JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/630137 630137 Research Article Properties of Functions in the Wiener Class BVp[a,b] for 0<p<1 Niu Yeli 1 http://orcid.org/0000-0002-8398-023X Wang Heping 2 Banaś Józef 1 School of Mathematical Sciences Capital Normal University Beijing 100048 China cnu.edu.cn 2 School of Mathematical Sciences BCMIIS Capital Normal University Beijing 100048 China cnu.edu.cn 2015 642015 2015 25 07 2014 16 10 2014 642015 2015 Copyright © 2015 Yeli Niu and Heping Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will investigate properties of functions in the Wiener class BVp[a,b] with 0<p<1. We prove that any function in BVp[a,b] (0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b]. We also obtain the explicit form of functions in BVp[a,b] and show that their derivatives are equal to zero a.e. on [a,b].

1. Introduction

Let 0<p<. We say that a real valued function f on [a,b] is of bounded p-variation and is denoted by fBVp[a,b], if (1)Vpf=supTk=1nfxk-fxk-1p1/p<,where the supremum is taken over all partitions T:a=x0<x1<<xn=b. When p=1, we get the well-known Jordan bounded variation BV[a,b]; and when 1<p<, we get Wiener’s definition of bounded p-variation. There are many other generalizations of BV, such as bounded Φ-variation in the sense of Young (see ) and Waterman’s Λ-bounded variation (see ). The class BVp and generalizations of BV have been studied mainly because of their applicability to the theory of Fourier series and some good approximative properties (see, e.g., ).

However, it should be mentioned that results of most papers deal mostly with the case p1. This is because that in this case BVp[a,b] is a Banach space with the norm fBVp=|f(a)|+Vpf (see, e.g., ). In the case 0<p<1, BVp[a,b] is no longer a Banach space and has not been studied as far as we know. Nevertheless, functions in BVp[a,b]  (0<p<1) have many interesting properties; for example, their derivatives are equal to zero a.e. on [a,b].

In this paper, we will investigate properties of functions in the class BVp[a,b] with 0<p<1. We will show that BVp[a,b] is a Frechet space with the quasinorm(2)qf=fap+Vpfp.We will get the Jordan type decomposition theorem which says that any function in BVp[a,b](0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b]. We also get the representation theorem which gives the explicit form of functions in BVp[a,b](0<p<1).

2. Statement of Main Results

Clearly, for any fixed p(0,1), the Wiener class BVp[a,b] is a linear space. We define the functional q on BVp[a,b] by(3)qf=fap+Vpfp=fap+supTk=1nfxk-fxk-1p,fBVpa,b.From the inequality (a+b)pap+bp  (a,b0,  0<p<1), we get that q(f+g)q(f)+q(g). It then follows that q is a quasinorm on BVp[a,b].

Our first result claims that BVp[a,b]  (0<p<1) equipped with the quasinorm q is a Frechet space.

Theorem 1.

The Wiener class BVp[a,b]  (0<p<1) equipped with the quasinorm q is a Frechet space.

From the inequality(4)i=1aip21/p2i=1aip11/p1,ai0,0<p1p2<,we get that, for any fBVp1[a,b],(5)Vp2fVp1f,which means that BVp1[a,b]BVp2[a,b]. Specially, for 0<p<1, BVp[a,b]BV1[a,b]BV[a,b]. This implies that BVp[a,b] functions are bounded, and the discontinuities of a BVp[a,b] function are simple and, therefore, at most denumerable (see [8, Theorem 13.7 and Lemma 13.2]). By the Jordan decomposition theorem, we know that every function f in BV[a,b] can be expressed as the difference of two increasing functions g and h defined on [a,b] (see [8, Corollary 13.6]). If fBVp[a,b]BV[a,b], we can require that the above increasing functions g and h are still in BVp[a,b]. This is our next theorem.

Theorem 2 (Jordan type decomposition theorem).

Any function in BVp[a,b]  (0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b].

Let t[a,b], d>0, and 0dd. We set(6)ht,d,dx=0,x<t,d,x=t,d,x>t.Then ht,d,d(x) is increasing on [a,b] with only one discontinuity point t. Also, (ht,d,dx)=0 for xt.

Let f be an increasing function in BVp[a,b](0<p<1). Denote by AA(f) the set of points of discontinuity of f. Then A is at most countable (see [8, Theorem 2.17]). Since f is increasing, we get that, for any tA, the right and left limits f(t+0) and f(t-0) of the function f at t exist, f(t+0)-f(t-0)>0, and 0f(t)-f(t-0)f(t+0)-f(t-0). For tA, we define(7)ht~(x)ht,f~(x)=ht,f(t+0)-f(t-0),f(t)-f(t-0)(x).

Our next theorem characterizes the form of an increasing function in BVp[a,b]. Any increasing function f in BVp[a,b] must be as follows:(8)fx=n=1Nhtn,dn,dnx+c,where N, tn[a,b], dn>0, dn[0,dn], and n=1Ndnp<.

Theorem 3.

(1) If f(x)=c+n=1Nhtn,dn,dn(x), where N, tn[a,b],  dn>0, and dn[0,dn], then fBVp[a,b]  (0<p<1) if and only if n=1Ndnp<. In this case,(9)n=1Ndnp1/pVpf2n=1Ndnp1/p.

(2) Let f be an increasing function in BVp[a,b](0<p<1). Then f(x)=tAht~(x)+c, where c is a constant, A is the set of points of discontinuity of f, and ht~(x) is defined by (7).

Finally, for an increasing function f in BVp[a,b]  (0<p<1), by Theorem 3 we have f(x)=tAht~(x)+c, where A is the set of points of discontinuity of f and at most countable. Since (ht~x)=0, a.e. x[a,b], by the Fubini term by term differentiation theorem (see [9, Proposition 4.6]), we get fx=0, a.e. x[a,b]. By Theorem 2, any function f in BVp[a,b] can be expressed as the difference of two increasing functions g(x) and r(x) in BVp[a,b]. Applying Theorem 3, we get the representation theorem of functions in BVp[a,b]  (0<p<1) as follows.

Corollary 4.

Let fBVp[a,b]  (0<p<1). Then f can be expressed in the following form:(10)fx=gx-rx=tA1ht,g~x-tA2ht,r~x+c,where c is a constant, g(x), r(x) are increasing functions in BVp[a,b], ht,g~(x) and ht,r~ are defined by (7), A1,A2A, and A1, A2, A are the sets of points of discontinuity of g, r, and f, respectively. Furthermore, f(x)=0, a.e. x[a,b].

3. Proofs of Theorems <xref ref-type="statement" rid="thm2.1">1</xref>–<xref ref-type="statement" rid="thm2.3">3</xref> Proof of Theorem <xref ref-type="statement" rid="thm2.1">1</xref>.

It suffices to prove that BVp[a,b] is complete. Let {fn} be a Cauchy sequence in BVp[a,b]; that is, q(fn-fm)=|fn(a)-fm(a)|p+(Vp(fn-fm))p0 as n,m. For any ξ[a,b], using the partition T:aξb and the definition of Vpf, we get that {fn(ξ)} is a Cauchy sequence in R and converges to a number denoted by f(ξ). For any ɛ>0, there exists an integer N such that q(fn-fm)ɛ for m,n>N. Let T:a=x0<x1<<xk=b be an arbitrary partition of [a,b]. Then(11)fma-fnap+i=1kfm-fnxi-fm-fnxi-1pqfn-fmɛ.Letting m, we get that(12)fa-fnap+i=1kf-fnxi-f-fnxi-1pɛ.Taking the supremum over all partitions T, we have q(f-fn)ɛ for n>N. This means that f=(f-fn)+fnBVp[a,b], and q(f-fn)0 as n. Hence, BVp[a,b]  (0<p<1) is complete. Theorem 1 is proved.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">2</xref>.

Suppose that fBVp[a,b]  (0<p<1). Since fBVp[a,b]BV[a,b], by the Jordan decomposition theorem (see [8, Corollary 13.6]), we have f(x)=g(x)-r(x), where g(x), r(x) are increasing functions on [a,b]. Indeed, we can choose g(x) to be Vax(f), the total variation function of f defined by(13)Vaxf=supTi=1nfxi-fxi-1,where the supremum is taken over all partitions T:a=x0<x1<<xn=x of [a,x], r(x)=Vax(f)-f(x). It suffices to show that g(x)=Vax(f)BVp[a,b]. For any fixed partition T:a=x0<x1<<xn=b, we note that(14)gxi-gxi-1p=Vxi-1xifp=supTij=1mifξi,j-fξi,j-1psupTij=1mifξi,j-fξi,j-1p,where the supremum is taken over all partitions Ti:xi-1=ξi,1<ξi,2<<ξi,mi=xi of [xi-1,xi]. It follows that(15)i=1ngxi-gxi-1pi=1nsupTij=1mifξi,j-fξi,j-1p=supTi,  1ini=1nj=1mifξi,j-fξi,j-1pVpfp,which implies gBVp[a,b]. This completes the proof of Theorem 2.

To prove Theorem 3, we introduce the next lemma.

Lemma 5.

If fBVp[a,b]C[a,b]  (0<p<1), then f is a constant function.

Proof.

It suffices to show that, for any d[a,b], f(d)=f(a). Assume that there exists d(a,b] such that f(d)f(a). Without loss of generality, we assume that f(a)<f(d). Since fC[a,b], there exist n-1 points ξ1,ξ2,,ξn-1 such that a=ξ0<ξ1<<ξn-1<ξn=d and f(ξi)=f(a)+((f(d)-f(a))/n)i. Hence,(16)Vpfpi=1nfξi-fξi-1p=n1-pfd-fap,as n, which implies that fBVp[a,b]. This leads to a contradiction. Lemma 5 is proved.

Proof of Theorem <xref ref-type="statement" rid="thm2.3">3</xref>.

(1) Without loss of generality, we may assume that N=. Let T:a=y0<y1<<ym=b be a partition of [a,b]. For j, 1jm, we note that (17)fyj-fyj-1p=n=1htn,dn,dnyj-htn,dn,dnyj-1p=nyj-1<tn<yjdn+ntn=yj-1dn-dn+ntn=yjdnpnyj-1tnyjdnp,where an empty sum denotes 0. It follows that(18)j=1mfyj-fyj-1pj=1mnyj-1tnyjdnp2n=1dnp.Taking the supremum over all partitions of [a,b], we obtain that(19)Vpfp2n=1dnp.

On the other hand, for any fixed m, by renumbering {tn}n=1m if necessary, we may assume that at1<t2<<tmb. We set yi=((ti+ti+1)/2)  (1im-1). Then T:a=y0<y1<y2<<ym-1<ym=b is a partition of [a,b]. It follows that(20)Vpfpj=1mfyj-fyj-1pj=1mnyj-1<tn<yjdnpj=1mdjp.Letting m, we get(21)Vpfn=1dnp1/p.Combining (19) with (21), we get (9). Hence, fBVp[a,b]  (0<p<1) if and only if n=1dnp<.

(2) Let f be an increasing function in BVp[a,b]  (0<p<1) and A the set of points of discontinuity of f on [a,b]. We set hf(x)=tAht~(x), where ht~(x) is defined by (7). Similar to the proof of (21), we have(22)tAft+0-ft-0pVpfp<.Applying the above proved result, we obtain that hf(x)BVp[a,b]. We set g(x)=f(x)-hf(x); then gBVp[a,b]. We will show that g(x) is continuous on [a,b].

Indeed, for x[a,b], we have(23)tAht~xtAft+0-ft-0tAft+0-ft-0p1/pVpf<.By Weierstrass M-test (see [10, Theorem 7.10]), we get that the series tAht~(x) converges uniformly on [a,b]. For x0[a,b]A, ht~(x)  (tA) is continuous at x0, so hf(x)=tAht~(x) is also continuous at x0. It follows that g(x) is continuous at x0 for x0[a,b]A.

For x0A, we set u(x)=tA{x0}ht~(x). Then u(x) is continuous at x0 and hf(x)=u(x)+hx0~(x). Hence,(24)hfx0+0=ux0+fx0+0-fx0-0,hf(x0-0)=u(x0),hfx0=ux0+fx0-fx0-0.Thus,(25)gx0+0=gx0=gx0-0=fx0-0-ux0,from which we can deduce that g is continuous at x0. Hence, g(x)C[a,b].

Since g(x)C[a,b]BVp[a,b], it follows from Lemma 5 that g(x) is a constant c. Thus f(x)=hf(x)+c=tAht~(x)+c. The proof of Theorem 3 is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors were supported by the National Natural Science Foundation of China (Project no. 11271263), the Beijing Natural Science Foundation (1132001), and BCMIIS.

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