Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
21 (2005), 161–167
www.emis.de/journals
ISSN 1786-0091
LOGARITHMIC SUMMABILITY OF FOURIER SERIES
G. TKEBUCHAVA
Abstract. A set of regular summations logarithmic methods is introduced.
This set includes Riesz and Nörlund logarithmic methods as limit cases. The
application to logarithmic summability of Fourier series of continuous and
integrable functions are given. The kernels of these logarithmic methods for
trigonometric system are estimated.
1. Introduction
In the literature it is known Riesz and Nörlund logarithmic summation methods
(see [H, HR, Z]). Applications of these methods to Fourier analysis are investigated
by many authors (see for example [S, Zh, Y, MS, GG, GT]. We construct the set of
logarithmic summation methods which in particular contains both mentioned methods. We study the application of these methods to Fourier series. The estimates of
kernels and Lebesgue functions in trigonometric case are obtained. Necessary and
sufficient condition which guarantee such logarithmic summability of Fourier series
for continuous and integrable functions in corresponding metric are established.
2. Construction of logarithmic summation methods
For any integers m and n such that 0 ≤ m ≤ n we put
(m−1
)
n
X Dk (x)
X
1
Dk (x)
(1)
Fm,n (x) ≡
+ Dm (x) +
l(m, n)
m−k+1
k−m+1
k=0
k=m+1
where
(2)
Dk (x) ≡
sin(k + 21 )x
2 sin x2
is Dirichlet kernel and
(3)
l(m, n) ≡
m−1
X
k=0
n
X
1
1
+1+
.
m−k+1
k−m+1
k=m+1
2000 Mathematics Subject Classification. 42A24, 40G05.
Key words and phrases. Fourier series, logarithmic summability, spaces of continuous and
integrable functions.
161
162
G. TKEBUCHAVA
Here and further we assume that if q < p then
that l(m, n) ³ ln(n + 2). We call
Pq
k=p
dk = 0 for any dk . It is clear
tm,n f ≡ Fm,n ∗ f
the logarithmic means of Fourier series of function f ∈ L1 [−π, π].
∞
∞
If we replace in (1) the sequence {Dn (x)}n=0 by arbitrary sequence {Sn }n=0 ,
then we define such logarithmic means in general case. It is easy to see that for each
fixed sequence of integers {mn } under condition 0 ≤ mn ≤ n these means tmn ,n
generate a regular logarithmic summation method. The set of all such methods
in particular includes Riesz (for mn = 0) and Nörland (for mn = n) logarithmic
methods. In the sequel C, Ci denote absolute positive constants.
3. Logarithmic means and their kernels
In following proposition we estimate the kernels Fm,n .
Theorem 1. There exist positive constants Ci , i = 1, 2, 0 < C1 ≤ C2 such that for
any integers m and n, 0 ≤ m ≤ n holds
¾
¾
½
½
ln2 (m + 2)
ln2 (m + 2)
1
(4)
C1 1 +
≤ kFm,n kL [−π,π] ≤ C2 1 +
.
ln(n + 2)
ln(n + 2)
In particular from Theorem 1 we obtain well-known results.
Corollary 1 (see [GT, Y]). For any n ≥ 0 we have
kFn,n kL1 [−π,π] ³ C ln(n + 2)
kF0,n kL1 [−π,π] ³ C.
Theorem 2. ³
The following
´ conditions are equivalent
√
a) mn = O exp ln n as n → ∞.
b) ktm,n f − f kC[−π,π] → 0 as n → ∞ ∀f ∈ C [−π, π],
c) ktm,n f − f kL1 [−π,π] → 0 as n → ∞ ∀f ∈ L1 [−π, π].
Theorem 2 is corollary of Theorem 1 because the condition a) is equivalent to
the boundedness of Lebesgue constants for method tmn ,n what is sufficient and
necessary condition for b) and c) (see [HR], Ch. 5.)
We can construct a logarithmic summation method with given possible growth
of logarithmic means. In particular we have
p
Theorem 3. For any τ (n) ∈ [1, ln n] and mn = [exp{ τ (n) ln n}] a logarithmic
means tmn ,n is such that
ktmn ,n f kC[−π,π] ≤ Cτ (n)kf kC[−π,π] .
Another corollaries of Theorem 1 for divergence of Fourier series of continuous
function one can obtain by well-known way using the uniform boundedness principle
(see [E], Ch. 10.3.2.)
LOGARITHMIC SUMMABILITY OF FOURIER SERIES
163
4. Proof of Theorem 1
Proof of Theorem 1. It is sufficient to prove the inequality (4) for n > 1. First note
that


m+1
n−m+1

X
X
Dm+1−j (x)
Dj+m−1 (x) 
1
(5) Fm,n (x) =
+ Dm (x) +

l(m, n) 
j
j
j=2
and
(6)
"Z
kFm,n kL1 = 2
Since |Dk (x)| ≤ k +
(7)
1
n+2
j=2
Z
|Fm,n (x)|dx +
0
1
2
#
π
1
n+2
|Fm,n (x)|dx ≡ 2(A1 + A2 ).
∀k ∀x ∈ [−π, π], then (see(1) , (2), (3) , (6))
1
Z n+2
A1 =
|Fm,n (x)|dx ≤ 1.
0
Further from (5) we get

 sin ¡m + 1 + 1 ¢ x m+1
X cos jx
1
2
Fm,n (x) =
x
l(m, n) 
2 sin 2
j
j=2
−
(8)
¡
¢ m+1
cos m + 1 + 12 x X sin jx
2 sin x2
j
j=2
n−m+1
sin(m − 1 + 12 )x X cos jx
2 sin x2
j
j=2

n−m+1
cos(m − 1 + 12 )x X sin jx 
.
+
2 sin x2
j 
+Dm (x) +
j=2
We have for A2 the following decomposition

¯
¯
¯
Z π
Z π ¯¯ sin(m + 1 + 1 )x ¯¯ ¯¯m+1
X cos jx ¯
1
2
¯
¯
¯
¯ dx
A2 =
|Fm,n (x)|dx ≤
x
¯¯
¯
1
 1 ¯
l(m,
n)
2
sin
j
¯
¯
2
n+2
n+2
j=2
¯
¯
¯¯
¯
Z π ¯
X sin jx ¯
¯ cos(m + 1 + 12 )x ¯ ¯m+1
¯
¯
¯ dx
¯
+
x
¯
¯¯
¯
1
2
sin
j
¯
¯
2
n+2
j=2
¯
¯
¯¯
¯
Z π ¯
X cos jx ¯
¯ sin(m − 1 + 12 )x ¯ ¯n−m+1
(9)
¯
¯
¯
¯ dx
+
x
¯
¯¯
¯
1
2
sin
j
¯
¯
2
n+2
j=2
¯
¯

¯¯
¯
Z π
Z π ¯

X sin jx ¯
¯ cos(m − 1 + 12 )x ¯ ¯n−m+1
¯¯
¯
¯ dx +
|D
(x)|
dx
+
m
¯¯
¯
1
1

2 sin x2
j ¯¯
¯ j=2
n+2
n+2
≡
1
{A2,1 + A2,2 + A2,3 + A2,4 + A2,5 } .
l(m, n)
By convention if m = 0 then A2,1 = A2,2 = 0 and if m = n then A2,3 = A2,4 = 0.
164
G. TKEBUCHAVA
Since for N ≥ 1 and−π ≤ x ≤ π we have the estimate ([Z], Ch.5.)
(10)
|
N
X
sin jx
j=1
then (see (9))
Z
(11)
A2,2 =
π
1
n+2
j
| ≤ C,
¯
¯
¯
¯¯X
¯
Z π
¯ cos(m + 1 + 12 )x ¯ ¯m+1
¯
sin
jx
¯¯
¯
¯ dx ≤ C
¯¯
¯
¯
1
2 sin x2
¯ j=2 j ¯
n+2
¯
¯
¯ 1 ¯
¯
¯
¯ 2 sin x ¯ dx
2
≤ C ln(n + 2)
and
Z
(12)
A2,4 =
π
1
n+2
¯
¯
¯¯ X
¯
¯
Z π
¯ cos(m − 1 + 12 )x ¯ ¯n−m+1
sin jx ¯¯
¯
¯¯
dx
≤
C
¯
¯¯
1
2 sin x2
j ¯¯
¯ j=2
n+2
¯
¯
¯ 1 ¯
¯
¯
¯ 2 sin x ¯ dx
2
≤ C ln(n + 2).
Moreover
Z
(13)
A2,1 =
π
1
n+2
¯
¯
¯
¯¯
¯
m+1
X 1Z π
X cos jx ¯
¯ sin(m + 1 + 21 )x ¯ ¯m+1
¯
¯¯
¯ dx ≤ C
|Dm+1 (x)|dx
¯
¯¯
¯
1
2 sin x2
j n+2
¯ j=2 j ¯
j=2
≤ C ln(m + 2)kDm+1 kL1 [−π,π] ≤ C ln2 (m + 2).
Now we estimate A2,3
¯
¯
¯
¯¯
¯
X cos jx ¯
¯ sin(m − 1 + 21 )x ¯ ¯n−m+1
¯
¯
¯ dx
¯
=
x
¯
¯¯
¯
1
2
sin
j
¯
¯
2
n+2
j=2
¯
¯
¯
¯¯
¯
1
Z m+2
X cos jx ¯
¯ sin(m − 1 + 21 )x ¯ ¯n−m+1
¯
¯
¯ dx
¯
=
x
¯
¯¯
¯
1
2
sin
j
¯
¯
2
n+2
j=2
¯
¯
¯¯
¯
Z π ¯
X cos jx ¯
¯ sin(m − 1 + 12 )x ¯ ¯n−m+1
¯
¯
¯
¯ dx = A2,3,1 + A2,3,2 .
+
x
¯
¯¯
¯
1
2
sin
j
¯
¯
2
m+2
j=2
Z
A2,3
(14)
π
Since (see [Z], Ch.5)
¯
¯
¯
¯X
¯ N cos jx ¯
¯ ≤ C + ln 1
¯
¯
¯
x
¯ j=1 j ¯
then
(15)
¯
¯
¯ s
¯
¯X cos jx ¯
¯
¯ ≤ C + ln(m + 2)
¯
¯
¯ j=2 j ¯
and we obtain the estimates
Z
(16)
∀x ∈ (0, π]
A2,3,1 ≤ (m + 2)
1
m+2
1
n+2
x∈[
∀N ≥ 1
1
, π]
m+2
µ
¶
1
C + ln
dx ≤ C + ln(m + 2)
x
LOGARITHMIC SUMMABILITY OF FOURIER SERIES
and
(17)
Z
A2,3,2 ≤ (C + ln(m + 2))
Since
(18)
Z
A2,5 =
π
1
m+2
165
¡
¢
1
dx ≤ C ln2 (m + 2) + ln(m + 2) .
2 sin x2
π
1
n+2
|Dm (x)| dx ≤ kDm kL1 [−π,π] ≤ C ln(m + 2),
then we have from (9), (11), (12), (13), (14), (16), (17), (18)
(19)
A2 ≤ C
ln2 (m + 2)
+ C.
ln(n + 2)
Finally (see (6), (7), (19)) we obtain the right estimate of (4).
Now we prove the left side of (4) From (8) it is evident that
l(m, n)Fm,n (x) = Dm (x) +
n−m+1
cos(m − 1 + 21 )x X sin jx
2 sin x2
j
j=2
−
m+1
m+1
cos(m + 1 + 12 )x X sin jx cos(m + 12 )x sin x X cos jx
+
2 sin x2
j
2 sin x2
j
j=2
j=2
−
n−m+1
m+1
X cos jx
cos(m + 12 )x sin x X cos jx
+
D
(x)
cos
x
m
x
2 sin 2
j
j
j=2
j=2
(20)
+ Dm (x) cos x
n−m+1
X
j=2
7
cos jx X
=
Bk .
j
k=1
By virtue of (10) for all x ∈ [0, π2 ] we obtain the following estimate
(21)
|Bk | ≤
C
x
k = 1, 2, 3.
1
By virtue of estimate (15) we get for x ∈ [ m+2
, π]
(22)
|Bk | ≤ C ln (m + 2)
k = 4, 5.
Since for s ≥ 2
s
X
cos jx
(23)
j=2
j
s−2
X
=
j=1
+
sin2 j+1
2
2 x
2
j(j + 1)(j + 2) 2 sin x2
sin2 sx
1
1
3
2
+ Ds (x) − − cos x
s(s − 1) 2 sin2 x2
s
4
then for m ≥ 3
B6 = Dm (x) cos x
m−1
X
j=1
(24)
j+1
sin2 2 x Dm (x) cos x sin2 m+1
1
2 x
+
2
2
x
j(j + 1)(j + 2) sin 2
m(m + 1) 2 sin x2
5
+ Dm (x) cos x
X
1
3
Dm+1 (x) − Dm (x) cos x − Dm (x) cos2 x =
B6,i .
m+1
4
i=1
166
G. TKEBUCHAVA
It is clear that for x ∈ [0, π] by analogical assertions it follows for j = 2, . . . , 5
C
(25)
|B6,j | ≤ .
x
√
Since for 0 ≤ k ≤ m + 1 and
½
¾
1
1
π 1
π
1
<x< √
x ∈ Jm = x : sin(m + )x ≥ ,
2
2
2 m+1
2 m+1
follows 0 ≤ kx ≤
π
2
and sin kx ≥
B6,1 ≥ Dm (x) cos x
(26)
≥C
2
π kx,
then we obtain
X
√
1≤k≤ m+1
sin(m + 12 )x
2 sin x2
sin2 k+1
1
2 x
2
k(k + 1)(k + 2) sin x2
X
√
1≤k≤ m+1
1
C
≥ ln(m + 2),
k
x
x ∈ Jm .
We note that if m = n then B7 = 0 and if m = n − 1 then B7 ≤
For m < n − 1 we have (see (23))
B7 = Dm (x) cos x
n−m−1
X
j=1
C
x
∀x ∈ [−π, π].
sin2 j+1
2
2 x
j(j + 1)(j + 2) 2 sin2 x2
x
sin2 n−m+1
Dm (x) cos x
2
2
x
(n − m + 1)(n − m) 2 sin 2
1
+ Dm (x) cos x
Dm−n+1 (x)
n−m+1
5
X
3
− Dm (x) cos x − Dm (x) cos2 x =
B7,i
4
i=1
+
(27)
and we obtain the same estimates for B7,i , i = 2, . . . , 5 as for B6,i , i = 2, . . . , 5. We
note also that B7,1 > 0. Thus for x ∈ Jm and m big enough we have (see (21-22),
(24-27))
½
¾
1
ln(m + 2) C
ln(m + 2)
|Fm,n (x)| ≥
C
− − C ln(m + 2) ≥ C
.
l(m, n)
x
x
xl(m, n)
It imply that for m big enough
Z
ln2 (m + 2)
(28)
kFm,n kL1 [−π,π] ≥
|Fm,n (x)|dx ≥ C
ln (n + 2)
Jm
and since for all m and n holds
(29)
kFm,n kL1 [−π,π]
¯Z
¯
≥ ¯¯
2π
0
¯
¯
Fm,n (x)dx¯¯ = 1
then from (28) and (29) follows the left side estimate in (4).
¤
References
[E] R. Edwards. Fourier series. A modern introduction. Vol. 1. 2nd ed., volume 64 of Graduate
Textes in Mathematics. Springer-Verlag, New York, Heidelberg. Berlin, 1979.
[GG] G. Gát and U. Goginava. Uniform and L-convergence of logarithmic means of Walsh-Fourier
series. to appear in Acta Math. Sinica.
[GT] U. Goginava and G. Tkebuchava. Convergence of logarithmic means of Fourier series. submitted.
LOGARITHMIC SUMMABILITY OF FOURIER SERIES
167
[H] G. Hardy. Divergent series. Oxford: At the Clarendon Press, 1949.
[HR] G. Hardy and W. Rogosinski. Fourier series. 3rd ed., volume 38 of Cambridge Tracts in
Mathematics and Mathematical Physics. Cambridge: At the University Press, 1956.
[MS] F. Móricz and A. Siddiqi. Approximation by Nörlund means of Walsh-Fourier series. J.
Approximation Theory, 70(3):375–389, 1992.
[S] O. Szász. On the logarithmic means of rearranged partial sums of a Fourier series. Bull. Am.
Math. Soc., 48:705–711, 1942.
[Y] K. Yabuta. Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz’s
logarithmic means. Tohoku Math. J., II. Ser., 22:117–129, 1970.
[Zh] L. Zhizhiashvili. Trigonometric Fourier series and their conjugates. Mathematics and its
Applications (Dordrecht). Kluwer Academic Publishers, 1996.
[Z] A. Zygmund. Trigonometric series. Vol. 1, 2nd ed. Cambridge: At the University Press,
1959.
Received March 16, 2005.
Department of Mechanics and Mathematics,
Tbilisi State University,
Chavchavadze str. 1, Tbilisi 0128,
Georgia
E-mail address: george tkebuchava@hotmail.com