Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
21 (2005), 155–160
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 ) .
Denote by Lipα the class of function f ∈ C ([0, 2π]) for which ω (δ, f ) ≤ c (f ) δ α
and let Sn (f, x) be the n-th partial sum of the trigonometric Fourier series of the
function f .
The estimation of Lebesgue (see [Dz, p. 116], or [Ba, Ch. 1])is well known
µ
¶
1
kf − Sn (f )kC ≤ cω
, f log (n + 2) .
n
Generalization of this estimation were studied by Chanturia [Ch], Oskolkov [Os],
Karchava [Ka]. The questions devoted to estimation the uniform deviation of f
from its partial Fourier sums with respect to the Walsh, Vilenkin (bounded and
unbounded case) systems were discussed by Fine [Fi], Onnewer [On] Tevzadze [Te],
Gát [Gá].
In this paper we consider the following characteristic of function f
ϕp (n, δ; f ) = sup sup
n
X
x |h|≤δ
i=1
|∆p (xi ; h, f )| ,
where xi = x + (i − 1) h.
There arises a question: for what subclasses of classes of C ([0, 2π]) the Lebesgue
estimate can be improved?
We prove that the following are true
2000 Mathematics Subject Classification. 41A10.
Key words and phrases. Fourier series, Lebesgue estimation, modulus of continuity.
155
156
T. KARCHAVA
Theorem 1. Let f ∈ C ([0, 2π]). Then
kf − Sn (f )kC ≤ c (p)
n
X
ϕp (k; π/n, f )
k2
k=1
.
Corollary 1. Let the function f has a finite number of intervals of monotonicity
on [0, 2π] and f ∈ Lip α, 0 < α < 1. Then
c (f, α)
.
nα
Corollary 2. Let the function f has a finite number of intervals of convexity or
concavity on [0, 2π], then
¶
µ
1
kf − Sn (f )kC ≤ c (f ) ω
;f .
n
kf − Sn (f )kC ≤
Corollary 3. Let the function f has a finite number of intervals of monotonicity
on [0, 2π]. Then
Zπ
1
ω (t, f )
kf − Sn (f )kC ≤ c (f )
dt.
n
t2
1/n
Proof of Theorem 1. Let Tn (x) be Vale-Poisson polynomial which provides best
approximation of function f in the space C ([0, 2π]) (see [Dz]), in particular
Zπ
Zπ
1
1
Tn (x) =
f (x + t) Vn (t) dt,
|Vn (t)| dt ≤ 1.
π
π
−π
−π
Denote
g (t) := f (x + t) + f (x − t) − 2f (x) − Tn (x + t) − Tn (x − t) + 2Tn (x) .
It evident that
ϕp (k, δ, Tn ) ≤ cϕp (k, δ, f )
and
n
i
k+1
X
(−1)
(−1)
c
c
=
+ Pk + Qn where Pk ≤ 2 , Qn ≤ .
i
2k
k
n
(1)
i=k+1
We write
µ
¶
Zπ
Zπ
Zπ
sin nt
sin nt
g (t) Dn (t) dt = g (t) Dn (t) −
dt + g (t)
dt.
(2)
t
t
0
0
0
Since
Dn (t) −
is bounded function and
sin nt
t
|g (t)| ≤ 4En (f ) ,
we have
(3)
¯
¯ π
¯Z
¶ ¯
µ
¯
¯
sin
nt
¯ g (t) Dn (t) −
dt¯¯ ≤ cEn (f ) .
¯
t
¯
¯
0
where En (f ) is best approximation of function f .
APPROXIMATION PROPERTIES OF PARTIAL SUMS OF FOURIER SERIES
On the other hand
¯
¯
¯Z
¯
π/n
Z
¯ π/n
sin nt ¯¯
sin nt
¯
(4)
g (t)
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)
0
sin nt
dt + γ,
t
where
γ ≤ cEn (f ) .
It is evident that
¯
¯ ¯
¯
¯ Zπ
¯ ¯n−1 π(k+1)/n
¯
Z
¯
¯
¯
sin nt ¯ ¯ X
sin nt ¯¯
¯
g (t)
dt¯ = ¯
g (t)
dt¯
¯
¯
¯ ¯
¯
t
t
¯π/n
¯ ¯ k=1 πk/n
¯
¯
¯
¯
¯
π/n µ
¶
Z
k
¯n−1
X
πk (−1) sin nt ¯¯
¯
=¯
dt¯
g t+
¯
¯
n
t + πk
n
¯ k=1 0
¯
¯
¯
¯
¯n−1 Zπ µ
¶
k
¯X
u + πk (−1) sin u ¯¯
du¯
= ¯¯
g
n
u + πk
¯
¯ k=1
(6)
0
¯
¯
¶
µ
k
¯n−1
u0 + πk (−1) ¯¯
¯X
≤ π¯
g
¯
¯
n
u + πk ¯
k=1
¯
!¯
¶Ã
µ
k
k
¯n−1
(−1)
(−1) ¯¯
u0 + πk
¯X
−
≤ π¯
g
¯
¯
¯
n
u + πk
πk
k=1
¯
¯
¶
µ
k
¯n−1
u0 + πk (−1) ¯¯
¯X
+π¯
g
¯
¯
n
πk ¯
k=1
= I + II,
where
¯ ¯
¯
¯n−1 µ
¶
¶
n−1 µ
k
k
¯X
(−1) ¯¯
u0 + πk
u + πk (−1) ¯¯ ¯¯ X
¯
g
sup ¯
g
¯=¯
¯.
n
u + πk ¯ ¯
n
u0 + πk ¯
u ¯
k=1
k=1
It is clear that
(7)
I ≤ kgkC
n−1
X
k=1
1
= O (En (f )) .
k2
Denote
uk :=
u0 + πk
.
n
157
158
T. KARCHAVA
Using Abel transformation and (1), for II we obtain
¯n−1
¯
k
¯X
(−1) ¯¯
¯
g (uk )
II = ¯
¯
¯
k ¯
k=1
¯
¯
n−1
¯n−2
X (−1)i ¯¯
¯X
(g (uk+1 ) − g (uk ))
≤¯
¯
¯
i ¯
i=k+1
k=1
¯
¯
n−1
X (−1)i ¯¯
1 ¯¯
+ ¯g (u1 )
¯
π¯
i ¯
i=1
¯n−2
Ã
!¯
k+1
¯X
¯
(−1)
¯
¯
≤¯
(g (uk+1 ) − g (uk ))
+ Pk + Qn ¯
¯
¯
2k
k=1
¯
¯
n−1
¯
X (−1)i ¯¯
¯
+ ¯g (u1 )
¯
¯
i ¯
i=1
¯n−2
¯
k+1 ¯
¯X
(−1)
¯
¯
≤¯
(g (uk+1 ) − g (uk ))
¯ + O (En (f ))
¯
2k ¯
k=1
¯
¯
k+1 ¯
¯n−2
³
π ´ (−1)
¯X
¯
=¯
∆1 u k ; , g
¯ + O (En (f )) .
¯
n
2k ¯
k=1
Iterating this inequality we obtain
¯
¯
k+1 ¯
¯n−2
³
π ´ (−1)
¯
¯X
II ≤ c (p) ¯
∆p uk ; , g
¯ + O (En (f ))
¯
n
2k ¯
k=1
¯ ¡
¢¯
n−2
X ¯∆p uk ; π , g ¯
n
≤ c (p)
+ O (En (f )) .
k
k=1
Using Abel transformation we obtain
¯ ¡
¢¯
k ¯
n−2
³ π ´¯
X ¯∆p uk ; π , g ¯ n−3
X 1 X
¯
¯
n
≤
∆
ui ; , g ¯
¯
p
k
k 2 i=1
n
k=1
k=1
+
n−2
1 X ¯¯ ³
π ´¯¯
¯∆p uk ; , g ¯ ,
n−2
n
k=1
consequently,
(8)
!
n
³ π ´
X
1
ϕp k, , g + O (En (f ))
II = O
k2
n
k=1
à n
!
³ π ´
X 1
=O
ϕp k, ; f
.
k2
n
Ã
k=1
Combining (5)–(8) we complete the proof of the Theorem 1.
Proof of Corollary 1. Since
µ
¶
µ ¶α
³ π ´
πk
πk
ϕ1 k, ; f ≤ c (f ) ω
; f ≤ c (f )
,
n
n
n
¤
APPROXIMATION PROPERTIES OF PARTIAL SUMS OF FOURIER SERIES
we have
159
¡
¢
n
n
³ π ´
X
X
ω πk
1
n ;f
ϕ
k,
;
f
≤
c
(f
)
1
k2
n
k2
k=1
k=1
≤
n
c (f ) X
nα
k=1
1
c (f, α)
≤
.
k 2−α
nα
¤
Proof of Corollary 2. Since the function f has a finite number of intervals of convexity or concavity on [0, 2π] we have
ϕ2 (k, δ; f ) = c (f ) ω (δ, f ) ,
consequently from Theorem 1 we get,
µ
¶
n
³ π ´
X
1
1
kf − Sn (f )kC ≤ c
ϕ2 k, ; f ≤ c (f ) ω
,f .
k2
n
n
k=1
¤
Proof of Corollary 3. From the fact that the function f has a finite number of
intervals of monotonicity on [0, 2π] we write
µ
¶
³ π ´
πk
ϕ1 k, ; f ≤ c (f ) ω
;f ,
n
n
then using Theorem 1 we complete the proof of Corollary 3 in the following way
¡
¢
n
n
³ π ´
X
X
ω πk
1
n ;f
ϕ1 k, ; f ≤ c (f )
kf − Sn (f )kC ≤ c (f )
k2
n
k2
k=1
k=1
≤ c (f )
1
n
n
X
k=1
ω
¡k
¢
n; f
2
(k/n)
1
1
≤ c (f )
n
n
Zπ
ω (t, f )
dt.
t2
1/n
¤
160
T. KARCHAVA
References
[Ba]
[Ch]
[Dz]
[Fi]
[Gá]
[Ka]
[On]
[Os]
[Te]
N. Bari. Trigonometric series. Nauka, Moscow, in Russian, 1961.
Z. Chanturia. Uniform convergence Fourier series. Mat. Sb., 100:534–554, 1976. in Russian.
V. Dziadzik. An introduction of theory of uniformly approximate of functions. Nauka,
Moscow, 1977. in Russian.
N.J. Fine. On the Walsh functions. Trans. Am. Math. Soc., 65:372–414, 1949.
G. Gát. Best approximation by Vilenkin-like systems. Acta Math. Acad. Paed.
Nyı́regyáziensis, 17:161–169, 2001.
T. Karchava. On the convergence of Fourier series. Reports of Enlarget session of the
Semminar of I. Vekua’s Institute of Applied Mathematics, 3:93–95, 1988. in Russian.
G. Onnewer. On uniform convergence for Walsh-Fourier series. Pacific. J. Math., 34:117–
122, 1970.
K. Oskolkov. Non-amplibiability of Lebesgue estimates for approximation by Fourier
sums with given modulus of continuity. Trudy Mat. Inst. Steklov, 112:337–345, 1971. in
Russian.
V. Tevzadze. On the uniform convergence of Walsh-Fourier series, volume 31 of Some
problems of Function Theory, pages 84–117. Tbilisi, 1986.
Received December 6, 2004.
Department of Mathematics and Computer Science,
Sokhumi Branch of Tbilisi State University,
Str. Jikia 10,
0128 Tbilisi, Georgia