On Uniform Convergence of Double Sine Series. Variation Double Sequences

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 51, HIKARI Ltd, On Uniform Convergence of Double Sine Series under Condition of p-supremum Bounded Variation Double 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 Correspondence should be addressed to: Moch Aruman Imron, Copyright 2013 Moch Aruman Imron et al. 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 The classical theorem on the uniform convergence of sine series with monotone decreasing coefficients have been proved by Chaundy and Jollife in Recently, the monotone decreasing coefficiets has been generalized by classes of Mean Bounded variation Sequences, Supremum Bounded Variation Sequence and p-supremum Bounded Variation Sequences. In two variables, class of Mean Bounded Variation Double Sequences and Supremum Bounded Variation Double Sequences were studied under the uniform convergence of double sine series. We shall generalize those results by defining class of p-supremum Bounded Variation Double Sequences and study uniform convergence of double sine series.

2 2536 Moch Aruman Imron, Ch. Rini Indrati and Widodo Mathematics Subject Classification: 42A20, 42A32, 42B99 Keywords: double sine series, p-supremum bounded variation double series, uniform convergence 1. Introduction The classical theorem on the uniform convergence of sine series with monotone decreasing coefficients have been proved by Chaundy and Jollife in 1916 [1, 12] as stated in Theorem 1.1. Theorem 1.1. Suppose that 0, is decreasing tending to zero. The necessary and sufficient conditions for the uniform convergence of the series is lim 0. sin 1.1 Many classes of sequences have been introduced by many reseachers such as Tikhonov [11], Zhou [10], Korus [7]. These classes are more general than the classes of monotone decreasing coeffecients (1.1). The definition of the classes of SBVS (Supremum Bounded Variation Sequences), SBVS (Supremum Bounded Variation Sequences), SBVS2 (Supremum Bounded Variation Sequences of second type) are the following. 0, 1 sup, / 2 0, sup, 1 where and C,, depend only, [x] the greatest integer that is less than or equal x. Imron et. al. [4] introduced classes of (p-supremum Bounded Variation Sequences) and 2 (p-supremum Bounded Variation Sequences of second type) stated in Definition1.2: Definition 1.2. Let and be two sequences of complex and positive numbers, respectively. A couple, is said to be (i) p-supremum Bounded Variation Sequences, written,, if there exist a positive constant C and 1 such that

3 On uniform convergence of double sine series 2537 / sup, 1, (ii) p- Supremum Bounded Variation Sequences of second type, written,, 2, if there exist a positive constant C and 0, tending monotonically to infinity depending only on, such that sup, 1, for 1, As the single sine series (1.1) we have double sine series of two variables. Let and consider the double sine series of the form 1.2 As an idea in one variable, to investigate the uniform convergence of double sine series, the coefficients of the series (1.2) are supposed to be member of class of general monotone double sequences. In two variables, Supremum Bounded Variation Double Sequences of first type (SBVSDS1) and Supremum Bounded Variation Double Sequences of second type (SBVSDS2) have been introduced by Korus [8]. Definition 1.3. A double sequence belongs to class SBVSDS1, if there exists C > 0 and integer 2 and,,, each one converges to infinity, all of them depend only such that where Δ max,, 1, Δ max, 1,, Δ sup,,,,,,,,,,. Definition 1.4. A double sequence belongs to class SBVSDS2, if there exists C > 0 and integer 1 and, converging to infinity, depending only such that

4 2538 Moch Aruman Imron, Ch. Rini Indrati and Widodo Δ sup,, 1, Δ sup, 1,, Δ sup,,. Related to double sine series of (1.2), Korus and Moricz [9] also consider the regular convergence of double sequence stated in Definition 1.5. Definition 1.5. A double sequence is regularly convergence if the sums converge to a finite number as m and n tend to infinity independently of each other, moreover, 1,2,3,, and, 1, 2, 3, are convergent. In the present paper, we construct classes of p-supremum Bounded Variation Double Sequences of first type 1 and p-supremum Bounded Variation Double Sequences of seconnd type 2 and investigate some of those two classes. Futhermore, we investigate the uniform regular convergence of the sine double series. 2. Uniform Convergence of Double Sine Series In this section, we construct class of p-supremum Bounded Variation Double Sequences First Type and Second Type. Furthemore we investigate the relations between those and also study other properties of class of p-supremum Bounded Variation Double Sequences. The Goal of this paper is to study the uniform regular corvergence of double sine series in Theorem Definition 2.1. Let and be two double sequences of complex and positive numbers, respectively. A couple, is said to be class of p-supremum Bounded Variation Double Sequences first type written, 1, if there exists C > 0 and integer 2 and,,, each one converges to infinity,all of them depend only such that (i) Δ / max,, 1,

5 On uniform convergence of double sine series 2539 (ii) Δ / max, 1,, (iii) Δ / sup,,, for 1. Definition 2.2. Let and be two double sequences of complex and positive numbers, respectively. A couple, is said to be class of p-supremum Bounded Variation Double Sequences second type written, 2, if there exists C > 0 and integer 1 and, converging to infinity, depending only such that (i) Δ / sup,, 1, (ii) Δ / sup, 1,, (iii) Δ / sup,,, for 1. Theorem 2.3. If 1, then 1 2. Proof: First, we prove that 1 2, let, 1 and put inf,,, 1, 1, 1,, for 1. From Definition 2.1 and Definition 2.2, 2, so we have 1 2. The second, we prove that 1 2. From idea of example [8] can be constructed in the following way. Set 2 for 1, 2, 3, and 0 if 1, 1 if, 0 if, 1 if 2, 0 if 2. We define the sequence, where, for every j. By Theorem 2.1. [6],, with constant 4 and.

6 2540 Moch Aruman Imron, Ch. Rini Indrati and Widodo The second example 0 if 1, if 0 if, / if 2 0 if 2. We define the sequence, where, for every j. By Theorem 2.4. [6],, 2 with constant 2 and /. Finally, we define, 2. Thus 0 for 1, 3. It is clear that for n=2, by Definition 1.3, 1. Futhermore the second and third condition of Definition 1.4 are satiesfied for 4 and /, this means, 2. Thus 1 2. Theorem 2.4. If 1, then 1 1. Proof: Given, 1 and for 1, (i) From proof Theorem 3.1 [4] by replacing to obtained Δ Δ Hence by Definition 2.1 we have Δ / Δ, max. (ii) Similarly by proof (i) and replacing to and Definition 2.1 we have Δ / Δ max. (iii) Similarly by proof (i) and replacing to with double sumation and Definition 2.1 we have Δ / Δ /

7 On uniform convergence of double sine series 2541 From (i),( ii) and (iii), sup, 1, then 1 1. Theorem 2.5. If 1, then 2 2. Proof: Similarly as proof of Theorem 2.4 and Definition 2.2. Definition 2.6. Let be double sequences of positive numbers. (i) A class of p-supremum Bounded Variation Double Sequences first type of, written 1, is defined as :, 1. (ii) A class of p-supremum Bounded Variation Double Sequences second type of, written 2, is defined as :, 2. By Theorem 2.3, Theorem 2.4 and Theorem 2.5 and straight from Definition 2.6 we have Corollary 2.7, Corollary 2.8 and Corollary 2.9. Corollary 2.7. If 1, then 1 2. Corollary 2.8. If 1, then 1 2. Corollary 2.9. If 1, then 1 2. Some properties of 2, 1, are stated below. Theorem 2.10 give a sufficient condition for a couple of sequences in 2 to be have bounded variation introduced by Hardy [3]. The notion of bounded variation of double sequence of real or complex term is defined as follows: The double sequence is bounded variation in, if (i). is for every fixed value of n and k, bounded variation in n and k. (ii). the summation is convergent. Theorem Let, 2, 1. If (i) for, sup decreasing monotone. (ii) for, sup decreasing monotone.,, (iii) for,, sup decreasing monotone.

8 2542 Moch Aruman Imron, Ch. Rini Indrati and Widodo then (i) / sup (ii) sup,, (iii) sup, holds respectively for p, 1. Proof. We prove this Theorem in three parts (i) Let, 2 and / sup decreasing monotone for. We denote for every / sup. Therefore / sup, 2. By Holder inequality obtained (2.1) and (2.2) obtained because of the decreasing monotone condition of Theorem 2.10 (i). (ii) The proof of (ii) is similar with the proof of (i) by replacing m to n and to. (iii) Let, 2 and sup decreasing monotone for,. We denote sup for every,, then we write

9 On uniform convergence of double sine series sup, 4. By Holder inequality obtained (2.3) and (2.4) obtained because of the decreasing monotone condition of Theorem 2.10 (iii). The proof of this Theorem is complete. Theorem Let, 2, 1. If (i) the summation sup (ii) the summation sup, tending to 0, for,, tending to 0, for,, (iii) the summation sup tending to 0, for. then, the sequence is bounded variation. Proof. (i) Given, 2 and let sup tending to 0 for. We denote sup, for fixed n, then 0, for. For every 0, there exists such that for and from the proof of Theorem 2.10 (i), we obtained, 2. Thus Therefore is bounded variation. (ii) Proof of (ii) similar with the proof of (i) by replacing m to n and to. Therefore is bounded variation.

10 2544 Moch Aruman Imron, Ch. Rini Indrati and Widodo (iii) Given, 2 and let sup tending to 0 for. We denote, sup, then 0, for. For every 0, there exists, such that for,, and from the proof of Theorem 2.10 (iii). we obtined Then we have. Thus 2 for,,., 4. Therefore from proof (i), (ii) and (iii), the double sequence, bounded variation. The proof of this Theorem is complete. is Lemma Let 0, and, 2, 1. condition (i), (ii) and (iii) of Theorem are satiesfied, then 0. Proof. From idea proof of Lemma 3 [7], for t be anbitrary natural numbers and 1, 2,, 2, we have. 2.5 Next, an analogous argument, for s be anbitrary natural numbers and 1, 2,, 2, we have. 2.6 Finally, let m, n be natural numbers. A double version of the above argument, for 1, 2,, 2 and 1, 2,, 2, we have If

11 On uniform convergence of double sine series If we add up all inequalities in (2.7) and from (2.5), (2.6), then by proof of Theorem 2.11, we have 3 for,,. Thus 0. Lemma If, 2, 1 and condition (i), (ii) and (iii) of Theorem are satisfied, then (i) 0, (ii) up 0, (iii) 0. Proof. (i) From proof of Theorem we have (ii) (iii). for any,. By condition (iii) of Theorem 2.11, therefore 0, as. From Lemma 2.12, we obtained 0. Further,, 2.8 for, 1, 2,3,. By (2.8) we have sup, 1, 2, 3,.Thus we have sup and an application of (i) yields 0. A similar argument by replacing to, obtained (iii). Theorem If, 2, 1 and condition (i), (ii) and (iii) of Theorem 2.11 are satisfied, then the series (1.2) converges uniformly, regularly at, for all 0,. Proof. To prove the uniformly regularly convergent of double sine series (1.2),

12 2546 Moch Aruman Imron, Ch. Rini Indrati and Widodo first by letting the single sine series with coefficients of double sequence sin, 1, 2, 3,, 2.9 sin, 1, 2, 3, Furthermore, by the condition Definition 2.2 (i) and (ii) it can be proved that for any 1,, 2 and for any 1,, 2. Hence by condition (i) and (ii) and Theorem 3.1. shown by Imron et. al [4] implies the uniform convergence of the series (2.9) and (2.10). The second, we show that the double series (1.2) is regularly convergence at (x,y) for all 0,. (i) Following an idea from Moricz [2], for, 0, 0 we represent the rectangular partial sums,,, 1, of series (1.2) to perform a double summation by part yields,,,, where conjugate Dirichlet kernel defined as and sin, 1 for 0,. Given condition (iii), by Lemma 2.13 we obtained thus Given condition (i), (ii) and (iii), by Lemmma , for and from (2.8) we have,. Further, 0 as. This implies that, for all 0, 2.13, 0 as, 2.14

13 On uniform convergence of double sine series 2547 uniformly in m. Similarly, for all 0,, 0 as, 2.15 uniformly in n. By (2.13), 0 as. Consequently by (2.11), (2.12), (2.14), (2.15) and (2.16), 0, as. (ii) For, 0, 0, From the first and second part of this proof and by Definition 1.5 it can be concluded that series (1.2) is regular convergence. Futhermore by Lemma 2.12, Lemma 2.13 and Theorem 1 [6], the regular convergence is uniformly at (x,y) for all 0,. The proof is complete. The uniform regular convergence of the series in (1.2) in class of 1 for p, 1, is stated in Corollary We abandon the proof, since it is similar to the proof of Theorem Corollary Let, 1, 1. If (i) the summation max 1 for, (ii) the summation max 1 for, (iii) the summation sup 1 for, then the series (1.2) is uniformly regularly convergence at, for all 0,. 3. Conclusions In this paper we have introduced the classes of 1 and 2. We have investigated that (i) The class of 1 and 2 is more general than class SBDS1 and class SBVDS2 respectively introduced by Korus [8] straight from Definition 2.1 and Definition 2.2. (ii) Uniformly regularly convergence of (1.2) in Theorem 2.14 is more general than Theorem 1 by Korus [8] and Theorem 1 by Korus and Moricz [9].

14 2548 Moch Aruman Imron, Ch. Rini Indrati and Widodo Ackowledgments. The authors gratefully acknowledge the support of the Department of Mathematics, Faculty of Mathematics and Sciences University of Brawijaya and Graduate School Department of Mathematics, Faculty of Mathematics and Sciences, University of Gadjah Mada. References [1] A. Zygmund, Trigonometric Series, Vol I, II, Second ed, Cambridge Univ. Press, [2] F. Moricz, On Integrability and L 1 -convergence of Double Trigonometric Series, Studia Math. 98 (3), [3] G. H.Hardy, On the convergence of certain multiple series, Proc. Camb. Phil.Soc, 19, 86-95, [4] M.A. Imron, C. 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). [5] M.A. Imron, C. R. Indrati and Widodo, Uniform Convergence of Trigonometric Series Under p-supremum Bounded Variation Condition, Proceeding of 3rd Annual Basic International Conference, M041-M043, FMIPA, UB, Malang, [6] M.A. Imron, C. R. Indrati and Widodo, Some properties of class of p-supremum bounded Variation sequences, Int.Journal of Math. Analysis, Vol 7, no.35, , [7] P. Korus, Remark On the uniform and L 1 -Convergence Of Trigonometric Series, Acta Math. Hungar, 128(4), [8] P. Korus, On the uniform Convergence Of Double sine series with generalized monotone coefficients, Periodica Math. Hungar, 63, , [9] P.Korus and F. Moricz, On the uniform convergence of Double sine series, Studia Math., , [10] 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. [11] S. Tikhonov, Best approximation and moduli of Smoothness computation and Equivalence Theorems, Journal of Approximation Theory, 153 (19-39), [12] T.W. Chaundy and A.E. Jollife, The Uniform Convergence of certain class trigonometric series, Proc. London, Soc. 15, , Received: August 25, 2013