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volume 5, issue 4, article 105, 2004.

Received 24 August, 2004;

accepted 29 September, 2004.

Communicated by:Hüseyin Bor

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Journal of Inequalities in Pure and Applied Mathematics

ON EMBEDDING OF THE CLASSH^ω

LÁSZLÓ LEINDLER

Bolyai Institute Jozsef Attila University Aradi vertanuk tere 1 H-6720 Szeged Hungary.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 155-04

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On Embedding of the ClassH^ω László Leindler

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J. Ineq. Pure and Appl. Math. 5(4) Art. 105, 2004

http://jipam.vu.edu.au

Abstract

In [4] we extended an interesting theorem of Medvedeva [5] pertaining to the embedding relationH^ω ⊂ ΛBV,whereΛBV denotes the set of functions of Λ-bounded variation, which is encountered in the theory of Fourier trigonomet- ric series. Now we give a further generalization of our result. Our new theorem tries to unify the notion ofϕ-variation due to Young [6], and that of the generalized Wiener classBV(p(n)↑)due to Kita and Yoneda [3]. For further references we refer to the paper by Goginava [2].

2000 Mathematics Subject Classification:26A15, 26A21, 26A45.

Key words: Embedding relation, Bounded variation, Continuity.

The author was partially supported by the Hungarian National Foundation for Scien- tific Research under Grant No. T 042462, and TS 44782.

1. Introduction

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)for0≤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

0≤h≤δ 0≤x≤1−h

|f(x+h)−f(x)|.

As usual, set

H^ω :={f ∈C:ω(f;δ) =O(ω(δ))}.

Ifω(δ) =δ^α, 0< α≤1we writeH^α instead ofH^δ^α.

Finally we define a new class of real functions f : [0,1] → R. For every k ∈ Nletϕ_k : [0,∞) → Rbe a nondecreasing function with ϕ_k(0) = 0;and letΛ :={λ_k}be a nondecreasing sequence of positive numbers such that

∞

X

k=1

1

λ_k =∞.

If a functionf : [0,1]→Rsatisfies the condition

(1.1) sup

N

X

k=1

ϕ_k(|f(b_k)−f(a_k)|)λ⁻¹_k <∞,

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where the supremum is extended over all systems of nonoverlapping subintervals(ak, bk)of[0,1],thenf is said to be ofΛ{ϕk}-bounded variation, and this fact is denoted byf ∈Λ{ϕ_k}BV.In the special cases when allϕ_k(x) =ϕ(x), we writef ∈Λ_ϕBV (see [4]), and ifϕ(x) =x^pwe use the notationf ∈Λ_pBV, and when p = 1, simply f ∈ ΛBV (see [5]). In the case λk = 1 and ϕ_k(x) = ϕ(x) for all k,then we get the class V_ϕ due to Young [6], finally if λ_k= 1andϕ_k(x) =x^p^k, p_k ↑,we get a class similar toBV(p(n)↑)(see [3]).

Medvedeva [5] proved the following useful theorem, among others.

Theorem 1.1. The embedding relationH^ω ⊂ΛBV holds if and only if

∞

X

k=1

ω(tk)λ⁻¹_k <∞ for any sequence{t_k}satisfying the conditions:

(1.2) t_k≥0,

∞

X

k=1

t_k ≤1.

In the sequel, the fact that a sequencet:={tk}has the properties (1.2) will be denoted byt ∈T. K andK_i 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 ⊂Λ_pBV always holds, furthermore that if0< p <1/α,thenH^α ⊂Λ_pBV is fulfilled if and only if for anyt ∈T,

∞

X

k=1

t^αp_k λ⁻¹_k <∞.

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Ifω(δ) is a general modulus of continuity then for 0 < p < 1we verified thatH^ω ⊂ΛpBV holds if and only if for anyt ∈T

(1.3)

∞

X

k=1

ω(t_k)^pλ⁻¹_k <∞.

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 ⊂Λ_ϕBV holds if and only if for anyt ∈T

∞

X

k=1

ϕ(ω(t_k))λ⁻¹_k <∞.

Remark 1.1. It would be of interest to mention that by Theorem 1.2 the re- striction 0 < p < 1claimed above, can be replaced by the weaker condition ω(δ)^p ∈ Ω, and then the embedding relation H^ω ⊂ Λ_pBV also holds if and only if (1.3) is true.

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2. Results

Our new theorem tries to unify and generalize all of the former results.

Theorem 2.1. Assume thatω(t)∈Ωand for everyk ∈N,ϕk(ω(δ))∈Ω.Then the embedding relationH^ω ⊂Λ{ϕ_k}BV holds if and only if for anyt ∈T

(2.1)

∞

X

k=1

ϕk(ω(tk))λ⁻¹_k <∞.

Our theorem plainly yields the following assertion.

Corollary 2.2. If for allk ∈N,p_k >0andω(δ)^p^k ∈Ω,that is, ifϕ_k(x) = x^p^k, thenH^ω ⊂Λ{x^p^k}BV holds if and only if for anyt ∈T

(2.2)

∞

X

k=1

ω(t_k)^p^kλ⁻¹_k <∞.

It is also obvious that ifω(δ) =δ^α, 0< α≤1,then (2.1) and (2.2) reduce to

∞

X

k=1

ϕ_k(t^α_k)λ⁻¹_k <∞and

∞

X

k=1

t^αp_k ^kλ⁻¹_k <∞,

respectively.

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3. Lemmas

In the proof we shall use the following three lemmas.

Lemma 3.1 ([1, p. 78]). Ifω(δ)∈Ωthen there exists a concave functionω^∗(δ) such that

ω(δ)≤ω^∗(δ)≤2ω(δ).

Lemma 3.2. Ifω(δ)∈Ωandt={tk} ∈T,then there exists a functionf ∈H^ω such that if

x0 = 0, x1 = t₁ 2, x_2n=

n

X

i=1

t_i andx_2n+1 =x_2n+t_n+1

2 , n≥1, then

f(x_2n) = 0andf(x_2n+1) =ω(t_n+1)for alln≥0.

A concrete function with these properties is given in [5].

Lemma 3.3. Ifω(t)∈Ωand for allk ∈N, ϕ_k(ω(t)) ∈Ωalso holds, further- more for anyt∈T the condition (2.1) stays, then there exists a positive number M such that for anyt∈T

(3.1)

∞

X

k=1

ϕk(ω(tk))λ⁻¹_k ≤M

holds.

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Proof of Lemma3.3. The proof follows the lines given in the proof of Theorem 2 emerging in [5]. Without loss of generality, due to Lemma3.1, we can assume that, for everyk, the functionsϕ_k(ω(δ))are concave moduli of continuity.

Indirectly, let us suppose that there is no number M with property (3.1).

Then for anyi∈Nthere exists a sequencet⁽ⁱ⁾ :={t_k,i} ∈T such that

(3.2) 2ⁱ <

∞

X

k=1

ϕ_k(ω(t_k,i))λ⁻¹_k <∞.

Now define

t_k :=

∞

X

i=1

t_k,i 2ⁱ .

It is easy to see thatt:={t_k} ∈T,and thus (2.1) also holds.

Since everyϕk(ω(ω))is concave, thus by Jensen’s inequality, we get that (3.3) ϕk(ω(tk)) =ϕk ω

∞

X

i=1

t_k,i 2ⁱ

!!

≥

∞

X

i=1

ϕ_k(ω(t_k,i)) 2ⁱ . Employing (3.2) and (3.3) we obtain that

∞

X

k=1

ϕ_k(ω(t_k))λ⁻¹_k ≥

∞

X

k=1

λ⁻¹_k

∞

X

i=1

ϕ_k(ω(t_k,i))2⁻ⁱ

=

∞

X

i=1

2⁻ⁱ

∞

X

k=1

ϕ_k(ω(t_k,i))λ⁻¹_k =∞, and this contradicts (2.1).

This contradiction proves (3.1).

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4. Proof of Theorem 2.1

Necessity. Suppose that H^ω ⊂ Λ{ϕ_k}BV, but there exists a sequence t = {t_k} ∈T such that

(4.1)

∞

X

k=1

ϕ_k(ω(t_k))λ⁻¹_k =∞.

Then, applying Lemma 3.2 with this sequence t = {t_k} ∈ T and ω(δ), we obtain that there exists a functionf ∈H^ωsuch that

|f(x2k−1)−f(x2k−2)|=ω(t_k)for allk ∈N. Hence, by (4.1), we get that

N

X

k=1

ϕ_k(|f(x2k−1)−f(x2k−2)|)λ⁻¹_k =

N

X

k=1

ϕ_k(ω(t_k))λ⁻¹_k → ∞,

that is, (1.1) does not hold ifb_k=x_2k−1anda_k =x_2k−2,thusf does not belong to the setΛ{ϕ_k}BV.

This and the assumptionH^ω ⊂Λ{ϕ_k}BV contradict, whence the necessity of (2.1) follows.

Sufficiency. The condition (2.1), by Lemma3.3, implies (3.1). If we consider a system of nonoverlapping subintervals(ak, bk)of[0,1]and taketk:= (bk−ak), then t := {t_k} ∈ T,consequently for this t(3.1) holds. Thus, iff ∈ H^ω,we always have that

N

X

k=1

ϕ_k(|f(b_k)−f(a_k)|)λ⁻¹_k ≤K

N

X

k=1

ϕ_k(ω(b_k−a_k))λ⁻¹_k ≤KM,

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and this shows thatf ∈Λ{ϕ_k}BV.

The proof is complete.

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References

[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 trigono- metric 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.