Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 4, Article ??, 2004
ON EMBEDDING OF THE CLASS H ω
LÁSZLÓ LEINDLER
B OLYAI I NSTITUTE
U NIVERSITY OF S ZEGED
A RADI VÉRTANÚK TERE 1
H-6720 S ZEGED , H UNGARY
leindler@math.u-szeged.hu
Received 24 August, 2004; accepted 29 September, 2004
Communicated by H. Bor
A BSTRACT. In [4] we extended an interesting theorem of Medvedeva [5] pertaining to the embedding relation H ω ⊂ ΛBV, where ΛBV denotes the set of functions of Λ-bounded variation,
which is encountered in the theory of Fourier trigonometric series. Now we give a further generalization of our result. Our new theorem tries to unify the notion of ϕ-variation due to Young [6],
and that of the generalized Wiener class BV (p(n) ↑) due to Kita and Yoneda [3]. For further
references we refer to the paper by Goginava [2].
Key words and phrases: Embedding relation, Bounded variation, Continuity.
2000 Mathematics Subject Classification. 26A15, 26A21, 26A45.
1. I NTRODUCTION
Let ω(δ) be a nondecreasing continuous function on the interval [0, 1] having the following
properties:
ω(0) = 0, ω(δ1 + δ2 ) ≤ ω(δ1 ) + ω(δ2 ) for 0 ≤ δ1 ≤ δ2 ≤ δ1 + δ2 ≤ 1.
Such a function is called a modulus of continuity, and it will be denoted by ω(δ) ∈ Ω.
The modulus of continuity of a continuous function f will be denoted by ω(f ; δ), that is,
ω(f ; δ) :=
sup |f (x + h) − f (x)|.
0≤h≤δ
0≤x≤1−h
As usual, set
H ω := {f ∈ C : ω(f ; δ) = O(ω(δ))}.
α
If ω(δ) = δ α , 0 < α ≤ 1 we write H α instead of H δ .
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant No. T 042462, and TS
44782.
155-04
2
L ÁSZLÓ L EINDLER
Finally we define a new class of real functions f : [0, 1] → R. For every k ∈ N let ϕk :
[0, ∞) → R be a nondecreasing function with ϕk (0) = 0; and let Λ := {λk } be a nondecreasing
sequence of positive numbers such that
∞
X
1
= ∞.
λ
k
k=1
If a function f : [0, 1] → R satisfies the condition
(1.1)
sup
N
X
ϕk (|f (bk ) − f (ak )|)λ−1
k < ∞,
k=1
where the supremum is extended over all systems of nonoverlapping subintervals (ak , bk ) of
[0, 1], then f is said to be of Λ{ϕk }-bounded variation, and this fact is denoted by f ∈ Λ{ϕk }BV.
In the special cases when all ϕk (x) = ϕ(x), we write f ∈ Λϕ BV (see [4]), and if ϕ(x) = xp
we use the notation f ∈ Λp BV, and when p = 1, simply f ∈ ΛBV (see [5]). In the case λk = 1
and ϕk (x) = ϕ(x) for all k, then we get the class Vϕ due to Young [6], finally if λk = 1 and
ϕk (x) = xpk , pk ↑, we get a class similar to BV (p(n) ↑) (see [3]).
Medvedeva [5] proved the following useful theorem, among others.
Theorem 1.1. The embedding relation H ω ⊂ ΛBV holds if and only if
∞
X
ω(tk )λ−1
k < ∞
k=1
for any sequence {tk } satisfying the conditions:
(1.2)
tk ≥ 0,
∞
X
tk ≤ 1.
k=1
In the sequel, the fact that a sequence t := {tk } has the properties (1.2) will be denoted by
t ∈ T. K and Ki will denote positive constants, not necessarily the same at each occurrence.
Among others, in [4] we showed that if 0 < α ≤ 1 and pα ≥ 1 then H α ⊂ Λp BV always
holds, furthermore that if 0 < p < 1/α, then H α ⊂ Λp BV is fulfilled if and only if for any
t ∈ T,
∞
X
−1
tαp
k λk < ∞.
k=1
If ω(δ) is a general modulus of continuity then for 0 < p < 1 we verified that H ω ⊂ Λp BV
holds if and only if for any t ∈ T
∞
X
(1.3)
ω(tk )p λ−1
k < ∞.
k=1
These latter two results are immediate consequences of the following theorem of [4].
Theorem 1.2. Assume that ϕ(x) is a function such that ϕ(ω(δ)) ∈ Ω. Then H ω ⊂ Λϕ BV holds
if and only if for any t ∈ T
∞
X
ϕ(ω(tk ))λ−1
k < ∞.
k=1
Remark 1.3. It would be of interest to mention that by Theorem 1.2 the restriction 0 < p < 1
claimed above, can be replaced by the weaker condition ω(δ)p ∈ Ω, and then the embedding
relation H ω ⊂ Λp BV also holds if and only if (1.3) is true.
J. Inequal. Pure and Appl. Math., 5(4) Art. ??, 2004
http://jipam.vu.edu.au/
O N E MBEDDING OF THE C LASS H ω
3
2. R ESULTS
Our new theorem tries to unify and generalize all of the former results.
Theorem 2.1. Assume that ω(t) ∈ Ω and for every k ∈ N, ϕk (ω(δ)) ∈ Ω. Then the embedding
relation H ω ⊂ Λ{ϕk }BV holds if and only if for any t ∈ T
∞
X
(2.1)
ϕk (ω(tk ))λ−1
k < ∞.
k=1
Our theorem plainly yields the following assertion.
Corollary 2.2. If for all k ∈ N, pk > 0 and ω(δ)pk ∈ Ω, that is, if ϕk (x) = xpk , then
H ω ⊂ Λ{xpk }BV holds if and only if for any t ∈ T
∞
X
(2.2)
ω(tk )pk λ−1
k < ∞.
k=1
α
It is also obvious that if ω(δ) = δ , 0 < α ≤ 1, then (2.1) and (2.2) reduce to
∞
X
ϕk (tαk )λ−1
k
k=1
< ∞ and
∞
X
k −1
tαp
k λk < ∞,
k=1
respectively.
3. L EMMAS
In the proof we shall use the following three lemmas.
Lemma 3.1 ([1, p. 78]). If ω(δ) ∈ Ω then there exists a concave function ω ∗ (δ) such that
ω(δ) ≤ ω ∗ (δ) ≤ 2ω(δ).
Lemma 3.2. If ω(δ) ∈ Ω and t = {tk } ∈ T, then there exists a function f ∈ H ω such that if
x0 = 0, x1 =
x2n =
n
X
t1
,
2
ti and x2n+1 = x2n +
i=1
tn+1
, n ≥ 1,
2
then
f (x2n ) = 0 and f (x2n+1 ) = ω(tn+1 ) for all n ≥ 0.
A concrete function with these properties is given in [5].
Lemma 3.3. If ω(t) ∈ Ω and for all k ∈ N, ϕk (ω(t)) ∈ Ω also holds, furthermore for any
t ∈ T the condition (2.1) stays, then there exists a positive number M such that for any t ∈ T
(3.1)
∞
X
ϕk (ω(tk ))λ−1
k ≤ M
k=1
holds.
Proof of Lemma 3.3. The proof follows the lines given in the proof of Theorem 2 emerging in
[5]. Without loss of generality, due to Lemma 3.1, we can assume that, for every k, the functions
ϕk (ω(δ)) are concave moduli of continuity.
J. Inequal. Pure and Appl. Math., 5(4) Art. ??, 2004
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4
L ÁSZLÓ L EINDLER
Indirectly, let us suppose that there is no number M with property (3.1). Then for any i ∈ N
there exists a sequence t(i) := {tk,i } ∈ T such that
∞
X
i
(3.2)
2 <
ϕk (ω(tk,i ))λ−1
k < ∞.
k=1
Now define
tk :=
∞
X
tk,i
2i
i=1
.
It is easy to see that t := {tk } ∈ T, and thus (2.1) also holds.
Since every ϕk (ω(ω)) is concave, thus by Jensen’s inequality, we get that
!!
∞
∞
X
X
tk,i
ϕk (ω(tk,i ))
(3.3)
ϕk (ω(tk )) = ϕk ω
≥
.
i
2
2i
i=1
i=1
Employing (3.2) and (3.3) we obtain that
∞
∞
∞
X
X
X
−1
−1
ϕk (ω(tk ))λk ≥
λk
ϕk (ω(tk,i ))2−i
k=1
=
k=1
i=1
∞
X
∞
X
2−i
i=1
ϕk (ω(tk,i ))λ−1
k = ∞,
k=1
and this contradicts (2.1).
This contradiction proves (3.1).
4. P ROOF OF T HEOREM 2.1
Necessity. Suppose that H ω ⊂ Λ{ϕk }BV, but there exists a sequence t = {tk } ∈ T such that
∞
X
(4.1)
ϕk (ω(tk ))λ−1
k = ∞.
k=1
Then, applying Lemma 3.2 with this sequence t = {tk } ∈ T and ω(δ), we obtain that there
exists a function f ∈ H ω such that
|f (x2k−1 ) − f (x2k−2 )| = ω(tk ) for all k ∈ N.
Hence, by (4.1), we get that
N
X
ϕk (|f (x2k−1 ) −
f (x2k−2 )|)λ−1
k
=
k=1
N
X
ϕk (ω(tk ))λ−1
k → ∞,
k=1
that is, (1.1) does not hold if bk = x2k−1 and ak = x2k−2 , thus f does not belong to the set
Λ{ϕk }BV.
This and the assumption H ω ⊂ Λ{ϕk }BV contradict, whence the necessity of (2.1) follows.
Sufficiency. The condition (2.1), by Lemma 3.3, implies (3.1). If we consider a system of
nonoverlapping subintervals (ak , bk ) of [0, 1] and take tk := (bk − ak ), then t := {tk } ∈ T,
consequently for this t (3.1) holds. Thus, if f ∈ H ω , we always have that
N
X
ϕk (|f (bk ) −
f (ak )|)λ−1
k
k=1
≤K
N
X
ϕk (ω(bk − ak ))λ−1
k ≤ KM,
k=1
and this shows that f ∈ Λ{ϕk }BV.
The proof is complete.
J. Inequal. Pure and Appl. Math., 5(4) Art. ??, 2004
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O N E MBEDDING OF THE C LASS H ω
5
R EFERENCES
[1] A.V. EFIMOV, Linear methods of approximation of continuous periodic functions, Mat. Sb., 54
(1961), 51–90 (in Russian).
[2] U. GOGINAVA, On the approximation properties of partial sums of trigonometric Fourier series,
East J. on Approximations, 8 (2002), 403–420.
[3] H. KITA AND K. YONEDA, A generalization of bounded variation, Acta Math. Hungar., 56 (1990),
229–238.
[4] L. LEINDLER, A note on embedding of classes H ω , Analysis Math., 27 (2001), 71–76.
[5] M.V. MEDVEDEVA, On embedding classes H ω , Mat. Zametki, 64(5) (1998), 713–719 (in Russian).
[6] L.C. YOUNG, Sur une généralization de la notion de variation de Wiener et sur la convergence des
séries de Fourier, C.R. Acad. Sci. Paris, 204 (1937), 470–472.
J. Inequal. Pure and Appl. Math., 5(4) Art. ??, 2004
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