Some Properties of Class of p-supremum. Bounded Variation Sequences

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 35, HIKARI Ltd, Some Properties of Class of p-supremum Bounded Variation Sequences Moch Aruman Imron Department of Math, Faculty of Math and Sciences University of Brawijaya, Malang and Graduate School, Faculty of Mathematics and Sciences University of Gadjah Mada, Yogyakarta, Indonesia Ch. Rini Indrati and Widodo Department of Math, Faculty of Math and Sciences University of Gadjah Mada,Yogyakarta, Indonesia Copyright 2013 Moch Aruman Imron, Ch. Rini Indrati and Widodo. 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. Abstract In this paper, we investigate some properties of class of p-supremum Bounded Variation Sequences. It is a generalization of Supremum Bounded Variation Sequences. The properties will be used to investigate uniform convergence of trigonometric series with p-supremum Bounded Variation Sequences coefficients. Keywords: p-supremum Bounded Variation Sequences, trigonometric series uniform convergence, 1. Introduction In Fourier analysis, it is well known that there are a great number of interesting results established by assuming monotonicity of coefficients. The following classical result in Theorem 1.1. was proved by Chaundy and Jollife [9]: Theorem 1.1. Suppose that 0, is nonincreasingly tending to zero.

2 1704 Moch Aruman Imron, Ch. Rini Indrati and Widodo The series sin converges uniformly in x if only if Theorem 1.1. has been generalized by many authors, to weaken the monotone conditions of the coefficients sequence in (1.1), by introducing classes GMS (General Monotone Sequences) [8], MVBVS (Mean Value Bounded Variation Sequences) [7], SBVS (Supremum Bounded Variation Sequences), and SBVS2 [5]. Here are the definition of them. Definition 1.2. A sequence 0, is said to be in class (i) GMS (General Monotone Sequences) if there exists a positive constant C, depending only on, such that, 1. (ii) MVBVS (Mean Value Bounded Variation Sequences) if there exist positive constant C and 2 depending only on, such that, 1, where [x] denotes the greatest integer that is less than or equal to x. (iii) SBVS (Supremum Bounded Variation Sequences) if there exist positive constant C and 1, depending only on, such that sup /, 1. (iv) SBVS2 if there exist positive constant C and 0, tending monotonically to infinity, depending only on, such that sup, 1. It has been shown that 2 [6, 8]. Further, Liflyand and Tikhonov [1, 2] defined a class of p-general Monotone Sequences as follows: Definition 1.3. Let and be two sequences of complex and positive numbers, respectively. The couple, determines a p-general Monotone Sequences, written,,if there exists a positive constant C and the relation /, 1, holds for p, 1. Let be a sequence of complex number and, where., if and only if [2]. As corollary, is more general than GMS. Futhermore, Imron, et. al. [5, 6] generalized MVBVS and SBVS to and, respectively, as stated in Definition 1.4. These generalization results give an opportunity to search the uniform convergence in a bigger class.

3 Class of p-supremum bounded variation sequences 1705 Definition 1.4. Let and be two sequences of complex and positive numbers, respectively. For 1, a couple, is said to be (i) p-meanvalue Bounded variation Sequences, written,, if there exist a positive constant C and 2 such that / /, 1, (ii) p- Supremum Bounded Variation Sequences, written,,, if there exist positive constant C and 1 such that sup, 1. The class of has been generalized to be 2, 1, as given in Definition 1.5. Definition 1.5. Let and be two sequences of complex and positive numbers, respectively. For 1, a couple, is said to be a p- Supremum Bounded Variation Sequences, written, 2, if there exist a positive constant C and 0, tending monotonically to infinity depending only on, such that / sup, 1. Imron, et. al. [5] have shown that 2, however, it has not proved the inclusion is inclusive or exclusive. In this present paper, we prove that the inclusion is exclusive, i.e., (Theorem 2.1) and In this paper, we also generalize some properties of p-supremum Bounded Variation Sequences class. We use the properties to investigate the uniform convergence of the sine and cosine series. 2. Main Result In this section, we investigate the proper subset relations between class of p-mean Value Bounded Variation Sequences and class of p-supremum Bounded Variation Sequences.Furthemore, we study the other properties of class of p-supremum Bounded Variation Sequences. Theorem 2.1. If 1, then. Proof: By Theorem 3.8. [4], it has shown that. Now let us show an example a couple, but,. For 1, 2, 3,, set 2 and

4 1706 Moch Aruman Imron, Ch. Rini Indrati and Widodo 0 1 1, 1 0, We define the sequence, where, for every k. Moreover, for 2 such that / 2 sup and for 2. / 2 sup / For 1 / 0 sup. Therefore,, by 4 and 2. On the other hand, we have 2 and / 2. As corollary, 2 for all 0,, and 2. This leads to a contradiction. Therefore,,. That means,. / Definition 2.2. Let be a sequence of positive numbers. (i) A class of p-mean Value Bounded Variation Sequences of, written, is defined as :,. (ii) A class of p-supremum Bounded Variation Sequences of, written, is defined as :,. Definition 2.2 and Theorem 2.1 implies corollary 2.3. Corollary 2.3. If 1, then.

5 Class of p-supremum bounded variation sequences 1707 Theorem 2.4. If 1, then 2. Proof. By Theorem 3.5. [4], it has been shown that 2. Now let us show an example for, 2 and,. For 1,2,3,, set 2 and 0 1 1, 0, / We define the sequence, where, for every k. For n, and 2 we have For n, / / 2 sup. / / 1 we have / sup /. So, 2 with 2 and /. On the other hand, we can show that,. To see this, calculate summation and Then sup / sup /. 1, sup / / 1 / 1 1 for all C > 0, 1 and. This leads to a contadiction. Therefore,, then 2.

6 1708 Moch Aruman Imron, Ch. Rini Indrati and Widodo Definition 2.5. Let be a sequence of positive numbers and 1,. A class of p-mean Value Bounded Variation Sequences two of, written 2, is defined as :, 2. By Theorem 2.4 and Defintioin 2.5, we have the following corollary. Corollary 2.6. If 1, then 2. Some properties of 2, 1, are stated below. Theorem 2.7. and Theorem 2.9. are the generalization of Lemma 3.1 and Lemma 3.4 in [8], respectively. Theorema 2.7. Let 1. If, 2, then for any integer n (i). / sup for any,,,2. (i)i. / sup for any,,2. Proof. (i). For each, can be stated as follows:: and,,, 1. Therefore. By using the Holder s inequality, we have / 1 / /. Since, 2, there exist positive constant C and 0,, as r, such that (ii). / 1 / / / sup for any,,,2. According to Theorem 2.1.i. we have / 1 / / / sup for any,,,2. Suppose 1,,2, then / sup / sup..

7 Class of p-supremum bounded variation sequences 1709 / sup. Summing up from 1 to m = 2n, we get 2 2 / sup. 2.1 By the same way, for 1,,, we get sup. 2.2 The addition of (2.1) and (2.2), we obtain sup for any,,2. Complete the proof. Corollary 2.8. Let 1. If,, then for any n and 1, we have (i). / sup / for any,,,2. (ii). / sup / for any,,2. Theorem 2.9 give a sufficient condition for a couple of sequences in 2 to be have bounded variation. Theorem 2.9. Let 1 If, 2 and / / sup, then is bounded variation. Moreover, for each, we have / / sup. Proof. Suppose /2 1. We have 1 /2 / / / 1 /2 1 / / /.. ] /. 2.3 By Holder s inequality, / /.

8 1710 Moch Aruman Imron, Ch. Rini Indrati and Widodo Since, 2, there exist positive constant C and 0,, as r, such that From (2.3) and (2.4), we have / sup. 2.4 / sup / / sup, where. Since / / sup, then Therefore is bounded variation. Corollary Let 1. If,, 1 and / / sup /, then is bounded variation. Moreover, /. / sup /. 3. Uniform Convergence of Sine and Cosine Series Dyachenko and Tikhonov [3] discuss the uniform convergence of class of to sine and cosine series. In this section, we discuss the uniform convergence of class of 2, 1. We consider the series and where is a given null sequence of complex numbers, i.e., 0 as. We define by f and g the sums of series (3.1) and (3.2), respectively, at the point where the series converge. Theorem 3.1. Let, 2, 1. /, / sup 1 the series (3.2) converges uniformly on 0, 2. If

9 Class of p-supremum bounded variation sequences 1711 Proof. (i). For 0, We denote / / sup. It is clear that 0. Let 0 be given, then there exists such that for. Let be nonincreasing null sequence such that, so there exists such that for. Given, 2 and / / sup 1, by Theorem 2.10 / / sup 3.3 Let us now estimate,, where, sin. By Abel s transformation, we get, Where sin and, By (3.3). To estimate A, for any 0, we can find,. Since then sin sin such that and (3.3) imply, if, 3.4 If, then we decompose A as Similar to (3.3) we get. Further where and. Since. 1 1 then 2. From, we obtain

10 1712 Moch Aruman Imron, Ch. Rini Indrati and Widodo From (3.3), (3.4) and (3.5), we get, So, if given 0 there exists, such that for, 2 The series (3.2) converges uniformly on 0,. (ii). For x = 0, 0 and from i, then (3.2) converges uniformly on 0, 2. The proof is complete. The uniform convergence of the series in (3.1) in class of 2, 1, is stated in Theorem 3.2. We abandon the proof, since it is similar to the proof of (i) in Theorem 3.1. Theorem 3.2. Let, 2, 1. if /, / sup 1 then the series (3.1) converges uniformly on 0, 2. Acknowledgements The authors gratefully acknowledge the support of the Department of Mathematics, Faculty of Mathematics and Sciences University of Brawijaya and the Graduate School Department of Mathematics, Faculty of Mathematics and Sciences, University of Gadjah Mada. References [1] E. Liflyand and S. Tikhonov, The Fourier Transforms of General Monotone Functions, Analysis and Mathematical Physics, Trends in Mathematics (Birchauser, 2009). [2] E. Liflyand and S. Tikhonov, A concept of general monotonicity and applications, Math Nachr, 284, No. 8-9, [3] M. Dyachenko and S. Tikhonov, General monotone sequences and convergence of trigonometric series, in: Topics in Classical Analysis and Applications in Honor of Daniel Waterman (World Scientific, Hakensack, NJ, 2008), pp [4] M.A. Imron, Ch. R. Indrati and Widodo, Sifat-sifat Barisan dan fungsi dari klas p-mean Value Bounded variation, Konferensi Nasional Matematika 16, Unpad, Bandung, 2012

11 Class of p-supremum bounded variation sequences 1713 [5] M.A. Imron, Ch. R. Indrati and Widodo, Relasi Inklusi pada Klas Barisan p-supremum Bounded variation, Jurnal Natural A,No 1, Vol 1, FMIPA, UB, Malang (Appear 2013). [6] P. Korus, Remark On the uniform And L 1 -Convergence Of Trigonometric Series, Acta Math. Hungar, 128(4), [7] S.P. Zhou, P. Zhou and D.S. Yu, Ultimate generalization to monotonicity for Uniform Convergence of Trigonometric Series, online:http//arxiv.org/abs/math/ v1. [8] S. Tikhonov, Best approximation and moduli of Smoothness computation and Equivalence Theorems, Journal of Approximation Theory, 153 (19-39), [9] T.W. Chaundy and A.E. Jollife, The Uniform Convergence of certain class trigonometric series, Proc. London, Soc. 15, , Received: April 19, 2013