**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 201*
RELATIONS BETWEEN SOME CLASSES OF FUNCTIONS OF GENERALIZED BOUNDED VARIATION AMIRAN GOGATISHVILI, USHANGI GOGINAVA AND GEORGE TEPHNADZE
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Zinta 25, 11567 Praha 1, Czech Republic E-mail:
[email protected] Department of Mathematics, Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia E-mail:
[email protected] Department of Mathematics, Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia E-mail:
[email protected] We prove inclusion relations between generalizing Waterman's and generalized Wiener's classes for functions.of two variable.
Abstract.
The notion of function of bounded variation was introduced by C. Jordan [16]. Generalized this notion N. Wiener [30] has considered the class BVp of functions. L. Young [31] introduced the notion of functions of Φ-variation. In [26] D. Waterman has introduced the following concept of generalized bounded variation. Definition
such that
∞ P
0.1. Let Λ = {λn : n ≥ 1} be an increasing sequence of positive numbers
n=1
(1/λn ) = ∞. A function f is said to be of Λ-bounded variation (f ∈ ΛBV ),
if for every choice of nonoverlapping intervals {In : n ≥ 1} we have ∞ X |f (In )| < ∞, λn n=1
2010 Mathematics Subject Classi cation : 42C10. Key words and phrases : Waterman's class, generalized Wiener's class. The reseach was supported by Shota Rustaveli National Science Foundation grant no.13/06 (Geometry of function spaces, interpolation and embedding theorems) The paper is in nal form and no version of it will be published elsewhere. [1]
2
A. GOGATISHVILI, U. GOGINAVA
where In = [an , bn ] ⊂ [0, 1] and f (In ) = f (bn ) − f (an ) . If f ∈ ΛBV, then Λ-variation of f is de ned to be the supremum of such sums, denoted by VΛ (f ) . Properties of functions of the class ΛBV as well as the convergence and summability properties of their Fourier series have been investigated in [22]-[29]. For everywhere bounded 1-periodic functions, Z. Chanturia [6] has introduced the concept of the modulus of variation. H. Kita and K. Yoneda [17] studied generalized Wiener classes BV (p (n) ↑ p) . They introduced 0.2. Let f be a nite 1-periodic function de ned on the interval (−∞, +∞). ∆ is said to be a partition with period 1 if there is a set of points ti for which
Definition
· · · t−1 < t0 < t1 < t2 < · · · < tm < tm+1 < · · · ,
(1)
and tk+m = tk + 1 when k = 0, ±1, ±2, ..., where m is any natural number. Let p (n) be an increasing sequence such that 1 ≤ p (n) ↑ p, n → ∞, where 1 ≤ p ≤ +∞. We say that a function f belongs to the class BV (p (n) ↑ p) if V (f, p (n) ↑ p) ≡ sup sup n≥1 ∆
m X
!1/p(n) p(n)
|f (Ik )|
k=1
1 : inf |Ik | ≥ n < +∞. k 2
We note that if p (n) = p for each natural number, where 1 ≤ p < +∞, then the class BV (p (n) ↑ p) coincides with the Wiener class Vp . Properties of functions of the class BV (p (n) ↑ p) as well as the uniform convergence
and divergence at point of their Fourier series with respect to trigonometric and Walsh system have been investigated in [9],[12],[18]. Generalizing the class BV (p (n) ↑ p) T. Akhobadze (see [1],[2]) has considered the BV (p (n) ↑ p, ϕ) and BΛ (p (n) ↑ p, ϕ)classes of functions. The relation between diferent classes of generalized bounded variation was taken into account in the works of M. Avdispahic [4], A. Kovocik [19], A. Belov (see [5], Z. Chanturia [7]), T. Akhobadze [3] and M. Medvedieva [21], Goginava [11, 13]. Let f be a real and measurable function of two variable of period 1 with respect to each variable. Given intervals J1 = (a, b), J2 = (c, d) and points x, y from I := [0, 1] we denote f (J1 , y) := f (b, y) − f (a, y),
f (x, J2 ) := f (x, d) − f (x, c)
and for the rectangle A = (a, b) × (c, d), we set f (A) = f (J1 , J2 ) := f (a, c) − f (a, d) − f (b, c) + f (b, d).
Let E = {Ii } be a collection of nonoverlapping intervals from I ordered in arbitrary way and let Ω be the set of all such collections E . For the sequence of positive numbers Λ = {λn }∞ n=1 we denote ΛV1 (f ) = sup sup y∈I {Ii }∈Ω
X |f (Ii , y)| i
λi
,
3
CLASSES OF FUNCTIONS
X |f (x, Jj )|
ΛV2 (f ) = sup sup x∈I {Jj }∈Ω
ΛV1,2 (f ) =
.
j
X X |f (Ii , Jj )|
sup {Ii },{Jj }∈Ω
i
j
λi λj
.
0.3. We say that the function f has bounded Λ-variation on I 2 := [0, 1] × [0, 1] and write f ∈ ΛBV , if Definition
ΛV (f ) := ΛV1 (f ) + ΛV2 (f ) + ΛV1,2 (f ) < ∞.
We say that the function f has bounded Partial Λ-variation and write f ∈ P ΛBV if P ΛV (f ) := ΛV1 (f ) + ΛV2 (f ) < ∞.
If λn ≡ 1 (or if 0 < c < λn < C < ∞, n = 1, 2, . . .) the classes ΛBV and P ΛBV coincide with the Hardy class BV and P BV respectively. Hence it is reasonable to assume that λn → ∞ and since the intervals in E = {Ii } are ordered arbitrarily, we will suppose, without loss of generality, that the sequence {λn } is increasing. Thus, in what follows we suppose that 1 < λ1 ≤ λ2 ≤ . . . ,
lim λn = ∞,
n→∞,
∞ X 1 = ∞. λ n=1 n
(2)
In the case when λn = n, n = 1, 2 . . . we say Harmonic Variation instead of Λvariation and write H instead of Λ (HBV , P HBV , HV (f ), etc). The notion of Λ-variation was introduced by Waterman [26] in one dimensional case and Sahakian [24] in two dimensional case. The notion of bounded partial variation (class P BV ) was introduced by Goginava [10]. These classes of functions of generalized bounded variation play an important role in the theory of Fourier series. We have proved in [14] the following theorem. Theorem
0.4 (Goginava, Sahakian).
Let
Λ = {λn = nγn } and γn ≥ γn+1 > 0, n =
1, 2, .... . 1) If ∞ X γn < ∞, n n=1
then
P ΛBV ⊂ HBV .
2) If, in addition, for some
δ>0 γn = O(γn[1+δ] )
and
then
(3)
as
∞ X γn = ∞, n n=1
n→∞
(4) (5)
P ΛBV 6⊂ HBV .
Dyachenko and Waterman [8] introduced another class of functions of generalized bounded variation. Denoting by Γ the the set of nite collections of nonoverlapping
4
A. GOGATISHVILI, U. GOGINAVA
rectangles Ak := [αk , βk ] × [γk , δk ] ⊂ T 2 we de ne Λ∗ V (f ) := sup
X |f (Ak )| λk
{Ak }∈Γ k Definition
f ∈ Λ BV if
.
0.5 (Dyachenko, Waterman). Let f be a real function on I 2 . We say that
∗
ΛV (f ) := ΛV1 (f ) + ΛV2 (f ) + Λ∗ V (f ) < ∞.
In [15] Goginava and Sahakian introduced a new class of functions of generalized bounded variation and investigated the convergence of Fourier series of function of that classes. For the sequence Λ = {λn }∞ n=1 we denote Λ# V1 (f ) = sup
sup
{yi }⊂T {Ii }∈Ω
Λ# V2 (f ) = sup
sup
{xj }⊂T {Jj }∈Ω Definition
Λ# BV , if
X |f (Ii , yi )| λi
i
X |f (xj , Jj | λj
j
, .
0.6 (Goginava, Sahakian). We say that the function f belongs to the class Λ# V (f ) := Λ# V1 (f ) + Λ# V2 (f ) < ∞.
The following theorem is proved in [15] Theorem
0.7.
a) If
λn log (n + 1) < ∞, n→∞ n
(6)
lim
then
Λ# BV ⊂ HBV. λ
b) If nn
↓ 0 and lim
n→∞
λn log (n + 1) = +∞, n
then
Λ# BV 6⊂ HBV.
In this paper we introduce new classes of bounded generalized variation. Let f be a function de ned on R2 with 1- periodic relative to each variable. ∆1 and ∆2 is said to be a partitions with period 1, if (i)
(i)
(i)
(i)
∆i : · · · < t−1 < t0 < t1 < · · · < t(i) mi < tmi +1 < · · · ,
satis es
(i) tk+mi
=
(i) tk
i = 1, 2
+ 1 for k = 0, ±1, ±2, ..., where mi , i = 1, 2 are a positive integers.
0.8. Let p (n) be an increasing sequence such that 1 ≤ p (n) ↑ p, n → ∞, where 1 ≤ p ≤ +∞. We say that a function f belongs to the class BV # (p (n) ↑ p) if
Definition
V1# (f, p (n) ↑ p)
5
CLASSES OF FUNCTIONS
≡ sup sup sup {yi }⊂I n≥1 ∆1
m1 X
!1/p(n) p(n)
|f (Ii , yi )|
i=1
1 : inf |Ii | ≥ n < +∞, i 2
and V2# (f, p (n) ↑ p) 1/p(n) m2 X 1 p(n) ≡ sup sup sup |f (xj , Jj )| : inf |Jj | ≥ n < +∞, j 2 {xj }⊂I n≥1 ∆2 j=1
where
(1) (1) (2) (2) Ii := ti−1 , ti , Jj := tj−1 , tj .
C I 2 and B I 2 are the spaces of continuous and bounded functions given on I 2 ,
respectively. In this paper we prove inclusion relations between Λ# BV and BV # (p (n) ↑ ∞) classes. In particular, the following are true Theorem
0.9. Λ# BV ⊂ BV # (p (n) ↑ ∞) lim
sup
n→∞ 1≤m≤2n
if and only if
m1/p(n) < ∞. m P (1/λj )
(7)
j=1 Theorem
0.10.
Let
f ∈ BV # (p (n) ↑ ∞) Corollary
∞ P
(1/λn ) = +∞ . Then there exists a functions C I 2 such that f ∈ / ΛBV # .
n=1 T
0.11. BV # (p (n) ↑ ∞) ⊂ Λ# BV
if and only if
Λ# BV = B I 2 .
Let us take an arbitrary f ∈ Λ# BV . Follow of the method of Paper Kuprikov in [20] we can prove that the following estimations Proof of Theorem 0.9.
m1 X
!1/p(n) |f (Ik , yk )|
p(n)
≤ Λ# V1 (f )
sup 1≤m≤2n
k=1
m1/p(n) m P
3nk−1 + 1
(11)
From (8) and (10) it is evident that 22nk−1 < m (nk ) ≤ 2nk . Two cases are possible: a) Let there exists a monotone sequence of positive integers {sk : k ≥ 1} ⊂ {nk : k ≥ 1} such that 22sk−1 < m (sk ) ≤ 2sk −sk−1 −1 . (12) Consider the function fk de ned by hk (2sk x − 2j + 1) , x ∈ [(2j − 1) /2sk , 2j/2sk ) −hk (2sk x − 2j − 1) , x ∈ [2j/2sk , (2j + 1) /2sk ) fk (x) = for j = m (sk−1 ) , ..., m (sk ) − 1 0, otherwise
where 1/2
hk =
1 2k
m(s Pk )
(1/λj )
.
j=1
Let f (x, y) =
∞ X
fk (x) fk (y) ,
k=2
where f (x + l, y + s) = f (x, y) ,
l, s = 0, ±1, ±2, ....
7
CLASSES OF FUNCTIONS
First we prove that f ∈ Λ# BV. For every choice of nonoverlapping intervals {In : n ≥ 1} we get Λ# V1 (f ; p (n) ↑ ∞) ≤
∞ X |f (Ij , yj )| j=1
≤4
∞ X
λj
m(si )
∞ X X 1 1 = 4. =4 i λ 2 j i=1 j=1
h2i
i=1
Analogously, we can prove that Λ# V2 (f ; p (n) ↑ ∞) ≤ 4.
Next, we shall prove that f ∈ / BV # (p (n) ↑ ∞) . By (11), (12) and from the construction of the function we get V1 (f ; p (n) ↑ ∞) p(sk ) 1/p(sk ) k )−1 m(s X 2j − 1 2j 2j 2j f ≥ , −f , 2sk 2sk 2sk 2sk j=m(sk−1 )
= =
m(sk )−1
j=m(sk−1 )
X
h2k
p(sk ) 1/p(sk ) 2j − 1 2j 2j fk − fk fk 2sk 2sk 2sk
(m (sk ) − m (sk−1 ))
1/p(sk )
1/p(s )
≥c
k m (sk ) ≥ c2k → ∞ m(s k) P 2k (1/λj )
as k → ∞.
j=1
Therefore we get f ∈ / BV # (p (n) ↑ ∞) . b) Let 2nk −nk−1 −1 < m (nk ) ≤ 2nk
for all k > k0 .
Consider the function gk de ned by dk (2nk x − 2j + 1) , x ∈ [(2j − 1) /2nk , 2j/2nk ) −dk (2nk x − 2j − 1) , x ∈ [2j/2nk , (2j + 1) /2nk ) gk (x) = for j = 2nk−1 −nk−2 , ..., 2nk −nk−1 −1 − 1 0, otherwise
where
1/2
dk =
1 2k
m(n Pk )
(1/λj )
.
j=1
Let g (x, y) =
∞ X k=k0 +2
gk (x) gk (y)
8
A. GOGATISHVILI, U. GOGINAVA
where l, s = 0, ±1, ±2, ....
g (x + l, y + s) = g (x, y) ,
For every choice of nonoverlapping intervals {In : n ≥ 1} we get ∞ X |f (Ij , yj )| j=1
≤4
λj
∞ X
d2i
j=1
i=k0 +1
≤4
i−1 −1 2ni −n X
1 λj
m(ni )
∞ X
d2i
i=k0 +1
X 1 < ∞. λj j=1
Analogously, we can prove that ∞ X |f (xj , Jj )| j=1
λj
< ∞.
Hence we have g ∈ Λ# BV. Next we shall prove that g ∈ / BV # (p (n) ↑ ∞) . By (8), (10), (11) and from the construction of the function we get V1# (g; p (n) ↑ ∞) n −n −1 p(nk ) 1/p(nk ) k−1 −1 2 k X g 2j − 1 , 2j − g 2j , 2j ≥ 2nk 2nk 2nk 2nk j=2nk−1 −nk−2
n −n −1 p(nk ) 1/p(nk ) k−1 −1 2 k X 2j gk 2j − 1 − gk 2j = gk n −n 2nk 2nk 2nk j=2
k−1
k−2
= d2k 2nk −nk−1 −1 − 2nk−1 −nk−2 1 ≥ d2k 2(nk −nk−1 )/p(nk ) 4 c2nk /p(nk ) ≥ m(n Pk ) 2k+2 (1/λj )
1/p(nk )
j=1 1/p(n )
≥c
k m (nk ) m(n Pk ) 2k (1/λj )
j=1
≥ c2 → ∞ as k → ∞. k
Therefore we get g ∈/ BV # (p (n) ↑ ∞) and the proof of Theorem 1 is complete. Proof of Theorem 0.10.
We choose a monotone increasing sequence of positive integers
{lk : k ≥ 1} such that l1 = 1 and
p (lk−1 ) ≥ ln k
for all
k ≥ 2.
(13)
9
CLASSES OF FUNCTIONS
Set (k = 1, 2, ...) l +1 l l l +1 2 k ck x − 1/2 k , if 1/2 k ≤ x ≤ 3/2 k lk +1 lk −1 lk +1 rk (x) = −2 ck x − 1/2 , if 3/2 ≤ x ≤ 1/2lk −1 0, otherwise
where −1/4 k X 1 ck = λ j j=1
and r (x, y) =
∞ X
rk (x) rk (y)
k=1
where l, s = 0, ±1, ±2, ....
r (x + l, y + s) = r (x, y)
It is easy to show that function r ∈ C I 2 .
First we show that r ∈ BV # (p (n) ↑ ∞) . Let {Ii } be an arbitrary partition of the interval I such that inf |Ii | ≥ 1/2l . For this xed l, we can choose integers lk−1 and i lk for which lk−1 ≤ l < lk holds. Then it follows that p (lk−1 ) ≤ p (l) ≤ p (lk ) and 1/2lk < 1/2l ≤ 1/2lk−1 .
By (13) and from the construction of the function r we obtain m X
=
|r (Ii , yi )|
p(l)
j=1
k X j=1
1/p(l)
X {i:2−lj ≤yi
