Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
28 (2012), 31–35
www.emis.de/journals
ISSN 1786-0091
APPROXIMATION PROPERTIES OF PARTIAL SUMS OF
FOURIER SERIES
T. KARCHAVA
Abstract. In this paper we find class of functions for which the Lebesgue
estimate can be improved.
Let C ([0, 2π]) denote the space of continuous function f with period 2π. If
f ∈ C ([0, 2π]) then the function
ωp (δ, f ) := sup sup |∆p (x; h, f )| , ω1 (δ, f ) := ω (δ, f )
x
|h|≤δ
is called the modulus of continuity of the function f , where
∆1 (x; h, f ) := f (x + h) − f (x) ,
∆p+1 (x; h, f ) = ∆p (x + h; h, f ) − ∆p (x; h, f ) .
Let Sn (f, x) be the nth partial sums of the trigonometric Fourier series of
the function f .
The estimation of Lebesgue (see [3], or [1]) is well-known
1
kf − Sn (f )k ≤ cω
, f log (n + 2) .
n
Generalization of this estimation were studied by Chanturia [2], Oskolkov [6],
Karchava [4, 5]. There arises a question: for what subclasses of C ([0, 2π]) the
Lebesgue estimate can be improved?
We prove that the following are true
Theorem 1. Let f ∈ C ([0, 2π]). Then
X ωk
[log n]
kf − Sn (f )kC ≤ c
k=1
1
,f
n
2k
.
2010 Mathematics Subject Classification. 41A10.
Key words and phrases. Fourier series, modulus of continuity.
31
32
T. KARCHAVA
Corollary 1. Let f ∈ C ([0, 2π]). Then
kf − Sn (f )kC ≤ c
max
1≤k≤[log log n]
Corollary 2. Let f ∈ C ([0, 2π]) and ωk
1
,f
n
ωk
1
,f
n
.
≤ M . Then
kf − Sn (f )kC → 0 as n → ∞.
2k
Corollary 3. Let f ∈ C ([0, 2π]) and ωk n1 , f ≤ l(k)
, where
∞
X
1
< ∞.
l
(k)
k=1
Then
kf − Sn (f )kC → 0 as n → ∞.
Proof. Let Tn (x) be Vale-Poisson polynomial which provides best approximation of function f in the space C ([0, 2π]) (see [3]), in particular
Zπ
Zπ
1
1
f (x + t) Vn (t) dt,
|Vn (t)| dt ≤ 1.
Tn (x) =
π
π
−π
−π
Set
g (t) := f (x + t) + f (x − t) − 2f (x) − Tn (x + t) − Tn (x − t) + 2Tn (x) .
It is evident that
ωp (δ, Tn ) ≤ cωp (δ, f )
and
n
X
(−1)i
(−1)k+1
=
+ Pk + Qn ,
i
2k
i=k+1
(1)
Pk ≤
c
,
k2
Qn ≤
c
.
n
We write
Zπ
Zπ
Zπ
sin nt
sin nt
(2)
g (t) Dn (t) dt = g (t) Dn (t) −
dt + g (t)
dt.
t
t
0
0
0
Since
Dn (t) −
sin nt
t
is bounded function and
|g (t)| ≤ 4En (f ) .
Hence
(3)
π
Z
g (t) Dn (t) − sin nt dt ≤ cEn (f ) ,
t
0
where En (f ) is best approximation of the function f .
APPROXIMATION PROPERTIS OF OF PARTIAL SUMS
On the other hand,
π/n
Z
Zπ/n
sin
nt
sin nt
g (t)
(4)
dt ≤ kgkc
dt ≤ cEn (f ) .
t
t
0
0
Combining (2)-(4) we have
Zπ
(5)
Zπ
g (t) Dn (t) dt =
0
g (t)
sin nt
dt + γ,
t
0
where
|γ| ≤ cEn (f ) .
It is evident that
Zπ
n−1 π(k+1)/n
Z
sin nt X
sin nt dt = dt
(6)
g (t)
g (t)
t
t
π/n
k=1 πk/n
n−1 Zπ/n k
X
(−1)
sin
nt
πk
dt
= g t+
n
t + πk/n
k=1
0
n−1 Zπ k
X
(−1)
sin
u
u
+
πk
du
= g
n
u + πk
k=1
0
n−1 k
X
(−1)
sin
u
u
+
πk
≤ π max g
0≤u≤π n
u
+
πk
k=1
n−1 X
(−1)k u0 + πk
≤ π
g
n
u0 + πk k=1
!
n−1 X
u0 + πk
(−1)k (−1)k
g
−
≤ π
n
u0 + πk
πk
k=1
n−1 X
u0 + πk (−1)k g
+π n
πk k=1
= I + II.
where
n−1 n−1 k k X
X
u
+
πk
u
+
πk
(−1)
(−1)
0
g
g
= sup .
n
u0 + πk n
u
+
πk
u
k=1
k=1
33
34
T. KARCHAVA
It is clear that
(7)
n−1
X
1
I ≤ kgkc
= O (En (f )) .
k2
k=1
Set
u0 + πk
.
n
Using Abel transformation and (1), for II we obtain
n−1
k
X
(−1)
II = g (uk )
πk k=1
n−1
n−2
n−1
X
X
X
(−1)i (−1)i ≤
(g (uk+1 ) − g (uk ))
+ g (u1 )
i i i=1
k=1
i=k+1
! n−2
n−1
k+1
i
X
X
(−1)
(−1)
≤
(g (uk+1 ) − g (uk ))
+ Pk + Qn + g (u1 )
2k
i
i=1
k=1
n−2
X
(−1)k+1 (g (uk+1 ) − g (uk ))
≤
+ O (En (f ))
2k
k=1
n−2
X
π (−1)k+1 =
∆1 uk ; , g
+ O (En (f ))
n
2k k=1
n−2
π (−1)k+1 1 X
∆1 u k ; , g
≤ + γ1
2 k=1
n
k
n−3
π (−1)k+1 1 X
∆2 uk ; , g
≤ 2
+ γ1 + γ2
2 k=1
n
k
p
n−3
k+1 X
π
(−1) X
1 ∆ p uk ; , g
γk
≤ ··· ≤ p +
2 k=1
n
2k k=1
uk :=
where
γk <
Consequently,
(8)
II ≤
ωp
1
,f
n
p
2
Since
ωi
1
,f
n
i
2
ωk
1
,f
n
2k
≤
log n +
.
p
X
ωk
k=1
ωj
1
,f
n
j
2
1
,f
n
k
2
(i ≥ j),
.
APPROXIMATION PROPERTIS OF OF PARTIAL SUMS
we can write
(9)
ωp
1
,f
n
p
2
p
≤
p
X
ωk
k=1
1
,f
n
k
2
35
.
Set p = [log n]. Then from (8) and (9) we obtain
[log n]
X ωk 1 , f
n
(10)
II ≤
.
k
2
k=1
Combining (6), (7) and (10) complete the proof of Theorem 1.
References
[1] N. K. Bari. Trigonometric series. Nauka, Moscow, 1961. In Russian.
[2] Z. A. Chanturiya. On uniform convergence of Fourier series. Mat. Sb. (N.S.), 100:534–
554, 1976. In Russian.
[3] V. K. Dzyadyk. Introduction into the Theory of Uniform Approximation of Functions by
Polynomials. Nauka, Moscow, 1977. In Russian.
[4] T. Karchava. On the convergence of Fourier series. Rep. Enlarged Sess. Semin. I. Vekua
Inst. Appl. Math., 3:93–95, 1988. In Russian.
[5] T. Karchava. Approximation properties of partial sums of Fourier series. Acta Math.
Acad. Paedagog. Nyházi. (N.S.), 21(2):155–160 (electronic), 2005.
[6] K. I. Oskolkov. The sharpness of the Lebesgue estimate for the approximation of functions with prescribed modulus of continuity by Fourier sums. Trudy Mat. Inst. Steklov.,
112:337–345, 389, 1971.
Received March 6, 2011.
Department of Mathematics and Computer Science,
Sokhumi University, Str. Jikia 10,
0128, Georgia.