Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation
The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.
Saved in:
| Date: | 2015 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2015
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1971 |
| 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_ | 1860507872722419712 |
|---|---|
| author | Goginava, U. Sahakian, A. Гогінава, У. Сахакян, А. |
| author_facet | Goginava, U. Sahakian, A. Гогінава, У. Сахакян, А. |
| author_sort | Goginava, U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:47:54Z |
| description | The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series. |
| first_indexed | 2026-03-24T02:16:13Z |
| format | Article |
| fulltext |
UDC 517.5
U. Goginava (Iv. Javakhishvili Tbilisi State Univ., Georgia),
A. Sahakian (Yerevan State Univ., Armenia)
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS
OF BOUNDED GENERALIZED VARIATION*
ЗБIЖНIСТЬ КРАТНИХ РЯДIВ ФУР’Є ФУНКЦIЙ
З ОБМЕЖЕНОЮ УЗАГАЛЬНЕНОЮ ВАРIАЦIЄЮ
The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the
convergence of multidimensional Fourier series.
Введено нову концепцiю Λ-варiацiї функцiй багатьох змiнних та вивчено її зв’язок зi збiжнiстю багатовимiрних
рядiв Фур’є.
1. Classes of functions of bounded generalized variation. In 1881 Jordan [11] introduced a class of
functions of bounded variation and applied it to the theory of Fourier series. Hereafter this notion was
generalized by many authors (quadratic variation, Φ-variation, Λ-variation etc., see [2, 12, 15, 17]). In
two-dimensional case the class BV of functions of bounded variation was introduced by Hardy [10].
For an interval T = [a, b] ⊂ R we denote by T d = [a, b]d the d-dimensional cube in Rd.
Consider a function f(x) defined on T d and a collection of intervals
Jk = (ak, bk) ⊂ T, k = 1, 2, . . . , d.
For d = 1 we set
f(J1) := f(b1)− f(a1).
If for any function of d− 1 variables the expression f(J1 × . . .× Jd−1) is already defined, then for
a function of d variables the mixed difference is defined as follows:
f
(
J1 × . . .× Jd
)
:= f
(
J1 × . . .× Jd−1, bd
)
− f
(
J1 × . . .× Jd−1, ad
)
.
Let E = {Ik} be a collection of nonoverlapping intervals from T ordered in arbitrary way and
let Ω = Ω(T ) be the set of all such collections E. We denote by Ωn = Ωn(T ) set of all collections
of n nonoverlapping intervals Ik ⊂ T.
For sequences of positive numbers
Λj = {λjn}∞n=1, lim
n→∞
λjn =∞, j = 1, 2, . . . , d,
and for a function f(x), x = (x1, . . . , xd) ∈ T d the (Λ1, . . . ,Λd)-variation of f with respect to the
index set D := {1, 2, . . . , d} is defined as follows:
{
Λ1, . . . ,Λd
}
V D
(
f, T d
)
:= sup
{Ijij }∈Ω
∑
i1,...,id
∣∣f(I1
i1
× . . .× Idid)
∣∣
λ1
i1
. . . λdid
.
* The research of U. Goginava was supported by Shota Rustaveli National Science Foundation, grant no. 31/48
(Operators in some function spaces and their applications in Fourier analysis).
c© U. GOGINAVA, A. SAHAKIAN, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2 163
164 U. GOGINAVA, A. SAHAKIAN
For an index set α = {j1, . . . , jp} ⊂ D and any x = (x1, . . . , xd) ∈ Rd we set α̃ := D \ α and
denote by xα the vector of Rp consisting of components xj , j ∈ α, i.e.,
xα = (xj1 , . . . , xjp) ∈ Rp.
By
{Λj1 , . . . ,Λjp}V α
(
f, xα̃, T
d
)
and f
(
I1
ij1
× . . .× Ipijp , xα̃
)
we denote respectively the (Λj1 , . . . ,Λjp)-variation over the p-dimensional cube T p and mixed
difference of f as a function of variables xj1 , . . . , xjp with fixed values xα̃ of other variables. The
(Λj1 , . . . ,Λjp)-variation of with respect to the index set α is defined as follows:{
Λj1 , . . . ,Λjp
}
V α(f, T p) = sup
xα̃∈T d−p
{
Λj1 , . . . ,Λjp
}
V α
(
f, xα̃ , T d
)
.
Definition 1. We say that the function f has total bounded (Λ1, . . . ,Λd)-variation on T d and
write f ∈ {Λ1, . . . ,Λd}BV (T d), if
{Λ1, . . . ,Λd}V (f, T d) :=
∑
α⊂D
{Λ1, . . . ,Λd}V α
(
f, T d
)
<∞.
Definition 2. We say that the function f is continuous in (Λ1, . . . ,Λd)-variation on T d and write
f ∈ C{Λ1, . . . ,Λd}V (T d), if
lim
n→∞
{
Λj1 , . . . ,Λjk−1 ,Λjkn ,Λ
jk+1 , . . . ,Λjp
}
V α
(
f, T d
)
= 0, k = 1, 2, . . . , p,
for any α ⊂ D, α := {j1, . . . , jp}, where Λjkn := {λjks }∞s=n.
Definition 3. We say that the function f has bounded partial (Λ1, . . . ,Λd)-variation and write
f ∈ P{Λ1, . . . ,Λd}BV (T d) if
P{Λ1, . . . ,Λd}V (f, T d) :=
d∑
i=1
ΛiV {i}
(
f, T d
)
<∞.
In the case Λ1 = . . . = Λd = Λ we set
ΛBV (T d) := {Λ1, . . . ,Λd}BV (T d),
CΛV (T d) := C{Λ1, . . . ,Λd}V (T d),
PΛBV (T d) := P{Λ1, . . . ,Λd}BV (T d).
If λn ≡ 1 (or if 0 < c < λn < C < ∞, n = 1, 2, . . .) the classes ΛBV and PΛBV coincide with
the Hardy class BV and PBV respectively. Hence it is reasonable to assume that λn →∞.
When λn = n for all n = 1, 2, . . . we say Harmonic Variation instead of Λ-variation and write
H instead of Λ, i.e., HBV, PHBV, CHV, etc.
For two variable functions Dyachenko and Waterman [5] introduced another class of functions of
generalized bounded variation.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS OF BOUNDED GENERALIZED . . . 165
Denoting by Γ the the set of finite collections of nonoverlapping rectangles Ak := [αk, βk] ×
× [γk, δk] ⊂ T 2, for a function f(x, y), x, y ∈ T, we set
Λ∗V (f, T 2) := sup
{Ak}∈Γ
∑
k
|f(Ak)|
λk
.
Definition 4 (Dyachenko, Waterman). We say that f ∈ Λ∗BV (T 2) if
ΛV (f, T 2) := ΛV1(f, T 2) + ΛV2(f, T 2) + Λ∗V (f, T 2) <∞.
In this paper we introduce a new classes of functions of generalized bounded variation and
investigate the convergence of Fourier series of function of that classes.
For the sequence Λ = {λn}∞n=1 we denote
Λ#Vs
(
f, T d
)
:= sup
{xi{s}}⊂T d−1
sup
{Isi }∈Ω
∑
i
∣∣f(Isi , x
i{s})
∣∣
λi
,
where
xi{s} :=
(
xi1, . . . , x
i
s−1, x
i
s+1, . . . , x
i
d
)
for xi :=
(
xi1, . . . , x
i
d
)
. (1)
Definition 5. We say that the function f belongs to the class Λ#BV (T d), if
Λ#V
(
f, T d
)
:=
d∑
s=1
Λ#Vs
(
f, T d
)
<∞.
The notion of Λ-variation was introduced by Waterman [15] in one-dimensional case, by Sahakian
[14] in two-dimensional case and by Sablin [13] in the case of higher dimensions. The notion of
bounded partial variation (class PBV ) was introduced by Goginava in [7]. These classes of functions
of generalized bounded variation play an important role in the theory Fourier series.
Remark 1. It is not hard to see that Λ#BV (T d) ⊂ PΛBV (T d) for any d > 1 and Λ∗BV (T 2) ⊂
⊂ Λ#BV (T 2).
We prove that the following theorem is true.
Theorem 1. Let d ≥ 2 and T = (t1, t2) ⊂ R. If
Λ = {λn} with λn =
n
logd−1(n+ 1)
, n = 1, 2, . . . , (2)
then
HV
(
f, T d
)
≤M(d)Λ#V (f, T d). (3)
Proof. We have to prove that for any α := {j1, . . . , jp} ⊂ D
sup
{Ijij }∈Ω
∑
i1,...,ip
|f(I1
i1
× . . .× Ipip , xα̃)|
i1 . . . ip
≤M(d)
d∑
s=1
Λ#Vs(f, T
d). (4)
To this end, observe that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
166 U. GOGINAVA, A. SAHAKIAN
∑
i1,..., ip
|f(I1
i1
× . . .× Ipip , xα̃)|
i1 . . . ip
=
∑
σ
∑
iσ(1)≤...≤iσ(p)
|f(I1
i1
× . . .× Ipip , xα̃)|
i1 . . . ip
, (5)
where the sum is taken over all rearrangements σ =
{
σ(k)
}p
k=1
of the set {1, 2, . . . , p}.
Next, we have
∑
i1≤...≤ip
∣∣f(I1
i1
× . . .× Ipip , xα̃)
∣∣
i1 . . . ip
=
∑
ip
1
ip
∑
i1≤...≤ip
∣∣f(I1
i1
× . . .× Ipip , xα̃)
∣∣
i1 . . . ip−1
. (6)
Taking into account that for the fixed ip, i1 ≤ . . . ≤ ip, there exists xip1 , . . . , x
ip
p−1 ∈ T such that∣∣f(I1
i1 × . . .× I
p
ip
, xα̃)
∣∣ ≤ 2d
∣∣∣f(Ipip , xip1 , . . . , xipp−1, xα̃
)∣∣∣
from (6) we obtain
∑
i1≤...≤ip
∣∣f(I1
i1
× . . .× Ipip , xα̃)
∣∣
i1 . . . ip
≤ 2d
∑
ip
∣∣f(Ipip , x
ip
1 , . . . , x
ip
p−1, xα̃)
∣∣
ip
∑
i1≤...≤ip
1
i1 . . . ip−1
≤
≤M(d)
∑
ip
logd−1(ip + 1)
ip
∣∣f(Ipip , x
ip
1 , . . . , x
ip
p−1, xα̃)
∣∣ ≤
≤M(d)Λ#Vip
(
f, T d
)
≤M(d)Λ#V
(
f, T d
)
.
Similarly one can obtain bounds for other summands in the right-hand side of (5), which imply (3).
Theorem 1 is proved.
Corollary 1. If the sequence Λ is defined by (2), then Λ#BV (T d) ⊂ HBV (T d).
Now, we denote
∆ :=
{
δ = (δ1, . . . , δd) : δi = ±1, i = 1, 2, . . . , d
}
(7)
and
πεδ(x) := (x1, x1 + εδ1)× . . .× (xd, xd + εδd),
for x = (x1, . . . , xd) ∈ Rd and ε > 0. We set πδ(x) := πεδ(x), if ε = 1.
For a function f defined in some neighbourhood of a point x and δ ∈ ∆ we set
fδ(x) := lim
t∈πδ(x), t→x
f(t), (8)
if the last limit exists.
Theorem 2. Suppose f ∈ Λ#BV (T d) for some sequence Λ = {λn}.
(a) If the limit fδ(x) exists for some x = (x1, . . . , xd) ∈ T d and some δ = (δ1, . . . , δd) ∈ ∆,
then
lim
ε→0
Λ#V (f, πεδ(x)) = 0. (9)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS OF BOUNDED GENERALIZED . . . 167
(b) If f is continuous on some compact K ⊂ T d, then
lim
ε→0
Λ#V
(
f, [x1 − ε, x1 + ε]× . . .× [xd − ε, xd + ε]
)
= 0 (10)
uniformly with respect to x = (x1, . . . , xd) ∈ K.
Proof. According to Definition 5, we need to prove that
lim
ε→0
Λ#Vs
(
f, πεδ(x)
)
= 0 (11)
for any s = 1, 2, . . . , d. Without loss of generality we can assume that s = 1 and δi = 1 for
i = 1, 2, . . . , d. Assume to the contrary that (11) does not holds:
lim
ε→0
Λ#V1
(
f, πεδ(x)
)
6= 0.
Then there exists a number α such that
Λ#V1
(
f, πεδ(x)
)
> α > 0 (12)
for any ε > 0.
Using induction on k = 1, 2, . . . , we construct positive numbers εk and the sequences of collec-
tions of nonoverlapping intervals
I1
i ⊂ (x1 + εk+1, x1 + εk), i = nk + 1, . . . , nk+1, (13)
and vectors
βi = (βi1, . . . , β
i
d) ∈ πεkδ(x), i = nk + 1, . . . , nk+1, (14)
as follows. By (12), for a fixed number ε1 > 0 we find a collection of nonoverlapping intervals
I1
i ⊂ (x1, x1 + ε1), i = 1, . . . , n1,
and vectors
βi = (βi1, . . . , β
i
d) ∈ πε1δ(x), i = 1, . . . , n1,
such that
n1∑
i=1
∣∣f(I1
i ;βi2, . . . , β
i
d)
∣∣
λi
> α. (15)
Now, suppose the number εk, intervals (13) and the vectors (14) for some k = 1, 2, . . . are
constructed. Since the limit fδ(x) exists, we can choose εk+1 satisfying
0 < εk+1 < εk, (x1, x1 + εk+1) ∩
(
nk⋃
i=1
I1
i
)
= ∅ (16)
and
nk∑
i=1
∣∣f(J1
i ; γi2, . . . , γ
i
d)
∣∣
λi
<
α
2
(17)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
168 U. GOGINAVA, A. SAHAKIAN
for any collection of nonoverlapping intervals
J1
i ⊂ (x1, x1 + εk+1), i = 1, . . . , nk,
and for any vectors
γi = (γi1, . . . , γ
i
d) ∈ πεk+1δ(x), i = 1, . . . , nk.
Further, according to (12) there is a collection of nonoverlapping intervals
J1
i ⊂ (x1, x1 + εk+1), i = 1, . . . , nk+1, (18)
and vectors
γi = (γi1, . . . , γ
i
d) ∈ πεk+1δ(x), i = 1, . . . , nk+1,
such that
nk+1∑
i=1
∣∣f(J1
i ; γi2, . . . , γ
i
d)
∣∣
λi
> α. (19)
Now, denoting
I1
i = J1
i , βi = γi for i = nk + 1, . . . , nk+1, (20)
from (17) and (19) we get
nk+1∑
i=nk+1
∣∣f(I1
i ;βi2, . . . , β
i
d)
∣∣
λi
>
α
2
. (21)
Intervals (13) and vectors (14) for k = 1, 2, . . . , are constructed.
By (16), (18) and (20), the intervals I1
i are nonoverlapping for i = 1, 2, . . . , while according
to (21),
∞∑
i=1
∣∣f(I1
i ;βi2, . . . , β
i
d)
∣∣
λi
=∞.
Consequently, Λ#V1(f, T d) = ∞. This contradiction completes proof of the statement (a) of Theo-
rem 2.
To prove statement (b), observe that (a) obviously implies (10) for any point x ∈ T d, where f is
continuous. Hence, we have to prove that (10) holds uniformly with respect to x ∈ K, provided that
f is continuous on the compact K ⊂ T d.
To this end let us assume to the contrary that (10) does not hold uniformly on K. Then there exist
δ > 0 and sequences
xi = (xi1, . . . , x
i
d) ∈ K and εi > 0, i = 1, 2, . . . , with εi → 0
such that
Λ#V
(
f ;
[
xi1 − εi, xi1 + εi
]
× . . .×
[
xid − εi, xid + εi
])
≥ δ > 0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS OF BOUNDED GENERALIZED . . . 169
Since K is compact we can assume without loss of generality that xi → x for some x =
= (x1, . . . , xd) ∈ K. Then obviously for each ε > 0 there is a number i(ε) such that
[xij − εi, xij + εi] ⊂ [xj − ε, xj + ε], j = 1, . . . , d, for i > i(ε).
Consequently,
Λ#V
(
f ; [x1 − ε, x1 + ε]× . . .× [xd − ε, xd + ε]
)
≥ δ > 0,
for any ε > 0, which is a contradiction.
Theorem 2 is proved.
Next, we define
v#
s (f, n) := sup
{xi}ni=1⊂T d
sup
{Isi }ni=1∈Ωn
n∑
i=1
∣∣f(Isi , x
i{s})
∣∣, s = 1, . . . , d, n = 1, 2, . . . ,
where xi{s} is as in (1). The following theorem holds.
Theorem 3. If the function f(x), x ∈ T d, satisfies the condition
∞∑
n=1
v#
s (f, n) logd−1(n+ 1)
n2
<∞, s = 1, 2, . . . , d,
then f ∈
{
n
logd−1(n+ 1)
}#
BV (T d).
Proof. Let s = 1, . . . , d be fixed. The for any collection of intervals {Isi }ni=1 ∈ Ωn and a
sequence of vectors {xi}ni=1 ∈ T d, using Abel’s partial summation we obtain
n∑
j=1
∣∣f(Isj , x
j{s})
∣∣ logd−1(j + 1)
j
=
=
n−1∑
j=1
(
logd−1(j + 1)
j
− logd−1(j + 2)
j + 1
)
j∑
k=1
∣∣f(Isk, xk{s})∣∣+
+
logd−1(n+ 1)
n
n∑
j=1
∣∣f(Isj , xj{s})∣∣ ≤
≤
n−1∑
j=1
(
logd−1(j + 1)
j
− logd−1(j + 2)
j + 1
)
v#
s (f, j) +
logd−1(n+ 1)
n
v#
s (f, n). (22)
Using the inequality
logd−1(n+ 1)
n
v#
s (f, n) ≤
∞∑
j=n
(
logd−1(j + 1)
j
− logd−1(j + 2)
j + 1
)
v#
s (f, j), (23)
from (22) we get{
n
logd−1(n+ 1)
}#
Vs(f, T
d) ≤ c
∞∑
n=1
v#
s (f, n) logd−1(n+ 1)
n2
<∞. (24)
Theorem 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
170 U. GOGINAVA, A. SAHAKIAN
2. Convergence of multiple Fourier series. We suppose throughout this section, that T =
= [0, 2π) and T d = [0, 2π)d, d ≥ 2, stands for the d-dimensional torus.
We denote by C(T d) the space of continuous and 2π-periodic with respect to each variable
functions with the norm
‖f‖C := sup
(x1,..., xd)∈T d
|f(x1, . . . , xd)|.
The Fourier series of the function f ∈ L1(T d) with respect to the trigonometric system is the
series
Sf(x1, . . . , xd) :=
+∞∑
n1,...,nd=−∞
f̂(n1, . . . ., nd)e
i(n1x1+...+ndxd),
where
f̂(n1, . . . ., nd) =
1
(2π)d
∫
T d
f(x1, . . . , xd)e−i(n1x1+...+ndxd)dx1 . . . dxd
are the Fourier coefficients of f.
In this paper we consider convergence of only rectangular partial sums (convergence in the
sense of Pringsheim) of d-dimensional Fourier series. Recall that the rectangular partial sums are
defined as follows:
SN1,...,Ndf(x1, . . . , xd) :=
N1∑
n1=−N1
. . .
Nd∑
nd=−Nd
f̂(n1, . . . ., nd)e
i(n1x1+...+ndx
d).
We say that the point x ∈ T d is a regular point of a function f, if the limit fδ(x) defined by (8)
exists for any δ ∈ ∆ (see (7)). For the regular point x we denote
f∗(x) :=
1
2d
∑
δ∈∆
fδ(x). (25)
Definition 6. We say that the class of functions V ⊂ L1(T d) is a class of convergence on T d, if
for any function f ∈ V
1) the Fourier series of f converges to f∗(x) at any regular point x ∈ T d,
2) the convergence is uniform on a compact K ⊂ T d, if f is continuous on K.
The well known Dirichlet – Jordan theorem (see [18]) states that the Fourier series of a function
f(x), x ∈ T, of bounded variation converges at every point x to the value [f(x+ 0) + f(x− 0)]/2.
If f is in addition continuous on T, then the Fourier series converges uniformly on T.
Hardy [10] generalized the Dirichlet – Jordan theorem to the double Fourier series and proved that
BV is a class of convergence on T 2.
The following theorem was proved by Waterman (for d = 1) and Sahakian (for d = 2).
Theorem WS (Waterman [15], Sahakian [14]). If d = 1 or d = 2, then the class HBV (T d) is
a class of convergence on T d.
In [1] Bakhvalov proved that the class HBV is not a class of convergence on T d, if d > 2. On
the other hand, he proved the following theorem.
Theorem B (Bakhvalov [1]). The class CHV (T d) is a class of convergence on T d for any
d = 1, 2, . . . .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS OF BOUNDED GENERALIZED . . . 171
Convergence of spherical and other partial sums of double Fourier series of functions of bounded
Λ-variation was investigated in deatails by Dyachenko [3, 4].
In [8, 9] Goginava and Sahakian investigated convergence of multiple Fourier series of functions
of bounded partial Λ-variation. In particular, the following theorem was proved.
Theorem GS. (a) If and Λ = {λn}∞n=1 with
λn =
n
logd−1+ε(n+ 1)
, n = 1, 2, . . . , d > 1,
for some ε > 0, then the class PΛBV (T d) is a class of convergence on T d.
(b) If Λ = {λn}∞n=1 with
λn =
n
logd−1(n+ 1)
, n = 1, 2, . . . , d > 1,
then the class PΛBV (T d) is not a class of convergence on T d.
In [5], Dyachenko and Waterman proved that the class Λ∗BV (T 2) is a class convergence on T 2
for Λ = {λn} with λn =
n
ln(n+ 1)
, n = 1, 2, . . . .
The main result of the present paper is the following theorem.
Theorem 4. (a) If Λ = {λn}∞n=1 with
λn =
n
logd−1(n+ 1)
, n = 1, 2, . . . , d > 1, (26)
then the class Λ#BV (T d) is a class of convergence on T d.
(b) If Λ = {λn}∞n=1 with
λn :=
{
nξn
logd−1(n+ 1)
}
, n = 1, 2, . . . , d > 1, (27)
where ξn → ∞ as n → ∞, then there exists a continuous function f ∈ Λ#BV (T d) such that the
cubical partial sums of d-dimensional Fourier series of f diverge unboundedly at (0, . . . , 0) ∈ T d.
Proof. The proof of the part (a) is based on the following statement, that in the case d = 2 is
proved by Sahakian (see formulaes (33) and (35) in [14]). For an arbitrary d > 2 the proof is similar.
Lemma S. Suppose f ∈ HV (T d) and x ∈ T d. If the limit fδ(x) exists for any δ ∈ ∆, then for
any ε > 0 ∣∣Sn1,...,ndf(x)− f∗(x)
∣∣ ≤M(d)
∑
δ∈∆
HV (f ;πεδ(x)) + o(1),
as ni →∞, i = 1, 2, . . . , d.
Moreover, the quantity o(1) tends to 0 uniformly on a compact K, if f is continuous on K.
Now, if the sequence Λ = {λn} is defined by (26) and f ∈ Λ#BV (T d), then Lemma S and
Theorem 1 imply that for any ε > 0∣∣Sn1,...,ndf(x)− f∗(x)
∣∣ ≤M(d)
∑
δ∈∆
Λ#V (f ;πεδ(x)) + o(1), (28)
which combined with Theorem 2 completes the proof of (a).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
172 U. GOGINAVA, A. SAHAKIAN
To prove part (b) suppose that Λ = {λn} is a sequence defined by (27). It is not hard to see that
the class C(T d) ∩ Λ#BV (T d) is a Banach space with the norm
‖f‖Λ#BV := ‖f‖C + Λ#BV (f).
Denoting
Ai1,...,id :=
[
πi1
N + 1/2
,
π(i1 + 1)
N + 1/2
)
× . . .×
[
πid
N + 1/2
,
π(id + 1)
N + 1/2
)
,
we consider the following functions:
gN (x1, . . . , xd) :=
N−1∑
i1,...,id=1
1Ai1,...,id (x1, . . . , xd)
d∏
s=1
sin(N + 1/2)xs,
for N = 2, 3, . . . , where 1A(x1, . . . , xd) is the characteristic function of a set A ⊂ T d.
It is easy to check that{
nξn
logd−1(n+ 1)
}#
Vs(gN ) ≤ c
N−1∑
i=1
logd−1(i+ 1)
iξi
= o(logdN)
and hence
‖gN‖Λ#BV = o(logdN) = ηN logdN,
where ηN → 0 as N →∞. Now, setting
fN :=
gN
ηN logdN
, N = 2, 3, . . . ,
we obtain that fN ∈ Λ#BV (T d) and
sup
N
‖fN‖Λ#BV <∞. (29)
Now, for the cubical partial sums of the d-dimensional Fourier series of fN at (0, . . . , 0) ∈ T d we
have that
πdSN,...,NfN (0, . . . , 0) =
=
1
ηN logdN
N−1∑
i1,...,id=1
∫
Ai1,..., id
d∏
s=1
sin2(N + 1/2)xs
2 sin(xs/2)
dx1 . . . dxd ≥
≥ c
ηN logdN
N−1∑
i1,...,id=1
1
i1 . . . id
≥ c
ηN
→∞ (30)
as N → ∞. Applying the Banach – Steinhaus theorem, from (29) and (30) we conclude that there
exists a continuous function f ∈ Λ#BV (T d) such that
sup
N
∣∣SN,...,Nf(0, . . . , 0)
∣∣ =∞.
Theorem 4 is proved.
The next theorem follows from Theorems 3 and 4.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
CONVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS OF BOUNDED GENERALIZED . . . 173
Theorem 5. For any d > 1 the class of functions f(x) x ∈ T d satisfying the following condition:
∞∑
n=1
v#
s (f, n) logd−1(n+ 1)
n2
<∞, s = 1, . . . , d,
is a class of convergence.
1. Bakhvalov A. N. Continuity in Λ-variation of functions of several variables and the convergence of multiple Fourier
series (in Russian) // Mat. Sb. – 2002. – 193, № 12. – P. 3 – 20.
2. Chanturia Z. A. The modulus of variation of a function and its application in the theory of Fourier series // Sov.
Math. Dokl. – 1974. – 15. – P. 67 – 71.
3. Dyachenko M. I. Waterman classes and spherical partial sums of double Fourier series // Anal. Math. – 1995. – 21. –
P. 3 – 21.
4. Dyachenko M. I. Two-dimensional Waterman classes and u-convergence of Fourier series (in Russian) // Mat. Sb. –
1999. – 190, № 7. – P. 23 – 40 (English transl.: Sb. Math. – 1999. – 190, № 7-8. – P. 955 – 972).
5. Dyachenko M. I, Waterman D. Convergence of double Fourier series and W-classes // Trans. Amer. Math. Soc. –
2005. – 357. – P. 397 – 407.
6. Goginava U. On the uniform convergence of multiple trigonometric Fourier series // East J. Approxim. – 1999. – 3,
№ 5. – P. 253 – 266.
7. Goginava U. Uniform convergence of Cesáro means of negative order of double Walsh – Fourier series // J. Approxim.
Theory. – 2003. – 124. – P. 96 – 108.
8. Goginava U., Sahakian A. On the convergence of double Fourier series of functions of bounded partial generalized
variation // East J. Approxim. – 2010. – 16, № 2. – P. 109 – 121.
9. Goginava U., Sahakian A. On the convergence of multiple Fourier series of functions of bounded partial generalized
variation // Anal. Math. – 2013. – 39, № 1. – P. 45 – 56.
10. Hardy G. H. On double Fourier series and especially which represent the double zeta function with real and
incommensurable parameters // Quart. J. Math. Oxford Ser. – 1906. – 37. – P. 53 – 79.
11. Jordan C. Sur la series de Fourier // C. r. Acad. sci. Paris. – 1881. – 92. – P. 228 – 230.
12. Marcinkiewicz J. On a class of functions and their Fourier series // Compt. Rend. Soc. Sci. Warsowie. – 1934. – 26. –
P. 71 – 77.
13. Sablin A. I. Λ-variation and Fourier series (in Russian) // Izv. Vysch. Uchebn. Zaved. Mat. – 1987. – 10. – P. 66 – 68
(English transl.: Sov. Math. Izv. VUZ. – 1987. – 31).
14. Sahakian A. A. On the convergence of double Fourier series of functions of bounded harmonic variation (in Russian)
// Izv. Akad. Nauk Armyan.SSR. Ser. Mat. – 1986. – 21, № 6. – P. 517 – 529 (English transl.: Sov. J. Contemp. Math.
Anal. – 1986. – 21, № 6. – P. 1 – 13).
15. Waterman D. On convergence of Fourier series of functions of generalized bounded variation // Stud. Math. – 1972. –
44, № 1. – P. 107 – 117.
16. Waterman D. On the summability of Fourier series of functions of Λ-bounded variation // Stud. Math. – 1975/76. –
54, № 1. – P. 87 – 95.
17. Wiener N. The quadratic variation of a function and its Fourier coefficients // Mass. J. Math. – 1924. – 3. – P. 72 – 94.
18. Zygmund A. Trigonometric series. – Cambridge: Cambridge Univ. Press, 1959.
Received 21.11.12
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
|
| id | umjimathkievua-article-1971 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:13Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/8ab9547e090a641f1074bff95e5d6d2a.pdf |
| spelling | umjimathkievua-article-19712019-12-05T09:47:54Z Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation Збіжність кратних рядів Фур'є функцій з обмеженою узагальненою варіацією Goginava, U. Sahakian, A. Гогінава, У. Сахакян, А. The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series. Введено нову концепцію Λ-варiацiї функцій багатьох змінних та вивчено її зв'язок зі з6іжністю багатовимiрних рядів Фур'є. Institute of Mathematics, NAS of Ukraine 2015-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1971 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 2 (2015); 163-173 Український математичний журнал; Том 67 № 2 (2015); 163-173 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1971/963 https://umj.imath.kiev.ua/index.php/umj/article/view/1971/964 Copyright (c) 2015 Goginava U.; Sahakian A. |
| spellingShingle | Goginava, U. Sahakian, A. Гогінава, У. Сахакян, А. Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title | Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title_alt | Збіжність кратних рядів Фур'є функцій з обмеженою узагальненою варіацією |
| title_full | Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title_fullStr | Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title_full_unstemmed | Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title_short | Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation |
| title_sort | convergence of multiple fourier series of functions of bounded generalized variation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1971 |
| work_keys_str_mv | AT goginavau convergenceofmultiplefourierseriesoffunctionsofboundedgeneralizedvariation AT sahakiana convergenceofmultiplefourierseriesoffunctionsofboundedgeneralizedvariation AT gogínavau convergenceofmultiplefourierseriesoffunctionsofboundedgeneralizedvariation AT sahakâna convergenceofmultiplefourierseriesoffunctionsofboundedgeneralizedvariation AT goginavau zbížnístʹkratnihrâdívfur039êfunkcíjzobmeženoûuzagalʹnenoûvaríacíêû AT sahakiana zbížnístʹkratnihrâdívfur039êfunkcíjzobmeženoûuzagalʹnenoûvaríacíêû AT gogínavau zbížnístʹkratnihrâdívfur039êfunkcíjzobmeženoûuzagalʹnenoûvaríacíêû AT sahakâna zbížnístʹkratnihrâdívfur039êfunkcíjzobmeženoûuzagalʹnenoûvaríacíêû |