Georgian Mathematical Journal
Volume 10 (2003), Number 2, 363–379
ESTIMATES OF FOURIER COEFFICIENTS
V. TSAGAREISHVILI
Abstract. Some well-known properties of the trigonometric system as well
as of the Haar and Welsh systems are generalized to general orthonormal
systems.
2000 Mathematics Subject Classification: 42A16, 42C10.
Key words and phrases: Orthonormal system, Fourier coefficients, modulus of continuity, bounded variation.
1. Introduction
In the theory of functions an important place is occupied by generalization
of properties of specific orthonormal series to general orthonormal systems.
Here we note only some of the authors who have significant results concerning
the mentioned problems: Marcinkiewicz [1], Stechkin [2], Olevskii [3], Bochkarev
[4], [5], Mitiagin [6], Kashin [7], McLaughlin [8].
It was proved that in many cases some properties of the well-known orthogonal systems are typical for general orthogonal systems (see, e.g., [2], [3], [4], [5]).
However, not all properties of the well-known orthogonal systems are extending
on general orthogonal systems. Therefore, in order to obtain well-known results
for general orthogonal systems, we need to impose specific conditions on the
given system.
2. Auxiliary Notation and Results
Let δ ∈ (0, 1]. If a function f ∈ C([0, 1]), then its modulus of continuity is
defined as follows
ω(δ, f ) = sup max |f (x) − f (x + h)|,
|h|≤δ 0≤x≤1−h
where 0 < h ≤ 1.
If a function f ∈ L2 ([0, 1]), then the integral modulus of continuity has the
form
µZ 1−h
¶ 21
ω2 (δ, f ) = sup
|f (x) − f (x + h)|2 dx .
|h|≤δ
0
We say that a function f ∈ Lip α if ω(δ, f ) = O(δ α ) as δ → 0.
Let (ϕn ) be an orthogonal system on [0, 1]. Then the Fourier coefficients with
respect to (ϕn ) for the function f ∈ L([0, 1]) are defined as follows
Z 1
cn (f ) =
f (x)ϕn (x) dx, n = 1, 2, . . . .
0
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
364
V. TSAGAREISHVILI
The best approximation with respect to the system (ϕn ) in the sense of
L2 ([0, 1]) is defined by the following equality:
°
°
n
°
°
X
°
°
En(2) (f ) = inf °f (x) −
am ϕm (x)° .
°
{am } °
m=1
2
If (ϕn ) is a complete system on [0, 1], then
! 21
̰
X
c2k (f )
.
En(2) (f ) =
k=n
In the sequel we will denote by (ψn ) either one of the orthonormal systems of
Haar or Walsh (see, e.g., [9, p. 53, 54]), or the trigonometric system. For these
systems the following results are valid. They are important for our purpose.
Theorem A. If
Z1
cn (f ) =
f (x)ψn (x) dx,
0
then the following
¡ ¡relations
¢¢ are valid:
1
a) cn (f ) = O ¡ω ¡n , f ¢¢for every f ∈ C([0, 1]);
b) cn (f ) = O ω2 n1 , f for every f ∈ L2 ([0, 1]).
Theorem B. For every function f ∈ L2 ([0, 1]) the relation
µ µ
¶¶
1
(2)
En = O ω2
,f
n
holds.
The relation a) of Theorem A for the Haar system was proved by B. Golubov
[10] and the relation b) was proved by P. Ul’yanov [11], for the Walsh system
it was proved by Fine [12].
Theorem B for the Haar system was proved by Ul’yanov [11]. As to the
trigonometric system, Theorems A and B are well-known (see, e.g., [13, p. 79]).
It is well-known that Theorems A and B are not valid for general orthogonal
systems. In fact, the following theorem holds.
Theorem C. Let f be a function from L2 ([0, 1]) and (cn ) be an arbitrary
sequence from `2 satisfying the condition
Z 1
∞
X
2
cn =
f 2 (x) dx.
n=1
0
Then there exists an orthonormal system (ϕn ) on [0, 1] such that
Z 1
cn =
f (x)ϕn (x) dx, n = 1, 2, . . . .
0
ESTIMATES OF FOURIER COEFFICIENTS
365
Theorem C is proved by A. Olevskii [3]. From this theorem follows that if
f ∈ L2 ([0, 1]) and the numbers cn satisfy the conditions
a)
∞
X
Z
c2n
1
=
|cn |
= +∞,
n→∞ ω2 ( 1 , f )
n
f 2 (x) dx and b) lim
0
n=1
then there exists an orthonormal system of functions (ϕn ) such that
Z
1
cn =
f (x)ϕn (x) dx,
n = 1, 2, . . . .
0
Therefore there exists a function f ∈ L2 ([0, 1]) such that
|cn (f )|
= +∞.
n→∞ ω2 ( 1 , f )
n
lim
The following propositions are valid (see [15]).
Lemma 1. Let the function f ∈ L2 ([0, 1]) take finite values at every point of
the interval [0, 1] and Φ ∈ L2 ([0, 1]). Then the following equality is valid:
Z
1
0
µ
¶¶ Z i
n−1 µ µ ¶
X
n
i
i+1
f (x)Φ(x) dx =
f
Φ(x) dx
−f
n
n
0
i=1
µ ¶¶
Z 1
n Z i µ
X
n
i
+
Φ(x) dx + f (1)
Φ(x) dx.
f (x) − f
i−1
n
0
i=1
n
(1)
Lemma 2. Let (ϕn ) be an orthonormal system of functions on [0, 1]. Let En
denote the¡ set of¢the natural numbers i (i = 1, 2, . . . , n) for which there exists a
point t ∈ i−1
, i such that
n n
Z
Z
t
sign
i−1
n
ϕn (x) dx 6= sign
0
ϕn (x) dx.
0
Then the inequality
¯ µZ
¯
¶ 21
¯
1
X ¯¯Z ni
¯
ϕ2n (x) dx
ϕn (x) dx¯ ≤
¯
¯
¯ 0
0
i∈En
holds.
366
V. TSAGAREISHVILI
3. Main Results
Let (an ) be a sequence of real numbers from `2 , Introduce the following notation:
N
+m
X
(m)
PN (x) ≡
Ã
eN ≡
ak ϕk (x),
k=N
∞
X
m = 1, 2, . . . ,
! 21
c2k (f )
,
k=N
¯ k
¯
¯
¯ N (m)
¯
HN ≡ sup
PN (x) dx¯ ,
¯
¯ 0
¯
m
k=1
à ∞
! 21
X
hN ≡
a2k
.
N
−1 ¯Z
X
k=N
Theorem 1. Let f ∈ C([0, 1]) and (ϕn ) be an orthonormal system of functions
R1
on [0, 1] satisfying ϕn (x) dx = 0, n = 1, 2, . . . .
0
Then the relation
µ µ
¶¶
1
eN = O ω
,f
N
holds if and only if the condition
HN = O(hN )
for every sequence (ak ) ∈ `2
is fulfilled.
Proof. Sufficiency. If cn (f ) =
R1
f (x)ϕn (x)dx, then
0
N
+m
X
c2k (f )
=
k=N
N
+m
X
Z
k=N
Z
1
=
f (x)
0
1
ck (f )
f (x)ϕk (x) dx
0
N
+m
X
k=N
Z
1
ck (f )ϕk (x) dx =
0
(m)
f (x)PN (x) dx.
Using the assertion of Lemma 1, we obtain
¶
µ
¶¶ Z i
Z 1
N
−1 µ µ
X
N
i
i+1
(m)
(m)
f (x)PN (x) dx =
f
PN (x) dx
−f
N
N
0
0
i=1
µ
¶¶
i µ
Z
Z
N
1
X N
i
(m)
(m)
+
f (x) − f
PN (x) dx + f (1)
PN (x) dx.
i−1
N
0
i=1
N
(2)
(3)
ESTIMATES OF FOURIER COEFFICIENTS
Estimate the right-hand side of equality (3). We have
¯
¯
¶
µ
¶¶ Z i
−1 µ µ
¯N
¯
X
N
i
i
+
1
¯
¯
(m)
PN (x) dx¯
f
−f
¯
¯
¯
N
N
0
i=1
¯
¯
¶
µ
¶¯ ¯Z i
N
−1 ¯ µ
¯
X
¯
¯ ¯ N (m)
i
i
+
1
¯
¯
¯
≤
PN (x) dx¯
¯
¯f N − f
¯
¯
¯
N
0
i=1
¯
¯
¶N
µ
¶
µ
−1 ¯Z i
¯
X
1
1
¯ N (m)
¯
,f
, f hN .
≤ω
PN (x) dx¯ ≤ O(1) ω
¯
¯ 0
¯
N
N
i=1
367
(4)
Applying the Hölder inequality, we obtain
¯N −1 Z i µ
¯
µ ¶¶
¯X N
¯
i
¯
¯
(m)
f (x) − f
PN (x) dx¯
¯
i−1
¯
¯
N
i=1
N
µ
µ
¶N
¶ µZ 1 ³
−1 Z i ¯
¯
´2 ¶ 21
X
N
1
1
¯ (m) ¯
(m)
≤ω
,f
,f
PN (x) dx
¯PN (x)¯ dx ≤ ω
i−1
N
N
0
i=1
N
! 21
µ
¶ ÃNX
µ
¶
+m
1
1
2
ck (f )
= O(1)ω
,f
≤ O(1)ω
, f eN .
(5)
N
N
k=N
Finally, since
¯Z
¯
¯
¯
R1
ϕn (x) dx = 0, n = 1, 2, . . . , from (3), (4) and (5) we have
0
1
0
¯
¯
¶ ¶
¶ ¶
µ µ
µ µ
1
1
, f eN + O ω
, f eN
≤O ω
N
N
µ µ
¶ ¶
1
=O ω
, f eN .
N
(m)
f (x)PN (x) dx¯¯
Hence, applying (2), we get
N
+m
X
c2k (f )
µ µ
¶ ¶
1
=O ω
, f eN ,
N
c2k (f )
µ µ
¶ ¶
1
=O ω
, f eN ,
N
k=N
i.e.,
∞
X
k=N
From (6) we have
Ã
∞
X
k=N
! 21
c2k (f )
µ µ
¶¶
1
=O ω
,f
.
N
Sufficiency of the theorem is proved.
Necessity. Let HN 6= O(hN ), i.e.,
HN
lim
= +∞.
N hN
(6)
368
V. TSAGAREISHVILI
Therefore there exist a sequence (an ) ∈ `2 and natural numbers mN ↑ ∞ such
that
(m )
H N
lim N
= +∞,
(7)
N
hN
where
¯
¯
N
−1 ¯Z k
¯
X
(mN )
¯ N (mN )
¯
HN
=
PN (x) dx¯
¯
¯ 0
¯
k=1
and
(m )
PN N (x)
N +mN
=
X
ak ϕk (x).
k=N
Consider the sequence of functions
¶
Z xµ
Z u
(mN )
sign
PN (t) dt du,
fN (x) =
0
Taking into account the condition
obtain
Z 1
0
N = 1, 2, . . . .
(8)
0
R1
ϕn (x) dx = 0, n = 1, 2, . . . , from (1) we
0
(m )
fN (x)PN N (x) dx
N
−1 µ
X
µ
¶¶ Z i
N
i+1
(m )
=
fN
− fN
PN N (x) dx
N
0
i=1
¶¶
µ
i µ
Z
N
X N
1
(m )
PN N (x) dx.
(9)
+
fN (x) − fN
i−1
N
i=1
N
i
N
¶
µ
Applying (8) we have
¯
¯
µ ¶¶
N Z i µ
¯
¯X
N
i
(mN )
¯
¯
PN (x) dx¯
fN (x) − fN
¯
i−1
¯
¯
N
i=1
N
N Z i
1 X N ¯¯ (mN ) ¯¯
≤
¯PN (x)¯ dx
N i=1 i−1
N

 12
N +mN
µZ 1 ³
¶ 12
´
X
2
1 
1
(m )
a2k  .
PN N (x) dx
=
≤
N
N
0
k=N
Taking into account the assertions of Lemma 2 and (8), we get
µ ¶
µ
¶¶ Z i
N
−1 µ
X
N
i
i+1
(m )
PN N (x) dx
fN
− fN
N
N
0
i=1
µ
µ
¶
µ
¶¶ Z i
X
N
i
i+1
(m )
=
fN
− fN
PN N (x) dx
N
N
0
i∈FN \EN
(10)
ESTIMATES OF FOURIER COEFFICIENTS
+
Xµ
i∈EN
µ
fN
i
N
¶
µ
− fN
i+1
N
¶¶ Z
i
N
0
(mN )
PN
369
(x) dx,
where FN = {1, 2, . . . , N }.
If i ∈ FN \ EN , then
µ ¶
µ
¶
Z i+1
Z u
N
i
i+1
(m )
fN
− fN
=−
sign
PN N (t) dt du
i
N
N
0
N
Z i
N
1
(m )
=−
sign
PN N (t)dt.
N
0
Therefore
¯
¯
¯
¯
µ µ ¶
µ
¶¶ Z i
X
¯
¯
N
i
i+1
(mN )
¯
fN
− fN
PN (x) dx¯¯
¯
N
N
0
¯i∈FN \EN
¯
¯Z i
¯
¯
1 X ¯¯ N (mN )
¯
=
PN (x) dx¯ .
¯
¯ 0
¯
N
(11)
(12)
i∈FN \EN
At last, as far as
¯ µ ¶
¶¯
µ
¯
¯
i
+
1
i
¯< 1
¯fN
−
f
N
¯ N
¯
N
N
and, by virtue of Lemma 2, we have
¯ µZ
¯
¯
´2 ¶ 21
1³
X ¯¯Z Ni (m )
(mN )
¯
N
PN (x) dx¯ ≤
PN (x) dx
¯
¯
¯ 0
0
i∈EN
 12 Ã

! 21
N +mN
∞
X
X
a2k  ≤
a2k
=
≡ hN .
k=N
N
N = 1, 2, . . . .
Using (12) and (15), we get
¯N −1 µ µ ¶
¯
µ
¶¶ Z i
¯X
¯
N
i
i+1
(mN )
¯
¯
fN
PN (x) dx¯
− fN
¯
¯
¯
N
N
0
i=1
¯Z i
¯
¯Z i
¯
¯
¯ N
¯
X
1 X ¯¯ N (mN )
1
(m
)
¯
¯
¯
≥
PN (x) dx¯ −
PN N (x) dx¯
¯
¯
¯ 0
¯ N
¯ 0
¯
N
≥
1 (mN ) hN
H
−
.
N N
N
(14)
k=N
Then, taking into account (13) and (14), we obtain
¯
¯
µ
¶¶ Z i
¯X µ µ i ¶
¯
N
i+1
1
(mN )
¯
¯
fN
− fN
PN (x) dx¯ ≤
h ,
¯
¯ N N
¯
N
N
0
i∈E
i∈FN \EN
(13)
i∈EN
(15)
370
V. TSAGAREISHVILI
Hence due to (7) we have
¯1
¯
¯R
¯
(mN )
¯
N ¯ fN (x)PN (x) dx¯¯
0
hN
and consequently
lim
(m )
H N
hN
≥ N
−
,
hN
hN
¯
¯1
¯
¯R
(mN )
N ¯¯ fN (x)PN (x) dx¯¯
0
hN
n→∞
= +∞.
(16)
Further, since
kfN kLip 1 = kfN kC + sup
x,y
|fN (x) − fN (y)|
,
|x − y|
then from (8) it follows that
kfN kLip 1 ≤ 2.
(17)
Finally, by virtue of the Banach–Steinhaus theorem (see (16) and (17)) there
exists a function f0 ∈ Lip 1 such that
¯1
¯
¯R
¯
(mN )
N ¯¯ fN (x)PN (x) dx¯¯
0
lim
= +∞.
(18)
n→∞
hN
As far as
¯Z
¯
¯
¯
¯
¯
¯ ¯Z 1
¯
N +mN
X
¯ ¯
¯
(mN )
f0 (x)PN (x) dx¯¯ = ¯¯
ak ϕk (x) dx¯¯
f0 (x)
0
¯
¯ 0
k=N
¯ ¯
¯
¯
¯ ¯N +mN
¯N +mN Z 1
¯
¯ ¯ X
¯ X
¯
= ¯¯
ak
f0 (x)ϕk (x) dx¯¯ = ¯¯
ak ck (f0 )¯¯
0
¯ k=N
¯ ¯ k=N
¯
 21

 21 
N +mN
N +mN
X
X
c2k (f0 ) < hN · eN ,
a2k  
≤
1
k=N
k=N
then taking into account that ω
¡1
N
¢
¡ ¢
, f0 = O N1 , from (18) we have
h ·e
¡ 1 N ¢N
= +∞,
n→∞ ω
, f 0 · hN
N
lim
i.e.,
e
¡ 1 N ¢ = +∞.
n→∞ ω
, f0
N
lim
¤
ESTIMATES OF FOURIER COEFFICIENTS
371
Theorem 2. Let the functions f and Φ be from L2 ([0, 1]). Then
¯Z
¯
¯
¯
1
0
¯
µ
¶
¯
1
f (x)Φ(x) dx¯¯ ≤ ω2
, f (Vn + 2kΦk2 )
n
¯Z 1
¯
Z 1
¯
¯
¯
+n
|f (x)| dx ¯
Φ(x) dx¯¯ ,
1
1− n
where Vn =
n−1
P ¯R
¯
Φ(x) dx¯.
k/n
0
¯
k=1
(19)
0
Proof. Applying the Abel transformation, we get
n
n Z
X
Z
k
n
k−1
n
k=1
=n
f (x) dx
ÃZ
n−1
X
k
n
k−1
n
k
n
k−1
n
k=1
Z
Φ(x) dx
Z
f (x) dx −
k
n
f (x) dx
Φ(x) dx
0
1
Φ(x) dx
f (x) dx
1
1− n
!Z
k
n
Z
1
+n
k+1
n
0
µ
¶¶
Z k
n
1
=n
Φ(x) dx
f (x) − f x +
dx
k−1
n
0
n
k=1
Z 1
Z 1
+n
f (x) dx
Φ(x) dx.
n−1 Z
X
k
n
µ
1
1− n
(20)
0
Since
Z
k
n
k−1
n
Z
f (x)Φ(x) dx − n
Z
=n
Z
=n
Z
=n
k
n
k−1
n
k
n
k−1
n
k
n
Z
dt
k−1
n
ÃZ
Z
k−1
n
k
n
k
n
k−1
n
k
n
k−1
n
k
n
k−1
n
Z
f (t) dt
k
n
k−1
n
Z
f (x)Φ(x) dx − n
Z
f (x) dt −
k
n
k−1
n
Φ(x) dx
Z
k
n
k−1
n
f (t) dt
!
k
n
k−1
n
Φ(x) dx
f (t) dt Φ(x) dx
(f (x) − f (t)) Φ(x) dx dt
(21)
and
Z
1
f (x)Φ(x) dx =
0
n Z
X
k=1
k
n
k−1
n
f (x)Φ(x) dx,
(22)
372
V. TSAGAREISHVILI
then, taking into account (20), (21) and (22), one has
Z
1
f (x)Φ(x) dx =
0
+n
n−1 Z
X
k=1
Z
n−1 Z
X
k−1
n
Z
k
n
k−1
n
k=1
µ
k
n
k=1
Z
1
f (x) dx
1
1− n
Φ(x) dx = n
0
µ
k
n
µ
f (x) − f
k−1
n
1
+n
k=1
f (x)Φ(x) dx − n
n Z
X
f (t) dt
k
n
k−1
n
Φ(x) dx
µ
¶¶
Z 1
Z k
Z 1
n
1
f (x) − f x +
dx
Φ(x) dx + n
f (x) dx
Φ(x) dx
k−1
1
n
0
1− n
0
n
ÃZ k
!
Z k
Z k
n
X
n
n
n
=
f (x)Φ(x) dx − n
f (t) dt
Φ(x) dx
+n
n−1 Z
X
k
n
k−1
n
k=1
µ
k
n
k=1
+n
n Z
X
µ
f (x) − f
k−1
n
1
x+
n
k−1
n
1
x+
n
n Z
X
Z
k
n
dx
Z
k
n
dx
f (x)Φ(x) dx
0
k
n
k−1
n
k=1
¶¶
¶¶
k−1
n
Z
k
n
k−1
n
(f (x) − f (t)) Φ(x) dx dt
Z
Z
1
Φ(x) dx+n
1
1− n
0
1
f (x) dx
Φ(x) dx.
0
Consequently,
¶¶
Z k
n
1
f (x)Φ(x) dx = n
dx
Φ(x) dx
f (x) − f x +
k−1
n
0
0
n
k=1
Z 1
Z 1
n Z k Z k
X
n
n
+n
(f (x) − f (t)) Φ(x) dx dt + n
f (x) dx
Φ(x) dx.
Z
n−1 Z
X
1
k=1
k−1
n
k−1
n
µ
µ
1
1− n
(k)
k
n
µ
k−1
n
Assuming ξn (t) =
Z
k
n
1
n
t+
k−1
,
n
(23)
0
we get
µ
¶¶
¶
µ
¶¶
Z µ µ
1
1
1
1 1
k−1
k
f (x) − f x +
dx =
f
t+
−f
t+
dt
n
n 0
n
n
n
n
µ
¶¶
Z µ
¡ (k) ¢
1 1
1
(k)
=
f ξn (t) − f ξn (t) +
dt.
n 0
n
Hence we have
¯Z k µ
µ
µ
¶¶ ¯¯
¶¯
Z 1¯
¯ n
¯ ¡ (k) ¢
¯
1
1
¯
¯ 1
(k)
¯f ξn (t) − f ξn (t) +
¯ dt
f (x) − f x +
dx¯ ≤
¯
¯
¯
¯ k−1
¯ n 0
n
n
n
µ
¶¯
µZ 1
¶ 21
Z ¯
1 ¯¯
1
1 1 ¯¯
2
f (t) − f t +
dt ≤ sup
≤
|f (t) − f (t + h)| dt
n 0 ¯
n ¯
n |h|≤ 1
0
n
µ
¶
1
1
,f .
(24)
= ω2
n
n
ESTIMATES OF FOURIER COEFFICIENTS
By virtue of the Hölder inequality we obtain
¯Z k Z k
¯
¯ n
¯
n
¯
¯
(f (x) − f (t)) Φ(x) dx dt¯
¯
¯ k−1 k−1
¯
n
n
! 12 ÃZ
Z k ÃZ k
≤
n
n
k−1
n
k−1
n
1
≤√
n
ÃZ
k
n
k−1
n
(f (x) − f (t))2 dx
Z
k
n
k−1
n
k−1
n
! 21
Φ2 (x) dx
! 21 ÃZ
(f (x) − f (t))2 dx dt
Applying the equality (see [11])
Z bZ b
Z
p
|f (x) − f (t)| dx dt = 2
a
k
n
a
b−a
µZ
k
n
k−1
n
b−ξ
373
dt
! 21
Φ2 (x) dx
.
(25)
¶
p
|f (y + ξ) − f (y)| dy dξ,
0
a
we get
Z
k
n
k−1
n
Z
k
n
k−1
n
Z
1
n
(f (x) − f (t))2 dx dt = 2
0
ÃZ
k
−ξ
n
k−1
n
!
|f (y + ξ) − f (y)|2 dy dξ. (26)
Therefore, by virtue of (25) and (26) we have
¯
¯ n Z k Z k
¯
¯X n
n
¯
¯
(f (x) − f (t)) Φ(x)dx dt¯
¯
k−1
k−1
¯
¯
n
n
k=1
ÃZ 1 Z k
! 21 ÃZ k
! 21
n
−ξ
X
n
n
n
2
(f (y + ξ) − f (y))2 dy dξ
·
Φ2 (x) dx
≤√
k−1
k−1
n k=1
0
n
n
ÃZ 1 n Z k
! 21 Ã n Z k
! 12
X n
n X
n
2
2
≤√
(f (y + ξ) − f (y)) dy dξ
·
Φ2 (x) dx
k−1
k−1
n
0 k=1
n
n
k=1
! 21 µZ
ÃZ 1 Z
¶ 21
1
1
n
2
(f (y + ξ) − f (y))2 dy dξ
Φ2 (x) dx
≤√
·
n
0
0
0
¶
µ
¶
µ
2kΦk2
1
2
1
1
, f · kΦk2 =
ω2
,f .
(27)
≤ √ · √ ω2
n
n
n
n
n
At last, taking into account (24), we have
¯
¯
µ
¶¶
Z k
n−1 Z k µ
¯X
¯
n
n
1
¯
¯
Φ(x) dx¯
f (x) − f x +
dx
¯
k−1
¯
¯
n
0
n
k=1
¯
¯
¡
µ
¶ X
n−1 ¯Z k
¯ ω 1 , f¢
1
1
2 n
¯ n
¯
≤ ω2
,f ·
Vn .
Φ(x) dx¯ =
¯
¯ 0
¯
n
n
n
k=1
374
V. TSAGAREISHVILI
Hence from (23) and (27) we get
¡1 ¢
¯Z 1
¯
µ
¶
¯
¯
ω
1
2kΦk
2 n ,f
2
¯
f (x)Φ(x) dx¯¯ ≤ n ·
Vn + n
ω2
,f
¯
n
n
n
0
¯Z 1
¯
Z 1
¯
¯
¯
+n
|f (x)|dx ¯
Φ(x) dx¯¯ .
1
1− n
0
Theorem 2 is proved completely.
¤
Theorem 3. Let (ϕn ) be an orthonormal system of functions on [0, 1] satisR1
fying the condition ϕn (x) dx = 0, n = 1, 2, . . . . Then
0
µ µ
¶¶
1
cn (f ) = O ω2
,f
n
for every f ∈ L2 ([0, 1])
if and only if
Vn = O(1),
where Vn =
n−1
P ¯R
¯
k
n
0
k=1
ϕn (x) dx|.
Proof. Sufficiency. Assuming in (19) that Φ(x) = ϕn (x), we obtain
¯Z 1
¯Z 1
¯
¯
¶
µ
Z 1
¯
¯
¯
¯
1
¯
¯
¯ ≤ ω2
¯.
|f
(x)|
dx
f
(x)ϕ
(x)
dx
,
f
(V
+
2kϕ
k
)+n
ϕ
(x)
dx
n
n
n
2
n
¯
¯
¯
¯
1
n
0
1− n
k
Since Vn =
n−1
P ¯ Rn
¯
k=1
ϕn (x) dx| = O(1) and
0
µ
R1
0
|cn (f )| = O ω2
0
ϕn (x) dx = 0, n = 1, 2, . . . , we have
µ
1
,f
n
¶¶
.
Necessity. If Vn 6= O(1), then as it is known (see [15]) there exists f0 ∈ Lip 1
such that
lim n|cn (f0 )| = +∞.
(28)
¡1
¢
¡ 1n→∞
¢
As far as ω2 n , f0 = O n , from (28) we have
cn (f0 )
¡
¢ = +∞.
n→∞ ω2 1 , f0
n
lim
Theorem 4. Let (ϕn ) be an orthonormal system on [0, 1] satisfying
Z1
ϕn (x)dx = 0,
n = 1, 2, . . . .
0
Then the relation
µ
e N = O ω2
µ
1
,f
N
¶¶
for every f ∈ L2 ([0, 1])
¤
ESTIMATES OF FOURIER COEFFICIENTS
375
holds if and only if
HN = O(hN )
for every sequence (an ) ∈ `2 .
Proof. Sufficiency. Let f ∈ L2 ([0, 1]). As it was shown above (see (2)), the
equality
Z 1
Z 1
N
+k
N
+k
X
X
(k)
2
cm (f ) =
f (x)
cm ϕm (x) dx =
f (x)PN (x) dx
(29)
0
m=N
0
m=N
holds. By virtue of Theorem 2, we get
¯
¯Z 1
µ
¶³
´
¯
¯
1
(k)
(k)
¯
¯ ≤ ω2
f
(x)P
(x)
dx
k
,
f
H
+
2kP
N
2 .
N
N
¯
¯
N
(30)
0
Since
µZ
1
(k)
kPN k2 =
0
³

Z
´2 ¶ 12
(k)
PN (x)
=
=
à N +k
X
à N +k
X
1
0
!2
cm (f )ϕm (x)
 21
dx
m=N
! 21
c2m (f )
≤ eN ,
m=N
and by virtue of the condition of Theorem 4 HN = O(eN ), from (30) we have
¯Z 1
¯
¶
µ
¯
¯
1
(k)
¯
¯
, f eN ,
f (x)PN (x) dx¯ ≤ c · ω2
¯
N
0
where c does not depend on N .
Finally, from (29) we get
N
+k
X
µ
c2m (f )
≤ c · ω2
m=N
hence
∞
X
1
,f
N
¶
eN ,
µ
c2m (f )
≤ c · ω2
m=N
From (31) we have
µ
eN = O ω2
µ
1
,f
N
1
,f
N
k = 1, 2, . . . ,
¶
eN .
(31)
¶¶
.
Thus the sufficiency is proved.
Necessity. Let HN 6= O(hN ). Then by virtue of Theorem 1 there exists a
function f0 ∈ Lip 1 such that
lim N eN = +∞.
¡
¢
¡ ¢
As far as for the function f0 the relation ω2 N1 , f0 = O N1 holds, from (31)
it follows
e
¡ N ¢ = +∞.
lim
n→∞ ω2 1 , f0
n
n→∞
376
V. TSAGAREISHVILI
Theorem is proved completely.
¤
Theorem 5. From every orthonormal system ϕn on [0, 1] satisfying the
R1
condition ϕn (x) dx = 0, n = 1, 2, . . . , one can choose a subsystem ψk = ϕnk
0
for which the following conditions are fulfilled:
¡ ¡
¢¢
1) cn (f ) = ¡O ¡ω2 n1 ¢¢
,f ,
2) eN = O ¡ω ¡N1 , f ¢¢,
3) eN = O ω2 N1 , f .
Proof. Let
εi (k) =
ÃZ
∞
X
i
k
!2
ϕs (x) dx
,
i = 1, 2, . . . , k.
0
s=1
By virtue of the Bessel inequality, εi (k) ≤ 1 for every i = 1, 2, . . . , k. Therefore
for each fixed k we can choose a number si (k) such that the inequality
ÃZ i
!2
∞
X
k
1
ϕs (x) dx < 3
k
0
s=si (k)
holds. Now if s(k) = max si (k), then
1≤i≤k
∞
X
ÃZ
s=s(k)
i
k
!2
ϕs (x) dx
1
.
k3
<
0
Hence for every i = 1, 2, . . . , k we have
¯
¯Z i
¯
¯ k
1
¯
¯
ϕs (x) dx¯ < 3/2
¯
¯ k
¯ 0
if s ≥ s(k). Assuming ϕs(k) = ψk , we get
¯Z i
¯
¯
¯ k
1
¯
¯
ψk (x) dx¯ < 3/2 .
¯
¯ 0
¯ k
This inequality implies the inequality
¯
¯
k−1 ¯Z i
¯
X
¯
¯ k
Vk =
ψk (x) dx¯ < 1,
¯
¯
¯ 0
n = 1, 2, . . . ,
(32)
i=1
and also the estimate
¯
¯ Ã
! 21 k−1  k+l ÃZ i
!2  12
k+l
k−1 ¯Z i X
k+l
¯
X
X
X
X
k
k
¯
¯

a2m
am ψm (x) dx¯ ≤
ψm (x) dx 
¯
¯ 0
¯
0
i=1
i=1
m=k
m=k
≤
à k+l
X
m=k
! 21
a2m
·
k−1
X
1
i=1
k
<
m=k
à k+l
X
m=k
! 12
a2m
.
ESTIMATES OF FOURIER COEFFICIENTS
377
is valid. Therefore
Hk = O(hk ).
(33)
Hence the validity of Theorem 5 follows from Theorems 1,2,3,4 and from the
relations (32) and (33).
¤
Remark 1. It follows from the proof of Theorem 1 that the relation
µ µ
¶¶
1
eN = O ω2
,f
N
(34)
holds for every f ∈ C([0, 1]) if and only if relation (34) holds for every f ∈ Lip 1.
Hence it follows
Corollary 1. Relation (34) holds for every f ∈ L2 ([0, 1]) if and only if the
condition
µ ¶
1
eN = O
N
holds for every function f ∈ Lip 1.
Corollary 2. The relation
µ
cn (f ) = O ω2
µ
1
,f
n
¶¶
holds for every f ∈ L2 ([0, 1]) if and only if the condition
µ ¶
1
cn (f ) = O
n
holds for every f ∈ Lip 1 (see [15]).
Remark 2. a) If (ϕn ) is a trigonometric system, then the inequality
¯
¯
n−1 ¯Z k
n−1
¯ X
X
1
1
¯
¯ n
Vn =
sin πnx dx¯ ≤
<
¯
¯
¯ 0
πn
π
k=1
k=1
is valid.
b) Let (χn ) be the Haar system (see [11]). Then if n = 2m + l (1 ≤ l ≤ 2m ),
we have
¯Z k
¯
¯ n
¯
m
2
¯
¯
χn (x) dx¯ ≤ 2− 2 < √ .
¯
¯ 0
¯
n
k
On the other hand, note that only one integral
zero when k = 1, 2, . . . , n. Thus
Rn
χn (x) dx is different from
0
¯
¯
n−1 ¯Z k
¯
X
2
¯ n
¯
Vn =
χn (x) dx¯ ≤ √ .
¯
¯ 0
¯
n
k=1
378
V. TSAGAREISHVILI
Further on, it is easy to verify that if n = 2s + l, in case l ≤ 2s the following
estimate is valid:
¯
¯
¯
¯
n+q
n−1 ¯Z k
n−1 ¯Z k X
¯ X
¯
X
n
n
¯
¯
¯
¯
Pn(q) (x) dx¯ ≡
am χm (x) dx¯
¯
¯
¯ 0
¯
¯
¯ 0
m=n
k=1
k=1
¯Z k n+q
¯
¯Z k
¯
n ¯
n ¯
¯
¯ X
X
s+1 X
n
2
¯
¯
¯
¯
(q)
≤
Pn (x) dx¯ .
am χm (x) dx¯ +
(36)
¯
¯
k
¯ 0
¯
¯
¯
s+1
m=n
k=1
k=1
2
Applying the Hölder inequality, we get
¯ Z
¯
! 12
à n+q
n ¯Z k
¯
1
X
X
¯
¯ n
a2m
|Pn(q) (x)| dx ≤
Pn(q) (x) dx¯ ≤
≤ hn .
¯
k
¯ s+1
¯
0
k=1
(37)
m=n
2
Then, if n + q > 2s+1 , then
¯
¯
¯
¯
k
k
n+q
n ¯Z
n ¯Z
2s+1
¯ X
¯
X
s+1 X
s+1 X
2
2
¯
¯
¯
¯
am χm (x) dx¯ ≤
am χm (x) dx¯
¯
¯
¯ 0
¯
¯ 0
¯
m=n
m=n
k=1
k=1
¯Z k
¯
n+q
n ¯
¯
X
¯ 2s+1 X
¯
+
am χm (x) dx¯ .
¯
¯ 0
¯
s+1
k=1
m=2
(38)
+1
Since m ≥ 2s+1 , we have
Z
k
2s+1
χm (x) dx = 0
0
and the second summand in (38) is equal to zero.
On the other hand, as far as 2s+1 − n ≤ 2s , we get
¯
¯
¯
¯
s+1
k
k
2s+1
n ¯Z
n ¯Z
¯ 2X
¯
X
X
¯
¯
¯ 2s+1 X
¯ 2s+1
am χm (x) dx¯ ≤
χm (x) dx¯
|am |
¯
¯
¯ m=n
¯
¯ 0
¯ 0
m=n
k=1
k=1
2s+1
≤
X
1
|am | · √ ≤
m
m=n
à 2s+1
X
! 12 Ã 2s+1 ! 12
X 1
a2m
≤ hn .
m
m=n
m=n
(39)
Finally, taking into account in (36) the inequalities (37), (38) and (39), we
obtain
¯
¯
n ¯Z k
¯
X
¯
¯ n (q)
Pn (x) dx¯ = O(hn ).
¯
¯
¯ 0
k=1
Consequently,
Hn = O(hn ).
c) Let now (ϕn ) be the Walsh system (see [12]). Then since
¯Z k
¯
¯ n
¯ 1
¯
¯
ϕn (x) dx¯ < ,
¯
¯ 0
¯ n
ESTIMATES OF FOURIER COEFFICIENTS
we get
379
¯
¯
n−1 ¯Z k
¯
X
n
¯
¯
Vn =
ϕn (x) dx¯ < 1.
¯
¯ 0
¯
k=1
In this case the inequality
Hn = O(hn )
is analogously proved as in case of the Haar system.
In conclusion, we can say that the efficiency of the conditions of Theorems 1,
2, and 3 is evident.
It should be noted that the above results were partially announced in [14].
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(Received 8.04.2002)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 0143, Georgia