Generalization of an elementary inequality in Fourier analysis

Appl. Math. J. Chinese Univ.2010, 25(1): 43-47Generalization of an elementary inequalityin Fourier analysis ;ZHOU Guan-zhenAbstract. The inequalitysup|”sinkx.≤3√元,plays an important role in Fourier analysis and approximation theory. It has recently beengeneralized by Telyakovskii and Leindler. This paper further generalizes and improves theirresults by introducing a new class of sequences called 7-piecewise bounded variation sequence(7-PBVS).81 IntroductionThe following inequality plays an important role in Fourier analysis and approximationtheory (see [4)):supp|它inkr≤3√元.(1.1)There are lots of generalizations of (1.1). Among them, Telyakovskii[2] established the inequalityin Theorem A which is more extensive than (1.1) and is used to study the properties of functionsof bounded variation.Theorem A. Let {nm} be a sequence of positive integers satisfying 1 =n1 < n2< ... and之⊥≤Am= 1,....(1.2)Lnj-nmfor a constant A > 1, then the inequality≤KAj=1| k=njholds for all x, where K is an absolute constant.Received: 2008-05-28.MR Subject Classification: 41A17, 41A25.Keywords: PBV condition, 7-PBVS, inequality.Digital Object Identifier(DOI): 10.1007/s11766010-2233-x.Supported by the National Science Foundation of China(60970151, 70973110), the Natural Science Founda-tion of Zhejiang Province( Y 7080068) and the Professor Foundation of Zhejiang Gongshang University(07-13).中国煤化工MHCNMH G.14Appl. Math. J. Chinese Univ.Vol. 25, No.1Let C= {cn}n=1 be a sequence of positive numbers. IfE IOcel≤K(C)emk=mfor allm= 1,2,-, where△ck =Ck-Ck+1 and K(C) is a positive constant depending only onC, then we say that the sequence C is of rest bounded variation, denoted by C∈RBVS. It isevident that every monotone decreasing null sequence is a sequence of rest bounded variation.Using the notion of RBVS, Leindler[1] generalized Telyakovskii's result toTheorem B. Let {nm} be a sequence of positive integers satisfying 1 = n1 < n2 < ... and thecondition (1.2). Suppose thatC= {cn}m1∈RBVS andnlCn|≤K,(1.3)then。|nj+1-1ECk simkx|≤K(C)Aj=1| k=nholds for all x.To further generalize Leindler's result, we introduce a new class of sequences.Definition 1. Let {nm} be a sequence of positive integers satisfying 1 = n1 < n2 < ... andthe condition (1.2). Let γ= {7n}n=1 be a positive sequence. A null sequenceC= {Cn}n=1 issaid to be of 7-piecewise bounded variation, denoted by C∈γPBVS, if there exists a positiveconstant K (C) depending only on C such that乞|Oca|≤K(C)m， nm _1 > 1for k= 1,2.... A mull sequence C= {cn}%1 is said to be of piecewise bounded variation on{nm}, denoted by C∈PBVS, if there exists a positive constant K(C) depending only on C中国煤化工MHCNMH G.ZHOU Guan-zhen.Generalization of an elementary inequality in Fourier analysis45such thatn-1》|Icr|≤ K(C)en|，nm-1+1 Ck sinkxj=1| k=nj|ni+1nj+1 - 1》Ck sinkx|+ 》》 Ck sin kxj=i+1| k=njI1+ I2+I3+ I4.(2.4)By Lemma 1, we haveI1+I2≤xk|cx|+ > k|cx|) ≤Knx≤Kπ.j=1 k=njk=ngBy Abel's transformation and the well-known inequality2 sinkx|≤-， q≥p≥0,we havekni+1-113=| 2 Ocx2sinkx+hn+1-1 E sinjx|≤→( E |Qoal+en+1-1| )k=n;\k=n+1Therefore, by (2.2) and Lemma 1 we have13≤T+(n+1t+1on+1-I)K(C)1_+._1n+ni1+n+1-1)≤nx≤K(C).(2.5)Similarly, we havenj+1-2nj+1- 1I4=之习Ock 2 sinkx +Cnj+1-1 2 sinjxj=i+1| h=nj k=njj=nj≤气点(三|IOcx| + |Cns+1-1|j=i+1\ k=n;K(O E (m, +7m+ +e-)j=i+1K(CE11 nj~ xni+1亡≤I≤K(C),(2.6)With (2.4)-(2.6), we have completed the proof of Theorem 1.Proof of Theorem 2. Since {nm } satisfies the Hadamard condition, we can insert the followingnew terms between nm-1 and nm (m = 1,2....):Nm-1√0rnm-1, (向)" m-...(0).nm-1,中国煤化工MHCNMH G.ZHOU Guan- zhen. .Generalization of an elementary inequality in Fourier analysis47where Nm= [ogN" - logN"-1] . Let {nm} be the new sequence which consists of all the termsin {nm} and the inserted terms such that1<√8<"k+1<δ，k= 1,2....(2.7)ngand C also belongs to PBVS on {nm}. Hence, we only need to prove the slightly strongerinequality≤K(C)A.2三For convenience, we write {n*} as {nm}. For all nm-1 < n≤nm, m= 1,2....(take no to be0), define7n= |cn|+ |Cnm-1+ |Cnm-1+1|.(2.8)By Theorem 1, (2.7) and (2.8), we only need to prove that ifC∈PBVS, then C∈γPBVSwithγ={7m} and 'n is defined by (2.8).If C satisfies condition (1.8) for nm-1 +1 < n≤nm, thenE IOcal=|cm-.|+ |ocal≤ K(C)m;k=nm-1+1and forn=nm-1+1，乙|Ock|= |Ocnm. .|s K(C)m. Hence C satisfes condition (1.4).If C satisfies condition (1.9) for nm-1+1≤n