����������� ������ 58 (2006), 91–96

UDK 517.28

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PROPERTIES OF FUNCTIONS OF GENERALIZED BOUNDED VARIATION

R. G. Vyas

Abstract. The class of functions of ΛBV^{(p)} shares many properties of functions of bounded variation. Here we have shown that ΛBV^{(p)} is a Banach space with a suitable norm, the intersection of ΛBV^{(p)}, over all sequences Λ, is the class of functions of BV^{(p)} and the union of ΛBV^{(p)}, over all sequences Λ, is the class of functions having right- and left-hand limits at every point.

Introduction. Looking to the feature of Jordan’s class the concept of bounded variation has been generalized in many ways and many interesting results are obtained in Analysis [1–6]. In most of the case these new notations have been introduced because of their applicability to the study of Fourier series. In 1972 Waterman [1] introduced the class of functions of ΛBV. In 1980 Shiba [4] generalized this class. He introduced the class ΛBV^{(p)} (p ≥ 1).

Definition. Given an interval I, and a sequence of non-decreasing positive real numbers Λ = {λ_{m}} (m = 1, 2, …) such that ∑_{m=1}^{n}(1/λ_{m}) diverges and 1 ≤ p < ∞, we say that f ∈ ΛBV^{(p)}(I) (that is f is a function of p-Λ-bounded variation over I) if

V_{Λ}(f, p, I) = sup_{{I_{m}}} V_{Λ}({I_{m}}, f, p, I) < ∞,

where V_{Λ}(\{I_{m}\}, f, p, I) = (∑_{m} \frac{|f(a_{m}) − f(b_{m})|^{p}}{λ_{m}} )^{1/p}, and \{I_{m}\} is a sequence of

non-overlapping subintervals I_{m} = [a_{m}, b_{m}] ⊂ I = [a, b].

Note that, if p = 1, one gets the class ΛBV (I); if λ_{m} ≡ 1 for all m, one gets the class BV^{(p)}; if p = 1 and λ_{m} ≡ m for all m, one gets the class Harmonic BV (I). If p = 1 and λ_{m} ≡ 1 for all m, one gets the class BV (I). Moreover, for any f ∈ BV^{(P)}(I) it follows f ∈ ΛBV^{(p)}(I).

D. Waterman [1, 2] has studied Fourier coefficients properties of this class. He also proved the following result.

AMS Subject Classification: 26A45.

Keywords and phrases: Banach space, right and left-hand limits, p-Λ-bounded variation.

Theorem A. The class of functions ΛBV (I) is a Banach space.

Perlman [6] has also studied some properties of this class. He proved the following results.

Theorem B. If f ∈ ΛBV (I) then f has right- and left-hand limits at every point of I.

Theorem C. If f ∈ ΛBV (I) for every sequence Λ, then f ∈ BV (I).

Theorem D. If a function f has a right- and left-hand limit at each point of I, then f ∈ ΛBV (I).

Here we have extended these results for the class of functions of ΛBV^{(p)}. We will show that the class of functions of ΛBV^{(p)} lies between the regulated functions and the class of functions of BV^{(p)}. That is, we will prove that the the union of ΛBV^{(p)}-functions over all sequences Λ are the regulated functions and the intersection of ΛBV^{(p)}-functions over all sequences Λ are the functions of BV^{(p)}.

Theorem 1. The class of functions of ΛBV^{(p)}(I) is a Banach space.

Theorem 2. If f ∈ ΛBV^{(p)}(I) then f has right- and left-hand limits at every point of I.

Theorem 1 and Theorem 2 generalize Theorem A and Theorem B respectively because p = 1 reduces the class ΛBV^{(p)}(I) to the class ΛBV (I).

Theorem 3. If f ∈ ΛBV^{(p)}(I), for every sequence Λ then f ∈ BV^{(p)}(I).

Hence the intersection of ΛBV^{(p)}(I), taken over all sequences Λ, is the class of functions of BV^{(p)}(I).

Theorem 4. If f is a continuous function over I, then f ∈ ΛBV^{(p)}(I) for some sequence Λ.

Theorem 5. If φ is a monotone function from I into [c, d] and f ∈ ΛBV^{(p)}[c, d] then f ∘φ ∈ ΛBV^{(p)}(I).

As a partial converse of Theorem 2 we have the following result.

Theorem 6. If f has right- and left-hand limits at every point of I, then f ∈ ΛBV^{(p)}(I) for some sequence Λ.

It follows from Theorem 2 and Theorem 6 that the union of ΛBV^{(p)}, taken over all sequences Λ, is the class of functions having right- and left-hand limits at every point.

Theorem 7. If g is continuous and F ∈ ΛBV^{(p)}(I), then g∘F ∈ Λ′BV^{(p)}(I) for some sequence Λ′.

To prove these results we need the following Lemmas.

Lemma 1. [6, Lemma 1] Let {a_{n}} be a sequence of positive numbers tending to zero. Then there exists a decreasing sequence {b_{n}} of positive numbers tending to zero such that ∑ b_{n} = ∞ and ∑ a_{n}b_{n} < ∞.

Lemma 2. [6, Lemma 3] Let {a_{n}} be a decreasing sequence of positive numbers. If {b_{n}} is a sequence of positive numbers tending to zero and {B_{n}} is the sequence {b_{n}} rearranged in decreasing order, then ∑ a_{k}b_{k} ≤ ∑ a_{k}B_{k}.

Lemma 3. [6, Theorem, p. 207] A function has right- and left-hand limits at each point if and only if it is the composition of a continuous function with a monotone function.

Lemma 4. [5, Lemma 1.6] If f ∈ ΛBV^{(p)}(I) then f is bounded over I.

Proof of Theorem 2. It is sufficient to prove the result for left-hand limits only. Suppose that there is a point x in (a, b] at which f does not have a left-hand limit. Then

L = \overline{lim}_{t→x−} > l = \underline{lim}_{t→x−} .

For δ = \frac{L−l}{3}, consider increasing sequences \{P_{n}\} and \{p_{n}\} converging to x such that f(P_{n}) ≥ L − δ and f(p_{n}) ≤ L + δ. We choose subsequences {Q_{n}} of {P_{n}} and {q_{n}} of {p_{n}} such that q_{1} < Q_{1} < q_{2} < Q_{2} < · · ·. Consider intervals I_{n} = [q_{n}, Q_{n}]; for all n, we get |f(I_{n})| ≥ (L − δ) − (l + δ) = δ.

Hence ∑ \frac{|f(I_{n})|^{p}}{λ_{n}} ≥ δ^{p}(∑ \frac{1}{λ_{n}}) = ∞, which contradicts our hypothesis. Hence the result follows.

Proof of Theorem 3. Let f∈ ΛBV^{(p)}(I) for at lest one choice of Λ = {λ_{n}}. Then, from Lemma 4, f is bounded over I that is m ≤ f ≤ M. To prove the result it is sufficient to prove that F = \frac{f−m}{M−m} belongs to BV^{(p)}(I).

Suppose that F is not in BV^{(p)}. Then there is a point x in I such that F is not of BV^{(p)} on any neighborhood of x. Let {a_{n}} be a sequence of positive numbers such that ∑ a_{n} = ∞. Then there is a partition P_{1} of I such that

∑_{J∈P_{1}} |F (J)|^{p} ≥ a_{1} + 2.

The point x is either an interior point of an interval in P_{1} or an endpoint of at most two intervals in P_{1}. Removing this one, or possibly two, intervals from P_{1}, for the remaining collection of intervals, say Q_{1}, since |F (x)| ≤ 1 for all x ∈ I, we get

∑_{J∈Q_{1}} |F (J)|^{p} ≥ a_{1}.

If Q_{1} has q_{1} intervals we get Q_{1} = {I_{k}^{1} | k = 1, 2, …, q_{1}}. Define λ_{k} = 1 for all k = 1, 2, …, q_{1}, and we have

∑_{1}^{q_{1}} \frac{|F (I_{k}^{1})|^{p}}{λ_{k}} ≥ a_{1}.

The result is true for the first step.

Assume the result is true for n steps and we have to prove the result for the next step. The one, or possible two, intervals that were removed from P_{n} to form Q_{n}, form a neighborhood U_{n} of x. Since F is not of BV^{(p)} on U_{n} there is a finite partition P_{n+1} of U_{n} so that

∑_{J∈P_{n+1}} |F (J)|^{p} ≥ (n + 1)a_{n+1} + 2.

The point x is either an interior point of one interval or an endpoint of at most two intervals in P_{n+1}. If we remove this one, or possible two, intervals from P_{n+1} and call the remaining collection of intervals Q_{n+1}, then

∑_{J∈Q_{n+1}} |F (J)|^{p} ≥ (n + 1)a_{n+1}.

If Q_{n+1} has q_{n+1} intervals we write Q_{n+1} = {I_{k}^{n+1} | k = 1, 2, …, q_{n+1}} and define λ_{r_{n}+i} = n + 1, for all i=1 to q_{n+1}, where r_{n} = ∑_{0}^{n} q_{k} and q_{0} = 0. We have

∑_{1}^{q_{n+1}} \frac{|F (I_{k}^{n+1})|^{p}}{λ_{r_{n}+k}} ≥ a_{n+1}.

Observe that the intervals of Q_{n+1} are within U_{n} and all the intervals of Q_{1} ∪ Q_{2} ∪ ⋯ ∪ Q_{n+1} are pairwise non-overlapping. Then

∑_{i=1}^{n+1} ∑_{k=1}^{q_{i}} \frac{|F (I_{k}^{i})|^{p}}{λ_{r_{i−1}+k}} ≥ ∑_{i=1}^{n+1} a_{i}.

Thus, we construct a sequence of non-decreasing positive numbers {λ_{k}} and a sequence {I_{k}^{n} | k = 1, 2, …, q_{n}; n = 1, 2, … } of non-overlapping subintervals of I such that \frac{1}{λ_{k}} decreases to zero, ∑ \frac{1}{λ_{k}} = ∞ and

∑_{i=1}^{∞} ∑_{k=1}^{q_{i}} \frac{|F (I_{k}^{i})|^{p}}{λ_{r_{i−1}+k}} = ∞.

Thus F is not in ΛBV^{(P)} for this particular sequence of λ’s which contradict our hypothesis. Hence the result follows.

Proof of Theorem 4. For δ > 0 and p ≥ 1 the p-modulus of continuity of f over I (that is ω(p, δ) over I) is defined as

ω(p, δ) = ∥ |T_{h}f − f|^{p}∥_{∞,I},

where (T_{h}f)(x) = f(x + h), ∀x. Clearly, ω(p, δ) is increasing and converges to zero as δ → 0 because of the uniform continuity of f on I.

Let I_{n} = [a_{n}, b_{n}] be a sequence of non-overlapping subintervals of I. Define

E_{m} = \{I_{k} | ω(p, \frac{b − a}{m}) ≥ |f(I_{k})|^{p} > ω(p, \frac{b − a}{m + 1})\}, m = 1, 2, ….

If |f(I_{k})|^{p} ≤ \frac{b−a}{m+1}, then

|f(I_{k})|^{p} = |f(b_{k}) − f(a_{k})|^{p} ≤ ω(p, |b_{k} − a_{k}|) ≤ ω(p, \frac{b − a}{m + 1}).

Thus I_{k} ∈ E_{m} only if |I_{k}| > \frac{b−a}{m+1}. Thus E_{m} contains at most m intervals. Also if I_{p} ∈ E_{r} and I_{q} ∈ E_{r+s} then

|f(I_{q})|^{p} ≤ ω(p, \frac{b − a}{r + s}) ≤ ω(p, \frac{b − a}{r + 1}) < |f(I_{p})|^{p}.

Thus by considering those intervals in E_{1}, then those in E_{2}, etc., and rearranging the intervals we get J_{k} such that

|f(J_{1})|^{p} ≥ |f(J_{2})|^{p} ≥ ⋯ ≥ |f(J_{n})|^{p} ≥ ⋯ → 0,  (1)

where

|f(J_{m})|^{p} ≤ ω(p, \frac{b − a}{m}).  (2)

Namely, if m is an integer for which |f(J_{m})|^{p} > ω(p, \frac{b−a}{m}), then

|f(J_{1})|^{p} ≥ |f(J_{2})|^{p} ≥ ⋯ ≥ |f(J_{m})|^{p} > ω(p, \frac{b − a}{m})

implies |J_{k}| > \frac{b−a}{m} (k = 1, 2, …, m), which is impossible since the intervals J_{k} (k = 1, 2, …, m) are non-overlapping and contained in [a, b]. Thus (2) holds for all m.

Since sequence ω(p, \frac{b−a}{n}) decreases to zero, from Lemma 1 we get a nondecreasing sequence of positive numbers {λ_{n}} such that

\frac{1}{λ_{n}} → 0, ∑ \frac{1}{λ_{n}} = ∞ and ∑ \frac{ω(p, \frac{b−a}{n})}{λ_{n}} < ∞.

Applying Lemma 2 to the sequences {λ_{n}} and {|f(I_{n})|^{p}} we get

∑ \frac{|f(I_{n})|^{p}}{λ_{n}} ≤ ∑ \frac{|f(J_{n})|^{p}}{λ_{n}} ≤ ∑ \frac{ω(p, \frac{b−a}{n})}{λ_{n}} < ∞.

Hence the result follows.

Since the convergence in the norm is the uniform convergence, the class of all continuous functions over I becomes a closed subspace of the class functions of ΛBV^{(p)}(I).

Proof of Theorem 5. Let I_{n} = [s_{n}, t_{n}] be a sequence of non-overlapping subintervals of I. Let J_{n} be the interval determined by the points φ(s_{n}) and φ(t_{n}). Then φ(I_{n}) ⊆ J_{n} ⊆ [c, d] and the intervals J_{n} are non-overlapping, which implies

∑ \frac{|f ∘φ(I_{n})|^{p}}{λ_{n}} = ∑ \frac{|f ∘φ(s_{n}) − f ∘φ(t_{n})|^{p}}{λ_{n}}

= ∑ \frac{|f[φ(s_{n})] − f[φ(t_{n})]|^{p}}{λ_{n}} = ∑ \frac{|f(J_{n}|^{p}}{λ_{n}} < ∞,

as f ∈ ΛBV^{(p)}. Hence the result follows.

Proof of Theorem 6. Since f has left- and right-hand limit at every point of I, from Lemma 3 we get f = F ∘φ where φ is a monotone function defined on I and F is a continuous function defined on the smallest closed interval, say [c, d], containing the range of φ. Let ψ be a linear one to one mapping from [c, d] onto I. Then g = ψ∘φ is a monotone function from I into I and h = f ∘ψ^{−1} is a continuous function defined on I. Hence the result follows from Theorem 4 and Theorem 5.

Proof of Theorem 7. Since F ∈ ΛBV^{(p)}(I), from Theorem 2 F has right- and left-hand limit at every point of I. From Lemma 3 we get F = h∘ϕ where h is a continuous function defined on I and ϕ is a monotone function from I into I. Then, we get g∘F = (g∘h)∘ϕ where g∘h is continuous and ϕ is monotone function. From Lemma 3, g∘F has a right- and left-hand limit at every point of I. Thus the result follows from Theorem 6.

REFERENCES

[1] D. Waterman, On the convergence of Fourier series of functions of bounded variation, Studia Math. 44 (1972), 107–117.

[2] D. Waterman, On Λ-bounded variation, Studia Math. 57 (1976), 33–45.

[3] J. R. Patadia and R. G. Vyas, Fourier series with small gaps and functions of generalized variations, J. Math. Anal. Appl. 182, 1 (1994),113–126.

[4] M. Shiba, On the absolute convergence of Fourier series of functions class ΛBV^{(p)}, Sci. Rep. Fukushima Univ. 30 (1980), 7–10.

[5] R. G. Vyas, On the absolute convergence of small gaps Fourier series of functions of ΛBV^{(p)}, J. Inequal. Pure Appl. Math. 6 (1),23 (2005), 1–6.

[6] S. Perlman, Functions of generalized variation, Fund. Math. 105 (1980), 199–211.

(received 10.09.2004, in revised form 29.10.2006)

Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara-390002, Gujarat, India.

E-mail: drrgvyas@yahoo.com