Convergence of Fourier series of functions $\text{Lip} 1$ with respect to general orthonormal systems
We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) $\{ \varphi_n(x)\}$ in order that the Fourier series in this system for any function from the class $\mathrm{L}\mathrm{i}\mathrm{p} 1$ be convergent almost everywhere on $[0, 1]$. It is sho...
Saved in:
| Date: | 2017 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1710 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507553427881984 |
|---|---|
| author | Gogoladze, L. Tsagareishvili, V. Гоголадзе, Л. Цагарейшвілі, В. |
| author_facet | Gogoladze, L. Tsagareishvili, V. Гоголадзе, Л. Цагарейшвілі, В. |
| author_sort | Gogoladze, L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:24:35Z |
| description | We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) $\{ \varphi_n(x)\}$
in order that the Fourier series in this system for any function from the class $\mathrm{L}\mathrm{i}\mathrm{p} 1$ be convergent almost everywhere on
$[0, 1]$. It is shown that the obtained conditions are best possible in a certain sense. |
| first_indexed | 2026-03-24T02:11:09Z |
| format | Article |
| fulltext |
UDC 517.5
L. Gogoladze, V. Tsagareishvili (I. Javakhishvili Tbilisi State Univ., Georgia)
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \bfL \bfi \bfp \bfone
WITH RESPECT TO GENERAL ORTHONORMAL SYSTEMS*
ЗБIЖНIСТЬ РЯДIВ ФУР’Є ФУНКЦIЙ \bfL \bfi \bfp \bfone
ВIДНОСНО ЗАГАЛЬНИХ ОРТОНОРМОВАНИХ СИСТЕМ
We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) \{ \varphi n(x)\}
in order that the Fourier series in this system for any function from the class \mathrm{L}\mathrm{i}\mathrm{p} 1 be convergent almost everywhere on
[0, 1]. It is shown that the obtained conditions are best possible in a certain sense.
Встановлено достатнi умови, якi повиннi задовольняти функцiї загальної ортонормованої системи (ЗОС) \{ \varphi n(x)\} ,
для того, щоб ряд Фур’є вiдносно вказаної системи для будь-якої функцiї з класу \mathrm{L}\mathrm{i}\mathrm{p} 1 збiгався майже скрiзь на
[0, 1]. Показано, що отриманi розв’язки є, в деякому розумiннi, найкращими.
1. Introduction. Let \{ \varphi n(x)\} be ONS on [0, 1],
\widehat \varphi n(f) =
1\int
0
f(x)\varphi n(x) dx, n = 1, 2, . . . ,
are Fourier coefficients of the function f(x) \in L(0, 1).
Suppose
Sn(a) =
1
n
n - 1\sum
i=1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(a, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| , (1)
where
\Phi n(a, x) =
n\sum
k=1
ak\lambda k\varphi k(x)
and (ak) and (\lambda k) are some sequences of numbers.
Besides,
Ln(x) =
1\int
0
\bigm| \bigm| \bigm| \bigm| n\sum
k=1
\varphi k(x)\varphi k(t)
\bigm| \bigm| \bigm| \bigm| dt (2)
is a Lebesgue function.
We have (see [1, p. 180 and 207]) the following theorem.
* This paper was supported by the Shota Rustaveli National Science Foundation, grant No. FR/102/5-100/14.
c\bigcirc L. GOGOLADZE, V. TSAGAREISHVILI, 2017
466 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 467
Theorem A (Kačmarž). Let (\lambda n) be a nondecreasing sequence of positive numbers, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \lambda n =
= +\infty and Ln(x) = O(\lambda n). If
\infty \sum
n=1
c2n\lambda n < +\infty , (3)
then the series
\infty \sum
n=1
cn\varphi n(x) (4)
converges a.e. on [0, 1].
Remark 1. It is well known (see [2, p. 202]) that for any ONS a.e.
Ln(x) = O(\gamma 1/2n ),
where \gamma n is a positive monotone increasing sequence satisfying the condition
\infty \sum
n=1
\gamma - 1
n <\infty .
Setting \gamma n = n2 we get Ln(x) = O(n).1
Let now
\lambda \prime n = \mathrm{m}\mathrm{i}\mathrm{n}(\lambda n, n),
where (\lambda n) is a sequence from Theorem A. Then \lambda \prime n will satisfy all conditions of Theorem A. In
addition,
\lambda \prime n \leq n.
Thus in Theorem A it can be assumed, without loss of generality, that \lambda n \leq n.
2. Formulation of the basic problem. S. Banach [3] proved that for any function f(x) \in
\in L2(0, 1) there exists ONS \{ \varphi n(x)\} such that the Fourier series of the function f(x) of this system
diverges a.e. [0, 1].
Besides, A. Olevski [4] proves that if f(x) \in L2(0, 1) is an arbitrary function and (an) \in \ell 2
is any sequence of numbers, then there exists ONS \{ \varphi n(x)\} such that an = c
\int 1
0
f(x)\varphi n(x) dx,
n = 1, 2, . . . , and c is some number.
Let Nk be an increasing sequence of natural numbers
an =
1
((Nk+1 - Nk)\lambda n)1/2
, Nk \leq n < Nk+1,
where
\infty \sum
n=1
(\lambda n)
- 1 <\infty .
Let f0(x) = 1, x \in [0, 1]. By virtue of A. Olevski’s theorem there exists ONS \{ \varphi n(x)\} such that
1The upper estimate is true and more exact, but in future it will be sufficient to use this estimate.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
468 L. GOGOLADZE, V. TSAGAREISHVILI
\widehat \varphi n(f0) =
1\int
0
f0(x)\varphi n(x) dx = c \cdot an, n = 1, 2, . . . .
Thus
\infty \sum
n=N1
\widehat \varphi 2
n(f0) = O(1)
\infty \sum
k=1
Nk+1 - 1\sum
n=Nk
1
(Nk+1 - Nk)\lambda n
= O(1)
\infty \sum
k=1
\lambda - 1
nk
< +\infty .
On the other hand,
\infty \sum
n=N1
\widehat \varphi 2
n(f0)\lambda n = c2
n\sum
k=1
Nk+1 - 1\sum
n=Nk
1
(Nk+1 - Nk)\lambda n
\lambda n = +\infty .
Consequently, in our case condition (3) is not satisfied for the Fourier coefficients of the functions
f0(x) = 1. Thus, “good” differential properties of functions do not provide the convergence of
Fourier series of these functions with respect to general ONS.
From the above example we can conclude that if we want the Fourier coefficients of smooth
functions to satisfy condition (3), it is necessary that the functions \varphi n(x) from ONS \{ \varphi n(x)\} satisfy
some conditions.
In the present paper we give certain conditions which are imposed on functions of ONS \{ \varphi n(x)\}
under which the Fourier coefficients of the function of class \mathrm{L}\mathrm{i}\mathrm{p} 1 satisfy condition (3).
Remark 2. As it was shown on an example, there exists ONS \{ \varphi n(x)\} for which the Fourier
coefficients f0(x) = 1 do not satisfy condition (3).
Thus if we want the Fourier coefficients of the function from class \mathrm{L}\mathrm{i}\mathrm{p} 1 to satisfy condition (3)
with respect to the system \{ \varphi n(x)\} we should require that
\infty \sum
n=1
\widehat \varphi 2
n(f0)\lambda n =
\infty \sum
n=1
\left( 1\int
0
\varphi n(x) dx
\right) 2 \lambda n < +\infty ,
for otherwise the formulation of the problem with respect to the functional class \mathrm{L}\mathrm{i}\mathrm{p} 1 has no meaning
since 1 \in \mathrm{L}\mathrm{i}\mathrm{p} 1.
3. Main results.
Theorem 1. Let \{ \varphi n(x)\} be ONS on [0, 1] and a sequence of nondecreasing numbers (\lambda n)
satisfies the following conditions:
(a) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \lambda n = +\infty ; Ln(x) = O(\lambda n) (see (2));
(5)
(b)
\sum \infty
n=1
\biggl( \int 1
0
\varphi n(x) dx
\biggr) 2
\lambda n < +\infty .
If for any sequence (an) \in \ell 2 the following condition:
Sn(a) = O(1)
\Biggl(
n\sum
k=1
a2k\lambda k
\Biggr) 1/2
(6)
holds, then the Fourier series of any function f(x) \in \mathrm{L}\mathrm{i}\mathrm{p} 1
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 469
\infty \sum
n=1
\widehat \varphi n(f)\varphi n(x) (7)
converges a.e. on [0, 1].
Proof. The equality (see [5])
1\int
0
f(x)\Phi (x) dx =
n - 1\sum
i=1
\biggl(
f
\biggl(
i
n
\biggr)
- f
\biggl(
i+ 1
n
\biggr) \biggr) i/n\int
0
\Phi (x) dx+
+
n\sum
i=1
i/n\int
i - 1
n
\biggl(
f(x) - f
\biggl(
i
n
\biggr) \biggr)
\Phi (x) dx+ f(1)
1\int
0
\Phi (x) dx (8)
holds, where f(x) and \Phi (x) are functions from L2(0, 1) and f(x) is finite at every point [0, 1].
We have the equality
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k =
1\int
0
f(x)
n\sum
k=1
\widehat \varphi k(f)\lambda k\varphi k(x) dx \equiv
\equiv
1\int
0
f(x)\Phi n(\widehat \varphi , x) dx, (9)
where \Phi n(\widehat \varphi , x) = \Phi n(a, x) for (an) = (\widehat \varphi n).
In equality (8) assume that \Phi n(a, x) = \Phi n(\widehat \varphi , x), we get
1\int
0
f(x)\Phi n(\widehat \varphi , x) dx =
n - 1\sum
i=1
\biggl(
f
\biggl(
i
n
\biggr)
- f
\biggl(
i+ 1
n
\biggr) \biggr) i/n\int
0
\Phi n(\widehat \varphi , x) dx+
+
n\sum
i=1
i/n\int
i - 1
n
\biggl(
f(x) - f
\biggl(
i
n
\biggr) \biggr)
\Phi n(\widehat \varphi , x) dx+ f(1)
1\int
0
\Phi n(\widehat \varphi , x) dx =
=M1 +M2 +M3. (10)
Let now f(x) \in \mathrm{L}\mathrm{i}\mathrm{p} 1. Then by (6) and Remark 1 we have
| M1| = O(1)
1
n
n - 1\sum
i=1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(\widehat \varphi , x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)Sn(\widehat \varphi ), (11)
| M2| = O(1)
1
n
n\sum
i=1
i/n\int
i - 1
n
| \Phi n(\widehat \varphi , x)| dx = O(1)
1
n
1\int
0
| \Phi n(\widehat \varphi , x)| dx =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
470 L. GOGOLADZE, V. TSAGAREISHVILI
= O(1)
1
n
\left( 1\int
0
\Phi 2
n(\widehat \varphi , x) dx
\right) 1/2 = O(1)
1
n
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda
2
k
\Biggr) 1/2
=
= O(1)
\surd
\lambda n
n
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k
\Biggr) 1/2
= O(1)
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k
\Biggr) 1/2
, (12)
| M3| = O(1)
\bigm| \bigm| \bigm| \bigm| n\sum
k=1
\widehat \varphi k(f)\lambda k
1\int
0
\varphi n(x) dx
\bigm| \bigm| \bigm| \bigm| =
= O(1)
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k
\Biggr) 1/2\left( n\sum
k=1
\left( 1\int
0
\varphi k(x) dx
\right) 2 \lambda k
\right)
1/2
=
= O(1)
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k
\Biggr) 1/2
. (13)
(Here condition b) of Theorem 1 is taken into account.)
Finally, in (10) taking into account (9), (11), (12) and (13), we get
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k = O(1)
\Biggl(
n\sum
k=1
\widehat \varphi 2
k(f)\lambda k
\Biggr) 1/2
,
i.e.,
\infty \sum
k=1
\widehat \varphi 2
k(f)\lambda k < +\infty .
Taking into account now the last inequality from the statement of Theorem A, it follows that
series (7) converges a.e. on [0, 1].
Theorem 1 is proved.
It should be noted that condition (6) is an important factor for the convergence of the Fourier
series of functions from class \mathrm{L}\mathrm{i}\mathrm{p} 1. We will show later what we will have if the condition is not
satisfied.
Theorem 2. Let \{ \varphi n(x)\} be ONS on [0, 1] and \lambda n \uparrow \infty . If for some sequence (bn) \in \ell 2
condition (6) does not hold, i.e.,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Sn(b)\Bigl( \sum n
k=1
b2k\lambda k
\Bigr) 1/2 = +\infty ,
then there exists the function f0(x) \in \mathrm{L}\mathrm{i}\mathrm{p} 1 such that
\infty \sum
n=1
\widehat \varphi 2
n(f0)\lambda n = +\infty .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 471
Proof. Let us assume from the beginning that
\infty \sum
k=1
\left( 1\int
0
\varphi k(x) dx
\right) 2 \lambda k < +\infty . (14)
Otherwise we will get that the Fourier coefficients of the function f0(x) = 1 do not satisfy condi-
tion (3) and thus Theorem 2 is valid.
If series (14) is convergent, then we will consider the sequence of functions
fn(x) =
x\int
0
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}
t\int
0
\Phi n(b, u) du dt, n = 1, 2, . . . . (15)
Assume in equality (10) f(x) = fn(x) and \Phi n(a, x) = \Phi n(b, x) we will have
1\int
0
fn(x)\Phi n(b, x) dx =
n - 1\sum
i=1
\biggl(
fn
\biggl(
i
n
\biggr)
- fn
\biggl(
i+ 1
n
\biggr) \biggr) i/n\int
0
\Phi n(b, x) dx+
+
n\sum
i=1
i/n\int
i - 1
n
\biggl(
fn(x) - fn
\biggl(
i
n
\biggr) \biggr)
\Phi n(b, x) dx+ fn(1)
1\int
0
\Phi n(b, x) dx =
= H1 +H2 +H3. (16)
Since for x \in
\biggl[
i - 1
n
,
i
n
\biggr]
\bigm| \bigm| \bigm| \bigm| fn(x) - fn
\biggl(
i
n
\biggr) \bigm| \bigm| \bigm| \bigm| \leq 1
n
,
using Remark 1 we obtain
| H2| \leq
1
n
n\sum
i=1
i/n\int
i - 1
n
| \Phi n(b, x)| dx =
1
n
1\int
0
| \Phi n(b, x)| dx \leq 1
n
\Biggl(
n\sum
k=1
b2k\lambda
2
k
\Biggr) 1/2
=
= O(1)
\surd
\lambda n
n
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
= O(1)
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
. (17)
Taking into account (14) we get
| H3| \leq | fn(1)|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = | fn(1)|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=1
bk\lambda k
1\int
0
\varphi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= O(1)
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2\left( n\sum
k=1
\left( 1\int
0
\varphi k(x) dx
\right) 2 \lambda k
\right)
1/2
=
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
472 L. GOGOLADZE, V. TSAGAREISHVILI
= O(1)
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
. (18)
For estimating M1 we will need the following lemma.
Lemma 1. Let In be a set of all i \in \{ 1, 2, . . . , n\} for each of which there exists the point
xin \in
\biggl[
i - 1
n
,
i
n
\biggr]
for which the condition
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}
xin\int
0
\Phi n(b, t) dt \not = \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}
i/n\int
0
\Phi n(b, t) dt
holds. Then (see Remark 2)
\sum
i\in In
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)
\sqrt{}
\lambda n
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
.
Proof. In view of condition of the lemma there exists the point yin \in
\biggl[
i - 1
n
,
i
n
\biggr]
such that\int yin
0
\Phi n(b, x) dx = 0. Then
\sum
i\in In
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\sum
i\in In
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
yin\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\sum
i\in In
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
yin
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\sum
i\in In
i/n\int
i - 1
n
| \Phi n(b, x)| dx \leq
1\int
0
| \Phi n(b, x)| dx \leq
\Biggl(
n\sum
k=1
b2k\lambda
2
k
\Biggr) 1/2
=
= O(1)
\sqrt{}
\lambda n
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
. (19)
The lemma is proved.
Assume now that En = \{ 1, 2, . . . , n - 1\} \setminus In. Then according to the property of En we get
n - 1\sum
i=1
\biggl(
fn
\biggl(
i
n
\biggr)
- fn
\biggl(
i+ 1
n
\biggr) \biggr) i/n\int
0
\Phi n(b, x) dx =
= -
n - 1\sum
i=1
(i+1)/n\int
i/n
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}
x\int
0
\Phi n(b, t) dt dx
i/n\int
0
\Phi n(b, x) dx =
= - 1
n
\sum
i\in En
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
\sum
i\in In
(i+1)/n\int
i/n
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}
x\int
0
\Phi n(b, t) dt dx
i/n\int
0
\Phi n(b, x) dx.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 473
Hence, in view of (19) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\sum
i=1
\biggl(
fn
\biggl(
i
n
\biggr)
- fn
\biggl(
i+ 1
n
\biggr) \biggr) i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq 1
n
\sum
i\in En
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
1
n
\sum
i\in In
| \Phi n(b, x) dx| =
=
1
n
n\sum
i=1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
2
n
\sum
i\in In
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq Sn(b) - O(1)
1
n
\sqrt{}
\lambda n
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
\geq Sn(b) - O(1)
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
. (20)
At last, applying (17), (18) and (20) from (16) we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
fn(x)\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq Sn(b) - O(1)
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2
.
From here
1\Bigl( \sum n
k=1
b2k\lambda k
\Bigr) 1/2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
fn(x)\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq Sn(b)\Bigl( \sum n
k=1
b2k\lambda k
\Bigr) 1/2 - O(1). (21)
According to the conditions of Theorem 2 from (21) we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1\Bigl( \sum n
k=1
b2k\lambda k
\Bigr) 1/2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
fn(x)\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Sn(b)\Bigl( \sum n
k=1
b2k\lambda k
\Bigr) 1/2 = +\infty . (22)
Evidently,
\| fn\| Lip 1 = \| fn\| C + \mathrm{s}\mathrm{u}\mathrm{p}
x,y
| fn(x) - fn(y)|
| x - y|
= 2.
Thus, in virtue of Banach – Steinhaus’ theorem (see (22)) there exists the function f0(x) \in \mathrm{L}\mathrm{i}\mathrm{p} 1 such
that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigm| \bigm| \bigm| \bigm| \int 1
0
f0(x)\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \Bigl( \sum n
n=1
b2k\lambda k
\Bigr) 1/2 = +\infty . (23)
Since \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
f0(x)\Phi n(b, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| n\sum
k=1
bk\lambda k
1\int
0
f0(x)\varphi n(x) dx
\bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
474 L. GOGOLADZE, V. TSAGAREISHVILI
=
\bigm| \bigm| \bigm| \bigm| n\sum
k=1
bk\lambda k \widehat \varphi k(f0)
\bigm| \bigm| \bigm| \bigm| \leq
\Biggl(
n\sum
k=1
b2k\lambda k
\Biggr) 1/2\Biggl( n\sum
k=1
\widehat \varphi 2
k(f0)\lambda k
\Biggr) 1/2
,
from (23) we get
\infty \sum
k=1
\widehat \varphi 2
k(f0)\lambda k = +\infty .
Theorem 2 is proved.
As is was stated above, the Fourier coefficients of functions from \mathrm{L}\mathrm{i}\mathrm{p} 1 do not, in general, satisfy
condition (3).
Theorem 3. From any ONS \{ \varphi n(x)\} one can isolate a subsystem \{ \varphi nk
(x)\} = \{ \psi k(x)\} with
respect to which for any sequence (an) \in \ell 2 the condition
Sn(a) =
1
n
n - 1\sum
i=1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
Pn(a, x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)
\Biggl(
n\sum
k=1
a2k\lambda k
\Biggr) 1/2
holds, where Pn(a, x) =
\sum n
k=1
ak\lambda k\psi k(x).
Proof. According to Parceval’s theorem
\infty \sum
m=1
\left( i/n\int
0
\varphi m(x) dx
\right)
2
\leq 1, i = 1, . . . , n,
by k(n) we denote the natural number for which
\infty \sum
m=k(n)
\left( i/n\int
0
\varphi m(x) dx
\right)
2
\leq 2 - 2n.
Thus, when m \geq k(n) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\varphi m(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 2 - n, i = 1, 2, . . . , n. (24)
Assume that k(n+ 1) \geq k(n) + 2n (k(n+ 1) > k(n)).
Let
\psi 2n+1(x) = \varphi k(n)+1(x), \psi 2n+s(x) = \varphi k(n)+s(x),
where 1 \leq s \leq 2n, n = 1, 2, . . . . In this way we obtain the sequence of functions (\psi m(x)) and for
m = 2n + l (1 \leq l \leq 2n) (see (24))\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\psi m(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
\varphi k(n)+l(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 2 - n, i = 1, 2, . . . , n. (25)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 475
The number j(i), i = 1, . . . , n, j = 1, . . . , 2m, is chosen so that
\bigm| \bigm| \bigm| \bigm| in - j(i)
2m
\bigm| \bigm| \bigm| \bigm| \leq 2 - m, n = 2m+ l.
Then by using (25), we get \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
2m+1 - 1\sum
s=2m
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
j(i)/2m\int
i/n
2m+1 - 1\sum
s=2m
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
j(i)/2m\int
0
2m+1 - 1\sum
s=2m
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2 - m/2
\left( 2m+1 - 1\sum
s=2m
a2s\lambda
2
s
\right) 1/2 + 2m+1 - 1\sum
s=2m
| as\lambda s| \cdot 2 - m. (26)
Assume now that n = 2d + n1, n1 < 2d, then according to (26)\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
d - 1\sum
m=0
2m+1 - 1\sum
s=2m
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
d - 1\sum
m=0
2 - m/2
\left( 2m+1 - 1\sum
s=2m
a2s\lambda
2
s
\right) 1/2 + d - 1\sum
m=0
2m+1 - 1\sum
s=2m
| as| \lambda s \cdot 2 - m \leq
\leq
d - 1\sum
m=0
2 - m/2
\sqrt{}
\lambda 2m+1
\left( 2m+1 - 1\sum
s=2m
a2s\lambda
2
s
\right) 1/2 + d - 1\sum
m=0
2 - m \cdot
\sqrt{}
\lambda 2m+1 \cdot 2m/2
\left( 2m+1 - 1\sum
s=2m
a2s\lambda s
\right) 1/2 \leq
\leq
\Biggl(
d - 1\sum
m=0
2 - m\lambda 2m+1
\Biggr) 1/2\left( d - 1\sum
m=0
2m+1 - 1\sum
s=2m
a2s\lambda s
\right) 1/2+
+
\Biggl(
d - 1\sum
m=0
2 - m\lambda 2m+1
\Biggr) 1/2\left( d - 1\sum
m=0
2m+1 - 1\sum
s=2m
a2s\lambda s
\right) 1/2 =
= O(1)
\left( 2d - 1\sum
k=1
a2k\lambda k
\right) 1/2 . (27)
Further (n < 2d+1), \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
n\sum
s=2m
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
476 L. GOGOLADZE, V. TSAGAREISHVILI
\leq
j(i)/2d\int
i/n
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
s=2d
as\lambda s\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| +
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
s=2d
as\lambda s
j(i)/2d\int
0
\psi s(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2 - d/2
\Biggl(
n\sum
s=2d
a2s\lambda
2
s
\Biggr) 1/2
+
n\sum
s=2d
| as| \lambda s2 - d \leq
\leq 2 - d/2
\sqrt{}
\lambda n
\Biggl(
n\sum
s=2d
a2s\lambda s
\Biggr) 1/2
+ 2 - d/2
\sqrt{}
\lambda n
\Biggl(
n\sum
s=2d
a2s\lambda s
\Biggr) 1/2
= O(1)
\Biggl(
n\sum
s=2d
a2s\lambda s
\Biggr) 1/2
. (28)
Finally, from (27) and (28) for any sequence (an) \in \ell 2 we have
Sn(a) =
1
n
n - 1\sum
i=1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
n\sum
m=1
am\lambda m\psi m(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)
\Biggl(
n\sum
m=1
a2m\lambda m
\Biggr) 1/2
. (29)
From (29) and Theorem 1 it follows that for any function f(x) \in \mathrm{L}\mathrm{i}\mathrm{p} 1 condition (3) holds in the
case of the subsequence \{ \psi n(x)\} .
Theorem 3 is proved.
Problems of efficiency. Condition (6) is said to be efficient if it is easily verified for classical
ONS (trigonometric system, Walsh [6] and Haar systems).
Theorem 4. Let \{ \varphi n(x)\} be ONS such that
x\int
0
\varphi n(x) dx = O
\biggl(
1
n
\biggr)
.
Then condition (6) is fulfilled.
Proof. For any i = 1, 2, . . . , n we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
n\sum
k=1
ak\lambda k\varphi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=1
ak\lambda k
i/n\int
0
\varphi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= O(1)
n\sum
k=1
| ak|
k
\lambda k = O(1)
\Biggl(
n\sum
k=1
\lambda k
k2
\Biggr) 1/2\Biggl( n\sum
k=1
a2k\lambda k
\Biggr) 1/2
= O(1)
\Biggl(
n\sum
k=1
a2k\lambda k
\Biggr) 1/2
.
From here and from Theorem 1 it follows that for the trigonometric and Walsh systems condi-
tion (6) is satisfied.
Theorem 5. If (\chi n) is a Haar system (see [2, p. 54]), then condition (6) is fulfilled.
Proof. Using the definition of the Haar function, we get (n = 2p + l, l < 2p)\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
2s+1 - 1\sum
k=2s
ak\lambda k\chi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 2 - s/2| ak(i)| \lambda k(i), where 2s \leq k(i) < 2s+1.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
CONVERGENCE OF FOURIER SERIES OF FUNCTIONS \mathrm{L}\mathrm{i}\mathrm{p} 1 . . . 477
Therefore, \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
2p - 1\sum
k=1
ak\lambda k\chi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
k=0
i/n\int
0
2k+1 - 1\sum
m=2k
am\lambda m\chi m(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
p - 1\sum
k=0
2 - k/2| ak(i)| \lambda k(i) \leq
p - 1\sum
k=0
| ak(i)|
\sqrt{}
\lambda k(i)
\sqrt{}
\lambda k(i)2
- k/2 \leq
\leq
\Biggl(
p - 1\sum
k=0
a2k(i)\lambda k(i)
\Biggr) 1/2\Biggl( p - 1\sum
k=0
2 - k\lambda k(i)
\Biggr) 1/2
\leq
\leq
\left( p - 1\sum
k=0
2k+1 - 1\sum
m=2k
a2m\lambda m
\right) 1/2\Biggl( p - 1\sum
k=0
2 - k\lambda k(i)
\Biggr) 1/2
= O(1)
\Biggl(
2p\sum
k=1
a2k\lambda k
\Biggr) 1/2
, i = 1, 2, . . . , n.
In a similar way it can be proved that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
n\sum
k=2p
ak\lambda k\chi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)
\Biggl(
n\sum
k=2p
a2k\lambda k
\Biggr) 1/2
.
At last we can conclude that
Sn(a) =
1
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
i/n\int
0
n\sum
k=1
ak\lambda k\chi k(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = O(1)
\Biggl(
n\sum
k=1
a2k\lambda k
\Biggr) 1/2
.
Theorem 5 is proved.
References
1. Kačmarž S., Šteı̆ngauz G. Theory of orthogonal series (in Russian). – Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit.,
1958.
2. Alexits G. Convergence problems of orthogonal series // Int. Ser. Monogr. in Pure and Appl. Math. – New York etc.:
Pergamon Press, 1961. – 20.
3. Banach S. Sur la divergence des séries orthogonales // Stud. Math. – 1940. – 9. – P. 139 – 155.
4. Olevskiı̆ A. M. Orthogonal series in terms of complete systems (in Russian) // Mat. Sb. (N. S.). – 1962. – 58 (100). –
P. 707 – 748.
5. Gogoladze L., Tsagareishvili V. Some classes of functions and Fourier coefficients with respect to general orthonormal
systems (in Russian) // Tr. Mat. Inst. Steklova. – 2013. – 280. – P. 162 – 174.
6. Fine N. J. On the Walsh functions // Trans. Amer. Math. Soc. – 1949. – 65. – P. 372 – 414.
Received 04.11.15,
after revision — 25.11.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
|
| id | umjimathkievua-article-1710 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:09Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/38/f53c3992aa1e70ecba041f02852c0338.pdf |
| spelling | umjimathkievua-article-17102019-12-05T09:24:35Z Convergence of Fourier series of functions $\text{Lip} 1$ with respect to general orthonormal systems Збiжнiсть рядiв Фур’є функцiй $\text{Lip} 1$ вiдносно загальних ортонормованих систем Gogoladze, L. Tsagareishvili, V. Гоголадзе, Л. Цагарейшвілі, В. We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) $\{ \varphi_n(x)\}$ in order that the Fourier series in this system for any function from the class $\mathrm{L}\mathrm{i}\mathrm{p} 1$ be convergent almost everywhere on $[0, 1]$. It is shown that the obtained conditions are best possible in a certain sense. Встановлено достатнi умови, якi повиннi задовольняти функцiї загальної ортонормованої системи (ЗОС) $\{ \varphi_n(x)\}$, для того, щоб ряд Фур’є вiдносно вказаної системи для будь-якої функцiї з класу $\mathrm{L}\mathrm{i}\mathrm{p} 1$ збiгався майже скрiзь на $[0, 1]$. Показано, що отриманi розв’язки є, в деякому розумiннi, найкращими. Institute of Mathematics, NAS of Ukraine 2017-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1710 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 4 (2017); 466-477 Український математичний журнал; Том 69 № 4 (2017); 466-477 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1710/692 Copyright (c) 2017 Gogoladze L.; Tsagareishvili V. |
| spellingShingle | Gogoladze, L. Tsagareishvili, V. Гоголадзе, Л. Цагарейшвілі, В. Convergence of Fourier series of functions $\text{Lip} 1$ with respect to general orthonormal systems |
| title | Convergence of Fourier series of functions $\text{Lip} 1$ with respect
to general orthonormal systems |
| title_alt | Збiжнiсть рядiв Фур’є функцiй $\text{Lip} 1$
вiдносно загальних ортонормованих систем |
| title_full | Convergence of Fourier series of functions $\text{Lip} 1$ with respect
to general orthonormal systems |
| title_fullStr | Convergence of Fourier series of functions $\text{Lip} 1$ with respect
to general orthonormal systems |
| title_full_unstemmed | Convergence of Fourier series of functions $\text{Lip} 1$ with respect
to general orthonormal systems |
| title_short | Convergence of Fourier series of functions $\text{Lip} 1$ with respect
to general orthonormal systems |
| title_sort | convergence of fourier series of functions $\text{lip} 1$ with respect
to general orthonormal systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1710 |
| work_keys_str_mv | AT gogoladzel convergenceoffourierseriesoffunctionstextlip1withrespecttogeneralorthonormalsystems AT tsagareishviliv convergenceoffourierseriesoffunctionstextlip1withrespecttogeneralorthonormalsystems AT gogoladzel convergenceoffourierseriesoffunctionstextlip1withrespecttogeneralorthonormalsystems AT cagarejšvílív convergenceoffourierseriesoffunctionstextlip1withrespecttogeneralorthonormalsystems AT gogoladzel zbižnistʹrâdivfurêfunkcijtextlip1vidnosnozagalʹnihortonormovanihsistem AT tsagareishviliv zbižnistʹrâdivfurêfunkcijtextlip1vidnosnozagalʹnihortonormovanihsistem AT gogoladzel zbižnistʹrâdivfurêfunkcijtextlip1vidnosnozagalʹnihortonormovanihsistem AT cagarejšvílív zbižnistʹrâdivfurêfunkcijtextlip1vidnosnozagalʹnihortonormovanihsistem |