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Uniform Convergence of Cosine Series with Coefficient from New Class of General Monotone Journal of Research in Mathematics Trends and Technology (JoRMTT) Vol. 01, No. 02, 2019 | 34-38 Journal of Research in Mathematics Trends and Technology Uniform Convergence of Cosine Series With Coefficient from New Class of General Monotone Moch Aruman Imron Department of Mathematics, University of Brawijaya, Malang, Indonesia Abstract. General monotone sequences class (GMS) has been introduced by Tikhonov. Then Bogdan Szal extended this class to the new class called new class of general monotone. By using non-negative sequence is called 훽 to controle the difference of sine series coefficient. By substituting 훽 of modulus of sine series coefficient, we study uniform convergence of new class of general monotone on cosine series. Keyword: New Class General Monotone, Cosine Series, Uniform Convergence Abstrak. Kelas general monotone sequences (GMS) telah diperkenalkan oleh Tikhonov. Lalu Bogdan Szal memperluas kelas ini ke kelas baru disebut new class of general monotone. Dengan menggunakan barisan non-negatif yang disebut 훽 untuk mengatur selisih dari koefisien deret sinus. Dengan mensubstitusi 훽 dari modulus koefisien deret sinus, kami pelajari konvergen seragam dari new class of general monotone pada deret kosinus. Kata Kunci: New Class General Monotone, Deret Kosinus, Konvergen Seragam Received 28 April 2019 | Revised 19 June 2019 | Accepted 27 July 2019 1. Introduction Chaundy and Jollife [6] have been discussed about the classic theorem, the necessary and sufficient condition of uniform convergence of ∞ ∑ 푎푛 sin 푛푥 (1) 푛=1 is lim 푛푎푛 = 0, if 푎 = {푎푖} is decreasing monotone and tending to zero. Tikhonov extended 푛→∞ decreasing monotone (1) to General Monotone (GM) by Tikhonov [5]. Definition 1.1. Let 푎 = {푎푖} be a complex sequence and let 훽 = {훽푖} be a non-negative real sequence, i.e. 훽푖 ≥ 0 for each i ∈ ℕ. The ordered pair (푎, 훽) is in 훽 general monotone, if there exists positive constant 퐶 and the following inequality stands for every 푛 ∈ ℕ *Corresponding author at: University of Brawijaya, Malang, Indonesia E-mail address: [email protected] Copyright ©2019 Published by Talenta Publisher, e-ISSN: 2656-1514, DOI: 10.32734/jormtt.v1i2.2834 Journal Homepage: http://talenta.usu.ac.id/jormtt Journal of Research in Mathematics Trends and Technology (JoRMTT) Vol. 01 , No. 02, 2019 35 2푛−1 ∑ |푎푘 − 푎푘+1| ≤ 퐶훽푛 푘=푛 Bogdan Szal [1] extended class of general monotone called class (훽, 푟) general monotone. Definition 1.2. If 푎 = {푎푖} complex sequence numbers and 훽 = {훽푖}, 훽푖 ≥ 0 for each i ∈ ℕ. An ordered pair (푎, 훽) is element in the class of ordered pair (훽, 푟) of general monotone (퐺푀(훽, 푟)), if we can find some positive constant 퐶 and the inequality 2푛−1 ∑ |푎푘 − 푎푘+푟| ≤ 퐶훽푛 푘=푛 holds for each 푛 ∈ ℕ and 푟 ∈ ℕ. For 푛+푟−1 [푐푛] |푎 | 훽∗ = 훽∗(푟) = ∑ |푎 | + ∑ 푘 , 푘 푘 푛 푘=푛 푘=[ ] 푐 and some 푐 > 1 and it can be obtained some result at follows ∗ ∗ Theorem 1.3. Let 푎 = {푎푖}, 푎푖 ≥ 0 for each 푖 ∈ ℕ and (푎, 훽 ) ∈ 퐺푀(훽 , 푟 ), where 푟 is natural numbers. If 푟 ≥ 1 and the sine series in point (1) is convergent uniformly to continuous function, then lim 푛푎푛 = 0. 푛→∞ ∗ ∗ Theorem 1.4. Let 푎 = {푎푖}, 푎푖 ≥ 0 for each 푖 ∈ ℕ and (푎, 훽 ) ∈ 퐺푀(훽 , 푟 ), 푟 ∈ ℕ. If 푟 = 2 and lim 푛|푎푛| = 0, then the sine series in point (1) converges uniformly. 푛→∞ Throught this paper, by using a new class of the general monotone [1] and let 훽푛 = 푎푛 ≥ 0, we discuss uniform convergence of this new class on cosine series like as [2], [3], [4]. 2. Materials and Methods This research is worked by study of literature and supporting paper from journals to get an understanding, then get results related to the research that published in the scientific journal. The results of this research are communicated in a journal. In summary the method of the research is discussing uniform convergence with coefficient belonging to the new class of the general monotone. 3. Results and Discussion Inathisbsection, we will discusscsome modification of new class generaldmonotone by taking a non-negative real sequence for controlling the sum of difference sine series coefficient. Let the series Journal of Research in Mathematics Trends and Technology (JoRMTT) Vol. 01 , No. 02, 2019 36 ∞ 푓(푥) = ∑푘=1 푎푘 cos 푘푥 (3.1) where 푎 = {푎푘} is a real sequence that tends to zero, i.e., 푎푘 → 0 as 푘 → ∞. We define sums of series f(x) where the point of the series converges. + Definition 3.1. A non-negative real sequence 푎 = {푎푖} is called element of class of 퐺푀(ℝ , 2) if we can find positive constant C, i.e. 퐶 > 0 and the relation 2푛−1 ∑ |푎푘 − 푎푘+2| ≤ 퐶풂풏 푘=푛 holds for each 푛 ∈ ℕ. + Lemma 3.2. If 풂 ∈ 퐺푀(ℝ , 2) and {푛푎푛} converges to 0, then ∞ lim 푛 ∑|푎푣 − 푎푣+2| = 0 푛→∞ 푣=푛 Proof. Let 푑푛 = sup(푣푎푣), then 푑푛 decreasing monotone and lim 푑푛 = lim 푚푎푚 = 0. For 푣≥푛 푛→∞ 푛→∞ 푎 ∈ 푀(ℝ+, 2), we have ∞ ∞ 2푠+1푛−1 푛 ∑|푎푣 − 푎푣+2| = 푛 ∑ ∑ |푎푣 − 푎푣+2| 푣=푛 푠=0 푣=2푠푛 ∞ ∞ 푎 푠 푛푎 ≤ ∑ 2푠푛 2 푛 ≤ ∑ 푛 = 2퐶′푛푎 . 2푠 2푠 푛 푠=0 푠=0 Thus ∞ lim 푛 ∑|푎푣 − 푎푣+2| = 0. ∎ 푛→∞ 푣=푛 + ∞ Theorem 3.3. Let 푎 ∈ 퐺푀(ℝ , 2) and lim 푛푎푛 = 0, if ∑푘=1 푎푘 converges then cosine series 푛→∞ (1) converges uniformly on closed interval [0, 휋]. ∞ Proof. Since ∑푘=1 푎푘 converges and {푎푘} is a null and non negative sequence, then ∞ lim ∑ 푎푘 = 0. (3.2) 푛→0 푘=푛 Let us calculate 푓(푥) − 푆푚(푓, 푥), where 푚 푆푚(푔, 푥) = ∑ 푎푗 cos 푗푥 푗=1 we get Journal of Research in Mathematics Trends and Technology (JoRMTT) Vol. 01 , No. 02, 2019 37 ∞ 푓(푥) − 푆푚−1(푓, 푥) = ∑ 푎푗 cos 푗푥 (3.3) 푘=푚 ∞ = ∑ ∆푎푘퐷푘(푥) − 푎푚퐷푚−1(푥) = 퐴 + 퐵 푘=푚 푚 ∞ where 퐷푚(푥) = ∑푗=1 cos 푗푥, 퐴 = ∑푘=푚 ∆푎푘퐷푘(푥) and 퐵 = −푎푚퐷푚−1(푥). 휋 휋 Let 푥 in interval (0, 휋], so we can find natural number 푀 satisfying 푥 ∈ ( , ]. Because of 푀 푀−1 1 푚 sin (푚 + ) 푥 2 1 휋 |∑ cos 푣푥| = | 1 | ≤ 1 ≤ , (3.4) 푣=1 2 sin 2푥 2sin 2푥 2푥 we have 휋 푀 |퐵| = |푎 퐷 (푥)| ≤ |푎 | ≤ |푎 |. 푚 푚−1 2푥 푚 2 푚 By (3.4) and 푎푖 nonnegative for each 푖 ∈ ℕ, we obtain ∞ ∞ ∞ ∞ ∞ 휋 푀 푀 |퐴| = | ∑ ∆푎 퐷 (푥)| ≤ | ∑ ∆푎 | ≤ | ∑ ∆푎 + ∑ ∆푎 | ≤ ∑ |푎 − 푎 | 푘 푘 2푥 푠 2 푠 푠 2 푠 푠+2 푘=푚 푠=푚 푠=푚 푠=푚+1 푠=푚 By proof of Lemma 3.2 we have |퐴| ≤ 푀퐶푎푚. Thus we get 푀 퐴 + 퐵 ≤ (푎 + 2퐶푎 ) (3.5) 2 푚 푚 For 푥 = 0 we have ∞ ∞ 푓(푥) = ∑ 푎푘 cos 0 = ∑ 푎푘 < ∞, 푘=1 푘=1 finally by (3.2) and (3.5), we get lim 푓(푥) − 푆푚(푓, 푥) = 0. 푛→∞ Then the series in point (3.1) converges uniformly on closed interval [0, 휋]. The proof is complete. 4. Conclusion In this paper we have concluded that, necessary and sufficient condition of uniform convergence + of cosine series with coefficient in new class of general monotone 퐺푀(ℝ , 2) is lim 푛푏푛 = 0. 푛→∞ Journal of Research in Mathematics Trends and Technology (JoRMTT) Vol. 01 , No. 02, 2019 38 Acknowledgements The author wants to say very thanks so much to the Department of Mathematics, FMIPA, University of Brawijaya that supports this article. REFERENCES [1] B. Szal, “A New Class of Numerical Sequences and its Applications to Uniform Convergence of Sine Series”, arXive: 0905.1294v1[math.CA], May 8th, 2009. [2] M. A. Imron, “Application of Approximation Theory for Sine Series with Coefficient from Class Supremum Bounded Variation Sequences of First Type”, AIP Conference Proceedings, vol. 1913 (020005), 2017. [3] M. A. Imron, “The Error Calculation of Sine Series Approximation with Coefficient from Class of General Monotone Order r”, AIP Conference Proceedings, vol. 2021 (060022), 2018. [4] M. A. Imron, “Application New class of p-Supremum Bounded Variation Double Sequences”, Journal of Engineering and Applied Sciences, vol. 13, no. 12, pp. 4279-4281, 2018. [5] S. Tikhonov, “Best approximation and moduli of Smoothness computation and Equivalence Theorems”, Journal of Approximation Theory, vol. 153, pp. 19-39, 2008. [6] T. W. Chaundy and A. E. Jollife, “The Uniform Convergence of Certain Class Trigonometric Series”, Proc. London, Soc. 15, pp. 214-116, 1916. Attribution-NonCommercial-ShareAlike 4.0 International License. Some rights reserved. .
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