General Mathematics Vol. 11, No. 3–4 (2003), 45–52
Some notes on absolute convergence of
Fourier series
Amelia Bucur
Dedicated to Professor Gheorghe Micula on his 60th birthday
Abstract
In this paper we study a necessary and sufficient condition of the
absolute convergence of a trigonometric Fourier series is established
for continuous 2π-periodic functions which in [−π, π] have a finite
number of intervals of convexity, and whose n-th Fourier coefficients
³ ³
´ ´
1 ; f /n where ω(δ; f ) is the continuity modulus of the
are O ω n
function f .
2000 Mathematical Subject Classification: 42A28, 42A16
Will use the following definition: a serie u0 + u1 + u2 + ... with real terms
is said to be absolutely convergent if the series |u0 | + |u1 | + |u2 | + ... of the
module of its terms is convergent.
45
46
Amelia Bucur
Let ω be an arbitrary modulus of continuity, i.e, a nondecreasing function
continuous on [0, 1], ω(0) = 0 and ω(δ1 + δ2 ) ≤ ω(δ1 ) + ω(δ2 ). Will use the
class of all functions f continuous on [−π, π] for which
ω(δ; f ) =
sup
|f (x1 ) − f (x2 )| = O(ω(δ)), 0 ≤ δ ≤ 1.
|x1 −x2 |≤δ
Let M be the class of all continuous 2π-periodic functions f for which
there exists a partitioning of the segment [−π, π] by the points −π =
x1 (f ) < ... < xm+1 (f ) = π such that f is convex, or convex, or linear,
on each segment [xk (f ), xk+1 (f )], k = 1, ..., m.
The Fourier coefficients of a function f with respect to the trigonometric
system will be denoted by an = an (f ), bn = bn (f ).
Problems parting to the absolute convergence of Fourier series have been
studied quite completely ([4], [5], [6], [7]).
This paper deals with one problem of the absolute convergence of trigonometric Fourier series of a function from class M .
The following fact is well known: the Fourier series of any 2π-periodic
continuous even function, convex on [−π, π], converges absolutely (see [7]).
We have obtained the following result:
Theorem 1. If f ∈ M , then for absolute convergence of the Fourier series
of the function f it is necessary and sufficient that
¶
µ
¶¯
∞ ¯ µ
X
¯1
¯
1
1
¯f xk (f ) +
¯ < ∞, k = 1, ..., n.
− f xk (f ) −
¯
¯n
n
n
n=1
Proof. Let f1 , f2 , f be continuous 2π-periodic functions defined as follows:
f1 (x) = 0 for x ∈ [−π, 0], f1 is convex or concave on a segment (0, 1), linear
Some notes on absolute convergence of Fourier series
47
on [1, π]; f2 (−π) = 0, f2 is linear on [−π, −1], f2 is convex or concave on
(−1, 0], f2 (x) = 0 for x ∈ (0, π]; f = f1 + f2 .
The theorem will be proved by showing that for Fourier series of f to
converge it is necessary and sufficient that
µ
¶¯
∞ ¯ µ ¶
X
¯
¯ 1
1
1
¯f
¯ · < +∞.
−f −
¯
¯ n
n
n
n=1
This follows from Wiener, s theorem and from the following facts: If the
function f is convex or concave on segment [a, b], then f is a lipschitz
function on any segment [c, d] entirely lying inside [a, b], and the Fourier
series of the functions f (x) and f (x + c) simultaneously converge or diverge
absolutely.
The function f1 is convex on [0, π] and continuous, which means that
it is absolutely continuous so that one can apply integration by parts and
Newton - Leibnitz formulas to obtain an (f ) = an (f1 ) + an (f2 ).
1
an (f1 ) =
π
Zπ
1
f1 (t) cos nt dt =
π
−π

1
sin nt ¯¯π
1
= f1 (t)
·
−π −
n
t
πn
Zπ
f1 (t)d
−π
Zπ

1
f10 (t) sin nt dt = −
nπ
−π
1
=−
πn
Z0
Zπ
f10 (t) sin nt dt =
−π
1
f10 (t) sin nt dt −
πn
−π
1
=−
πn
sin nt
=
n
Zπ
f10 (t) sin nt dt =
0
Z1
Zπ
Z1/n
1
1
0
0
f1 (t) sin nt dt −
f10 (t) sin nt dt.
f1 (t) sin nt dt −
πn
πn
0
1/n
1
48
Amelia Bucur
The derivative f 0 of the convex or concave function f is monotonous and
therefore, applying the second theorem of the mean value, we obtain:
¯
¯
¯ Z1
¯
¯
¯
¯
¯
0
¯ f1 (t) sin nt dt¯ =
¯
¯
¯1/n
¯
¯
¯
¯
¯
¶ Z²
Z1
¯
¯ 1 µ1
1
¯
¯
0
0
= ¯ f1
+0
sin nt dt +
f1 (t − 0) sin nt dt¯ ≤
¯
¯ πn
n
πn
¯
¯
²
1/n
¯ µ
¶¯
¯
1 ¯¯ 0 1
1
1
≤
+ 0 ¯¯ + 2 |f10 (1 − 0)| with < ² < 1.
2 ¯ f1
n
n
πn
πn
Wherever we come across expressions of the form f 0 (x±0), the left and right
limits are considered with respect to the set at whose points the derivative
f 0 exists.
For the convex (concave) function f we have the relation
µ
f (x3 ) − f (x2 )
f (x2 ) − f (x1 )
≥ f 0 (x2 ± 0) ≥
x2 − x1
x3 − x2
f (x2 ) − f (x1 )
f (x3 ) − f (x2 )
≥ f 0 (x2 ± 0) ≥
x2 − x1
x3 − x2
¶
where x1 < x2 < x3 . Therefore
¶
µ
µ ¶
1
1
¯ µ
¶¯ f1
− f1
¯
¯ 0 1
n
n+1
¯f1
≤
6= 0 ¯¯ ≤
¯
1
1
n
−
n n+1
µ µ ¶
µ
¶¶
1
1
2
≤ (n + 1) f1
− f1 − f1
.
n
n+1
Hence
µ
µ ¶
¶¯
µ
¶¶
∞ µ
∞ ¯
X
X
¯ 0 1
¯ 1
1
1
¯f1
¯
f1
+0 ¯ 2 ≤2
− f1
< +∞.
¯
n
n
n+1
n
n=1
n=1
Some notes on absolute convergence of Fourier series
∞
P
n=1
49
Since f1 is linear on the segment [², π], we have |f10 (1 − 0)| ≤ β and
∞
P
|f10 (1 − 0)|/n2 =
β/n2 < ∞.
n=1
0
The function
¯ π f1 is linear
¯ on the segment [1, π], i.e. f1 (t) = const = β,
¯R
¯
β
so that 1/n ¯¯ f10 (t) sin nt¯¯ ≤ 2 .
n
1
1/n
∞
1 R f 0 (t) sin nt dt + γ , where P |γ | < +∞.
Finally, an (f1 ) = − nπ
n
n
1
n=1
0
1
If we introduce the notation In = − πn
an (f1 ) = In + γn , In = an (fr ) − γn .
1/n
R
0
f10 (t) sin nt dt, then
Since the function f1 has a bounded variation, we have
∞
f1 (x) =
a0 (f1 ) X
+
[an (f1 ) cos nx + bn (f1 ) sin nx].
2
n=1
By substituting here x = 0 we obtain
∞
P
∞
P
an (f1 ) < ∞. Therefore
∞
P
In =
n=1
n=1
(an (f1 ) − γn ) < ∞.
n=1
One can easily verify that the values In do not change their sign for
∞
P
(In ) < +∞. Since |an (f1 )| ≤ |In | + |γn |, we
sufficiently large n. Thus
obtain
∞
P
n=1
n=1
|an (f1 )| < +∞.
In a similar manner we shall show that
∞
P
n=1
|an (f2 )| < ∞. We have
∞
P
|an (f )| = |an (f1 ) + an (f2 )| ≤ |an (f1 )| + |an (f2 )| and
|an (f )| < +∞.
n=1
Now we consider the coefficients bn (f ). We have bn (f ) = bn (f1 ) + bn (f2 ).
1
bn (f1 ) =
π
Zπ
−1
f1 (t) sin nt dt =
π
−π
1
cos nt ¯¯π
1
= − f1 (t)
−π +
π
n
πn
Zπ
f1 (t)d
cos nt
=
n
−π
Zπ
−π
1
f10 (t) cos nt dt =
πn
Zπ
f10 (t) cos nt dt =
−π
50
Amelia Bucur
1
=
πn
Zπ
f10 (t) cos nt
1
dt = +
πn
0
Z1/n
Z1
1
0
f1 (t) cos nt dt +
f10 (t) cos nt dt+
πn
0
1
+
πn
1/n
Zπ
f10 (t) cos nt dt.
1
The function f1 is linear on the segment [1, π], i.e. f10 (t) = const = β,
so that
¯
¯ π
¯
¯Z
¯
¯
1¯
0
¯≤ β
f
cos
nt
dt
1
¯ n2
¯
n¯
¯
1
1 < ² < 1):
Again applying the theorem of the mean, we obtain (with n
¯
¯
¯
¯ Z1
¯1
¯
¯
¯
0
f
cos
nt
dt
¯
¯=
1
¯n
¯
¯ 1/n
¯
¯
¯
¯ µ
¯
1
²
¶
Z
Z
¯
1 ¯¯ 0 1
¯
cos nt dt + f10 (1 − 0) cos nt dt¯ ≤
= ¯f1
+0
¯
n¯
n
¯
¯
²
1/n
¯ µ
¶¯
¯
1
1 ¯¯ 0 1
+ 0 ¯¯ + 2 |f10 (1 − 0)| < +∞.
≤ 2 ¯f1
n
n
n
1/n
∞
R 0
P
1
Therefore bn (f1 ) = + πn
f1 (t) cos nt dt + δn ,
|δn | < +∞. But,
n=1
0
1
πn
Z1/n
Z1/n
Z1/n
1
1
f10 (t) cos nt dt = −
f10 (t)(1 − cos nt − 1)dt =
f10 (t)dt−
πn
πn
0
1
−
πn
0
0
µ ¶
Z1/n
Z1/n
1
1
nt
1
f1
f10 (t)(1 − cos nt)dt =
−
f10 (t) · 2 sin2 dt,
πn
n
πn
2
0
0
Some notes on absolute convergence of Fourier series
51
¯
¯
¯
¯
¯
¯
Z1/n
Z1/n
¯ 1
¯
¯ 2 nt ¯
2
nt
0
2
0
¯
¯ = 2|In |.
dt¯ ≤
f1 2 sin
|f1 (t)| · ¯¯sin
¯ πn
2 ¯¯ πn
2¯
¯
0
As we have seen, above
0
∞
P
|In | < ∞ and therefore
n=1
1
f1
bn (f1 ) =
πn
where
∞
P
n=1
µ ¶
µ ¶
1
1
1
f
− Cn =
− Cn ,
n
πn
n
|Cn | < +∞.
In a similar manner it will be shown that
¶
µ
1
1
bn (f2 ) =
f −
+ Pn ,
πn
n
where
∞
P
n=1
|Pn | < +∞.
Since bn (f ) = bn (f1 ) + bn (f2 ), we have
½ µ ¶
µ
¶¾
∞
X
1
1
1
0
bn (f ) = bn (f1 ) + bn (f2 ) +
f
+f −
+ γn ,
|γn0 | < ∞,
πn
n
n
n=1
and the proof is completely.
References
[1] Bucur, A., On some series of functions, Italian Journal of Pure and
Applied Mathematics (accepted for publication in 7.10.2002).
[2] Karchava, J. T., On absolute convergence of Fourier series, Georgian
Mathematical Journal, vol. 4, 1997, 333 - 340.
[3] Leladze, D., On some properties of multiple conjugate trigonometric
series, Georgian mathematical Journal, 1, nr.3, 1994, 287 - 302.
52
Amelia Bucur
[4] Nikolshy, S. M., A course of Mathematical Analysis, vol. 1, 2, Mir
Publishers Moscow, 1975, English translation, Mir Publishers, 1981.
[5] Roşculeţ, M., Serii trigonometrice şi aplicaţii, Ed. Academiei Române,
Bucureşti, 1986 (in Romanian).
[6] Vescan, A., Sumabilitatea seriilor, Editura Tehnică Bucureşti, 1965 (in
Romanian).
[7] Zygmund, A., Trigonometric series, v. 1, Cambridge University Press,
1959; Russian translation: Mir, Moscow, 1965.
Department of Mathematics
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, nr. 5 - 7
550012 - Sibiu, România
E-mail: amelia.bucur@ulbsibiu.ro