn (x)) be ONS on [0, 1] satisfying conditions:
∞
P
1)
ϕ
b2n (1)(log log n)2 < +∞,
n=1
2) for every sequence of polynomials (Pn (x)) (see (1.1) and (1.2))
∞ X
A(Pn )
n=1
kPn kL2
2
log n < +∞.
Then for any f (x) ∈ V
∞
X
ϕ
b2n (f )(log log n)2 < +∞.
n=1
Proof. Suppose that
Pn (x) =
2n+1
X−1
k=2n
ϕ
bk (f )(log log k)2 ϕk (x).
228
LARRY GOGOLADZE AND VAKHTANG TSAGAREISHVILI
Analogously to Theorem 2.1 we have (see (2.5))
Z 1
f (x)Pn (x) dx ≤
0
21 2n+1
X−1
n
A(Pn )
1
ϕ
b2 (1)(log log k)2
×
≤ 2kf kV log n
+ 2− 2 +
kPn kL2
log n k=2n k
2n+1
21
X−1
2
2
ϕ
bk (f )(log log k)
×
,
k=2n
According to the equality
2n+1
X−1
Z
1
ϕ
b2k (f )(log log k)2 =
f (x)
0
k=2n
Z
=
2n+1
X−1
ϕ
bk (f )(log log k)2 ϕk (x) dx
k=2n
1
f (x)Pn (x) dx
0
we conclude that
∞
X
ϕ
b2n (f )(log log n)2 < +∞.
n=1
Theorem 2.3 is proved.
Theorem 2.4. Let (ϕn (x)) be ONS on [0, 1] satisfying conditions of Theorem 2.3. Then for any f (x) ∈ A and α > 0
lim σnα (f, x)
n→∞
exists a.e. on [0, 1].
The validity of Theorem 2.4 follows from Theorems 2.3 and Theorem B.
Theorem 2.5. Suppose that for some ONS (ϕn (x)) the following conditions
are satisfied:
R1
1) 0 ϕn (x) dx = 0, n = 1, 2, . . . ;
2) there exist a sequence of polynomials (Pn (x)) (see (1.1) and (1.2)) such
that
nA(Pn )
lim
= +∞.
n→∞ kPn kL2
Then there exists a function f0 (x) ∈ V such that
∞
X
ϕ
b2k (f0 ) log2 n = +∞.
n=1
Proof. Consider the sequence of linear functionals
Z 1
n
f (x)Pn (x) dx,
(2.6)
Cn (f ) =
kPn kL2 0
CONVERGENCE ALMOST EVERYWHERE OF FOURIER SERIES . . .
where Pn (x) =
2n+1
P−1
l=2n
rem 2.5.
Let
229
ck ϕk (x) and (ck ) ∈ `2 satisfies condition 2) of Theo-

in 
0 for x ∈ 0, 2n ,
fn (x) = 1 for x ∈ in2+1
n ,1 ,

linear and continuous on in , in +1 2n
2n
(2.7)
be the sequence of functions from V . Obviously,
kfn kV = 2,
n = 1, 2, . . . .
Now, consider (2.2) with f (x) = fn (x). One can find
Z
Z
1
fn (x)Pn (x) dx = −
(2.8)
0
in
2n
Pn (x) dx
0
Z
in +1
2n
+
in
2n
fn (x) − fn
in + 1
2n
Pn (x) dx.
Since (see (2.7))
Z in +1
2n
i
+
1
n
(2.9) Pn (x) dx ≤
fn (x) − fn
n
in
2
2n
−n
2
Z
1
≤2
Pn2 (x) dx
21
= 2− 2 kPn kL2 ,
n
0
from (2.8) we get
Z
0
1
n
fn (x)Pn (x) dx ≥ A(Pn ) − 2− 6 kPn kL2 .
Taking into account condition 2) of Theorem 2.5 we obtain
Z
nA(Pn )
n
n 1
fn (x)Pn (x) dx ≥
A(Pn ) − n2− 2 .
kPn kL2 0
kPn kL2
Consequently (see (2.6)),
(2.10)
lim |Cn (fn )| = +∞.
n→∞
From (2.10) and Banach-Steinhaus theorem it follows that there exists a function f0 (x) ∈ A such that
(2.11)
lim |Cn (f0 )| = +∞.
n→∞
230
LARRY GOGOLADZE AND VAKHTANG TSAGAREISHVILI
Next, by virtue of Hölder’s inequality we get
Z 1
2n+1
Z 1
X−1
f0 (x)Pn (x) dx = ck
f0 (x)ϕk (x) dx
0
0
k=2n
2n+1 −1 1 2n+1 −1
2n+1 −1
21
X
X
2
X
c2k
ck ϕ
bk (f0 ) ≤
ϕ
b2k (f0 )
= k=2n
≤
Therefore,
2n+1
X−1
kPn kL2
n
ϕ
b2k (f0 ) log k
2
12
k=2n
2n+1
X−1
k=2n
ϕ
b2k (f0 ) log2 k
k=2n
21
.
k=2n
Z
n 1
= Cn (f0 ).
f
(x)P
(x)
dx
≥
0
n
kPn kL2 0
Then from (2.11) we have
lim
n→∞
2n+1
X−1
ϕ
b2k (f0 ) log2 k = +∞.
k=2n
Theorem 2.5 is completely proved.
Theorem 2.6. If (ϕn (x)) is ONS on [0, 1] satisfying conditions 1) and 2) of
Theorem 2.5, then there exists a function f0 (x) ∈ A such that
∞
X
ϕ
b2n (f0 )(log log n)2 = +∞.
n=1
Theorem 2.6 is proved analogously to Theorem 2.5.
R1
Remark 2.1. If in Theorem 2.5 we change condition 0 ϕn (x)dx = 0 by the
∞
P
inequality
ϕ
b2n (1) log2 n < +∞, then Theorem 2.5 will also be valid.
n=1
3. Efficiency of Theorem 2.1
√
√
Consider trigonometric system ( 2 cos 2πnx, 2 sin 2πnx) on [0, 1] for every
sequences (ak ) and (bk ). Then
Z inn 2n+1 −1
2 X √
A(Pn ) = 2(ak sin 2πkx + bk cos 2πkx) dx
0
k=2n
1
12 2n+1
2n+1
X |ak | + |bk |
X−1
X−1 1 2
2
2
≤2
≤4
(ak + bk )
k
k2
k=2n
k=2n
k=2n
2n+1 −1
≤ 2− 2 +2 kPn kL2 .
n
Consequently, for the trigonometric system conditions of Theorem 2.1 are valid.
CONVERGENCE ALMOST EVERYWHERE OF FOURIER SERIES . . .
231
References
[1] G. Alexits. Convergence problems of orthogonal series. International Series of Monographs in Pure and Applied Mathematics, Vol. 20. Pergamon Press, New York, 1961.
[2] S. Banach. Sur la divergence des séries orthogonales. Studia Math., 9:139–155, 1940.
[3] V. Tsagareishvili. On the Fourier coefficients for general orthonormal systems. Proc. A.
Razmadze Math. Inst., 124:131–150, 2000.
Received February 13, 2011.
I. Javakhishvili Tbilisi State University
2 University St.,
Tbilisi 0186,
Georgia
E-mail address: lgogoladze1@hotmail.com
E-mail address: cagare@ymail.com