On Uniform Convergence of Sine and Cosine Series. under Generalized Difference Sequence of. p-supremum Bounded Variation Sequences

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1 International Journal of Matheatical Analysis Vol. 10, 2016, no. 6, HIKARI Ltd, On Unifor Convergence of Sine and Cosine Series under Generalized Difference Sequence of p-supreu Bounded Variation Sequences Moch. Aruan Iron Departent of Matheatics, Faculty of Matheatics and Natural Sciences University of Brawijaya, Malang, Indonesia Copyright 2015 Moch. Aruan Iron. This article is distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract Let sine series defined on [0,2π]. It is shown that if (a, β) in p-supreu bounded variation sequences and n 1 1 p 2 sup b(n) k= β k = o(1) for 1 p <, with a = {a n } is sequence of coefficients of sine series, {b(n)} [0, ) tending onotonically to infinity depending only on {a n }, and β = {β n } is sequence of real non-negatif, then the sine series converges uniforly on [0, 2π]. We weaken this condition to so called generalized difference sequence of p-supreu bounded variation sequences and study the properties that class. It will be shown that unifor convergence of sine and cosine series under that class is fulfilled. Matheatics Subject Classification: 40A05, 42A16, 46A45 Keywords: generalized difference sequence, p-supreu bounded variation, sine and cosine series 1. Introduction It is well known that there are several interesting classical theores in Fourier analysis having assuptions deterined by certain onotonicity of the coefficients.

2 246 Moch. Aruan Iron The following classical convergence result can be found in Zygund [1], Chaundy and Jollife [10]. Theore 1.1. Suppose that {a k } [0,) is nonincreasingly tending to zero. A necessary and sufficient conditions for the unifor convergence of the series k=1 a k sin kx (1.1) is li k a k = 0. k Tikhonov [8] introduced a class of General Motonotone Sequences (GMS) as follows: A sequence a = {a n }, a n C, is said to be GMS if there exists C > 0 such that the relation 2n 1 k=n a k a k+1 C a n holds for n N. He used a class of GMS to weaken the onotonicity condition of coefficient (1.1). The class of GMS was further generalized by Zhou et. al [9] to class of Mean Value Bounded Variation Sequences (MVBVS). A sequence a = {a n }, a n C, is said to be MVBVS if there exist C > 0 and λ 2 such that [λn] k=[n/λ] 2n 1 k=n a k a k+1 C a n k holds for n N, where [x] the greatest integer that a less then or equal to x. He proved that Theore 1.2 also valid when the condition a GMS is replaced by a MVBVS. Further, Liflyand and Tikhonov [2] defined the class of p-general Monotone Sequences (GMS p ). Let a = {a n } and β = {β n } be two sequences of coplex and non-negative nubers, respectively, a couple (a, β) GMS p if there exist C > 0 such that the relation ( 2n 1 k=n a k a k+1 p ) 1/p Cβ n holds for n N and 1 p <. Then Dyachenko and Tikhonov [6] proved the following Theore: Theore 1.2. If (a, β) GMS 1 and li a k = 0, then necessary condition k for the unifor convergence of the series (1.1) is li k β k = 0. k For β n = a n, (a, β) GMS 1 if and only if a GMS (see Liflyand and Tikhonov [2]). As corollary, GMS p ore general than GMS. In 2010 Korus [7] introduces class of SBVS (supreu bounded variation sequences) and SBVS2 (supreu bounded variation sequence of second type). A sequence a = {a n }, a n C is said to be SBVS if there exists C > 0 and γ 1 such that the relation 2n 1 k=n a k a k+1 C sup 2 a n k= k [n/γ] holds for n N.

3 On unifor convergence of sine and cosine series 247 The sequence a = {a n }, a n C is said to be SBVS2 if there exists C > 0 and {b(k)} [0, ) tending onotonically to infinity, such that 2n 1 k=n a k a k+1 C sup 2 a n k= k, n 1. b(n) Furtherore, Iron, et. Al [4], generalized MVBVS and SBVS to MVBVS p (p- Mean Value Bounded Variation Sequences) and SBVS p (Supreu Bounded variation Sequences), respectively. Let a = {a n } and β = {β n } be two sequences of coplex and non-negative nubers, respectively, a couple (a, β) MVBVS p if there exist C > 0 and λ 2 such that [λn] k=[n/λ], ( 2n 1 k=n a k a k+1 p ) 1/p C β k for p, 1 p < and (a, β) SBVS p if there exist C > 0 and γ 1 such that ( 2n 1 k=n a k a k+1 p ) 1/p C ( sup n [n/γ] 2 k= β k ), for p, 1 p <. A little odification of definition of SBVS p is class SBVS2 p (Iron et. al. [4]). The couple (a, β) is said to be p-supreu Bounded Variation Sequences Second type, written (a, β) SBVS2 p if there exist C > 0 and {b(k)} [0, ) tending onotonically to infinity depending only on {a k }, such that ( 2n 1 k=n a k a k+1 p ) 1/p C ( sup 2 β n k= k ), b(n) for p, 1 p <. Iron, et al. [3] have shown that GMS p MVBVS p SBVS p SBVS2 p and they used a class of SBVS2 p to weaken the onotonicity condition of coefficient (1.1) as stated in Theore 1.3. Theore 1.3. If (a, β) SBVS2 p and n 1 1 p 2 sup b(n) k= β k = o(1) for p, 1 p <, then the series (1.1) is unifor convergence. Furtherore Iron, et. al. [5] have intoduced the class of generalized difference sequence SBVS2 p ( n ) and proved that class of SBVS2 p (β, n ) is a separable Banach space. The definition of that class as follows: Definition 1.4. A couple (a, β) is said to be p-supreu Bounded Variation Sequences of second type order n, written (a, β) SBVS2 p ( n ), if there exist C > 0 and {b k } [0, ) tending onotonically to infinity depending only on {a k }, such that for ( 2 1 k= n a k p ) 1/p C ( sup 1 p < and n = 1,2,3,. k=i β k ), 1

4 248 Moch. Aruan Iron Theore 1.5. Let n N be given, then SBVS p ( n ) SBVS2 p ( n ) for p, 1 p <. Theore 1.6. Let n N be given, then SBVS2 p ( n ) SBVS2 p ( n+1 ) for p, 1 p <. In the present paper, we investigate the unifor convergence of sine and cosine series under generalized difference p-supreu bounded variation condition. 2. Soe Properties Class of Generalized Difference In this section, we study soe properties of class SBVS2 p ( n ) to investigate the unifor convergence of sine and cosine series under condition of this class. Theore 2.1. Let n N be given. If (a, β) SBVS2 p ( n ) and { 1 1/p sup k=i β k holds for p, 1 p <. Proof. We denote } is decreasing onotone, then v= n a v C ( 1 1/p sup α = 1 1/p sup k=i β k k=i β k ) for every N. Given (a, β) SBVS2 p ( n ) then we write 2 s+1 1 v=2 s n a v = n a v v= s=0 s=0 [( n a v p)1/p ((2 s+1 1) (2 s 1)) 1 1/p ] 2s+1 n 1 v=2 s s=0 α 2 s 2 s s=0 2 s (2 s ) 1 1/p ( n a v p C = C ( 1 1 p sup C k=i β k 2s+1 1 v=2 s s=0 α 2 s ), C = 2C. Lea 2.2. Let n N be given. If (a, β) SBVS2 p ( n ) and 1 1/p sup k=i β k = o(1), for, then v= n a v = o(1) for. Proof. Let α = 1 1/p sup k=i β k and we denote d = sup(vα v ), v then d decreasing onotone and d = o(1) for. By Theore 2.1. ) 1/p

5 On unifor convergence of sine and cosine series 249 v= n a v Cd = o(1).. Corollary 2.3. Let n N be given, then SBVS2 p SBVS2 p ( n ), for 1 p <. GMS p MVBVS p SBVS p Proof. By Theore 2.3 in Iron et. al. [3] we have GMS p MVBVS p SBVS p SBVS2 p and by Theore 1.5 and 1.6 we have SBVS2 p SBVS2 p ( n ). 3. Unifor Convergence of Sine and Cosine Series In this section, we investigate the unifor convergence of sine and cosine series under condition of class of SBVS2 p ( n ). We consider the series and k=1 a k cos kx (3.1) k=1 a k sin kx (3.2) where a = {a k } is a given null sequence of coplex nubers, i.e., a k 0 as k. We define by f(x) and g(x) the sus of series (3.1) and (3.2) respectively at the point where the series converges. Theore 3.1. Let β and n be given. Suppose that (a, β) SBVS2 p ( n ) with 1 1/p sup k=i β k = o(1), for and (i) for n = 1 the series (3.1) converges uniforly on [0, π], if and only if k=1 a k converges.. (ii) k= a k a k n 1 a k+1 = o(1), for n 2, the series (3.1) converges uniforly on [0, π], if and only if k=1 a k converges, for p, 1 p <. Proof. (i) The part if is obvious. We will show the part if only. Since k=1 a k converges and {a k } is a null sequence, we can construct {v n } be nonincreasing null sequence such that j=n a j v n. For every ε > 0, there exists n 0 N such that v, so there exists n 0 N such that j= a j v < ε (3.3) for n 0. Let us now estiate g(x) S (g, x), where By Abel s transforation, we get S (g, x) = j=1 a j cos jx.

6 250 Moch. Aruan Iron g(x) S 1 (g, x) = k= a j cos jx (3.4) = a k D k (x) a D 1 (x) = A + B k= where D (x) = j=1 cos jx, A = k= a k D k (x) and B = a D 1 (x). B = a D 1 (x) a v= a By Lea 2.2, we have B v= a = o(1). (3.5) To estiate A, then we decopose A as k=+n, A = k= a k D (x) + a k D +N (x) = P + Q k=+n. where P = k= a k D (x) and Q = a k D +N (x) For any x (0, π] we can find N N such that x ( π then D (x) = v=1 cos vx = sin(+1 2 )x sin1 2 x 2 sin x v=1 cos vx π. Therefore x sin 1 2 x π x k=+n. Q = k=+n a k D +N (x) π a x k N+1, π N ]. Since By Lea 2.2, we have If > N, then By Lea 2.2, we have If N, then Q N+1 o(1) o(1) (3.6) k=. P = k= a k D k (x) π a x k P N + 1 o(1) o(1) (3.7) N 1 P = k= a k D (x) k= k a k + k= a k D (x) k = L + M, where L = k= k a k and M = N 1 a k D (x) k k=.

7 On unifor convergence of sine and cosine series 251 Since L = k= k a k = k= a k + ( 1)a ( + N 1)a N +N k=+1 a k + N k=n a k + k= a k v +1 + N k=n a k + a k By Lea 2.2. we have Fro D k (x) k k 2 x, we obtain M x k= k 2 a k k=. L = k= k a k v o(1). (3.8) = x( 2 a + ( + 1) 2 a ( + N 1) 2 a ) = x( 2 a + ( ) a ( + N 1) 2 a ) j= j= ] j=+1 ] π N [2 a j + a j + π N [( + 1) a j + + ( + N 1) a = π N [2 j= a j + k= k a j j=k ]. By Lea 2.2. we have M π o(1) + π o(1) N N k= 2 π o(1) N Fro, (3.6), (3.7), (3.8) and (3.9) if > N, we and if N, we get Fro (3.4) and (3.10) 2πo(1) (3.9) A = P + Q 2 o(1) get A = L + M + Q v o(1) + 2π. o(1). (3.10) f(x) S (f, x) = A + B v +1 + (3 + 2π) o(1) Thus, if given ε > 0 by (3.3) there exists n 0 such that for n 0 f(x) S (f, x) A + B ε + (3 n + 2πn) o(1) Then series (3.1) converges uniforly on [0, π]. (ii) Since fro proof (i) then a s = a s a s n 1 a s+1 + n a s (3.11)

8 252 Moch. Aruan Iron B = a D 1 (x) a v= a = v= a v a v n 1 a v+1 + n a v. v= a v a v n 1 a v+1 + n a v v=. By Lea 2.2. and condition Theore 3.1. (ii), we have B 2o(1) (3.12) Fro proof (i) and (3.11) we have Q π a x k k=+n k=+n k=+n + π x k=+n n a k. = π x a k+1+ 2 a s n 1 a k+1 + n a k π x a k+1+ 2 a s n 1 a k+1 By Lea 2.2. and condition Theore 3.1. (ii), we have If Q N + 1 o(1) o(1) (3.13) + N > N, then k= P = k= a k D k (x) π a x k π a x s= s+1+ 2 a s n 1 a s+1 + n a s + π x n a s s=. By Lea 2.2. and condition Theore 3.1.(ii) P N + 1 o(1) + o(1) 2o(1) (3.14) If N, fro proof (i) then P = L + M, where L = k= k a k and M = N 1 a k D (x) k k=. By replacing a k with (3.11), condition Theore 3.1.(ii) and Lea 2.2. we have L = k= k a k v o(1). (3.15) Fro D k (x) k k 2 x, we obtain M x k= k 2 a k = π N [2 j= a j + k= k a j j=k ]. By replacing a k with (3.11), condition theore 3.1.(ii) and Lea 2.2. we have M 2π o(1) + 2π o(1) 4π N N k= o(1) N 4πo(1). (3.16)

9 On unifor convergence of sine and cosine series 253 Fro (3.13), (3.14), (3.15) and (3.16) if > N, we get and if N, we get A = P + Q 2 o(1) A = L + M + Q v o(1) + 4π. o(1). (3.17) Fro (3.4) and (3.17), we have f(x) S (f, x) = A + B v +1 + (3 + 4π) o(1) Thus, if given ε > 0 by (3.3) there exists n 0 such that for n 0 f(x) S (f, x) A + B ε + (3 n + 2πn)o(1), then the series (3.1) converges uniforly on [0, π]. The proof is coplete. Theore 3.2. Let β and n be given. Suppose that (a, β) SBVS2 p ( n ) with 1 1/p sup k=i β k = o(1), for and (i) for n = 1, then the series (3.2) converges uniforly on [0, π]. (ii) k= a k a k n 1 a k+1 series (3.2) converges uniforly on [0, π]. for p, 1 p <. = o(1), for n 2, then the Proof. (i) Let (a, β) SBVS2 p ( n ) for fix n N. Let us now estiate g(x) S (g, x), where S (g, x) = j=1 a j sin jx. Now we calculate g(x) S 1 (g, x) = a k sin kx k=. To estiate k= a k sin kx, for any x (0, π] we can find N N such that x ( π, π ]. Since sin vx N+1 N v=1 = cos1 2 x cos(+1 2 )x 2 sin x sin 1 π, 2 x x if N, then D (x) = sin vx and we decopose where The first v=1 π x k= a k sin kx = A + B A = k= a k sin kx, B = k=+n a k sin kx part tends to 0, since k= a k sin kx x ka k k= k= s=k = x k a s = x k= k s=k a s. (3.18)

10 254 Moch. Aruan Iron By Lea 2.2, we have A xn o(1) πo(1). The second part, by Abel s transforation, we get B = a k D +N (x) k=+n a +N D (x) = S + T (3.19) where D (x) = j=1 sin jx and S = S π x a +N and T π x a +N. Therefore a k D k=+n k (x), T = a +N D (x), B 2π x a +N 2(N + 1) a +N 2( + N) s=+n a s 2( + N) s=+n a s. (3.20) By Lea 2.2, we have B 2 o(1).thus Therefore, A + B (2 + π) o(1), x (0, π]. g(x) S 1 (g, x) A + B (2 + π)ε. At x = 0, S (g, 0) = 0, thus S (g, x) converges uniforly on [0, π]. (ii) For n 2 and fro (3.18), we have A = x k a s k= s=k and fro (3.11) we have A = x k a s a s n 1 a s+1 + n a s k= k= s=k s=k ) x k( a s a s n 1 a s+1 + n a s xn o(1) + (C 1 1/p sup j=i β j ) πo(1) + C o(1). Fro (3.19) we have B = a k D +N (x) a +N D (x) = S + T k=+n and fro (3.20) we have B 2( + N) s=+n a s By (3.11) we obtain B 2( + N) a s a s n 1 a s+1 + n a s s=+n 2( + N) s=+n a s a s n 1 a s+1 +2( + N) n a s s=+n

11 On unifor convergence of sine and cosine series 255 By condition Theore 3.2. (i) and Lea 2.2, we have B 2 o(1) + 2 o(1). Therefore A + B (4 + C + π) o(1), x (0, π] and g(x) S 1 (g, x) A + B (4 + C + π)ε. At x = 0, S (g, 0) = 0, thus S (g, x) converges uniforly on [0, π]. 4. Conclusions In this paper we have investigated that (i) Let n N be given. If (a, β) SBVS2 p ( n ), 1 1/p sup k=i β k = o(1), for and (a) for n = 1 then series (3.1) converges uniforly on [0, π], for 1 p <, if and only if k=1 a k converges. (b) s= a s a s n 1 a s+1 = o(1), for n 2,, then series (3.1) converges uniforly on [0, π], for 1 p <, if and only if k=1 a k converges. (ii) Let n N be given. If (a, β) SBVS2 p ( n ), 1 1/p sup k=i β k = o(1), for and (a) for n = 1 then series (3.2) converges uniforly on [0, π], for 1 p <. (b) s= a s a s n 1 a s+1 = o(1), for n 2,, then series (3.2) converges uniforly on [0, π], for 1 p <. Acknowledgents. The authors gratefully acknowledge the support of the Departent of Matheatics, Faculty of Matheatics and Natural Sciences University of Brawijaya. References [1] A. Zygund, Trigonoetric Series, Vol. I, II, Second ed., Cabridge Univ. Press, [2] E. Liflyand and S. Tikhonov, A concept of general onotonicity and applications, Math. Nachr., 284 (2011), no. 8-9,

12 256 Moch. Aruan Iron [3] M.A. Iron, C.R. Indrati and Widodo, Unifor Convergence of Trigonoetric Series Under p-supreu Bounded Variation Condition, Proceeding of 3rd Annual Basic International Conference, FMIPA. UB, Malang, (2013), M041- M043. [4] M.A. Iron, C.R. Indrati and Widodo, Soe Properties of Class of p-supreu Bounded Variation Sequences, Int. Journal of Matheatical Analysis, 7 (2013), no. 35, [5] M.A. Iron, C.R. Indrati and Widodo, On Generalized difference sequence space over class of p-supreu Bounded Variation Sequences, Int. J. Appl. Math. Stat., 52 (2014), no. 1, [6] M. Dyachenko and S. Tikhonov, General onotone sequences and convergence of trigonoetric series, Chapter in: Topics in Classical Analysis and Applications in Honor of Daniel Wateran, World Scientific, Hakensack, New Jersey, 2008, [7] P. Korus, Reark on the unifor and L 1 -convergence of trigonoetric series, Acta Math. Hungar., 128 (2010), no. 4, [8] S. Tikhonov, Best approxiation and oduli of soothness: coputation and equivalence theores, Journal of Approx. Theory, 153 (2008), [9] S.P. Zhou, P. Zhou and D.S. Yu, Ultiate generalization to onotonicity for Unifor Convergence of Trigonoetric Series, Science China Matheatics, 53 (2010), [10] T.W. Chaundy and A.E. Jollife, The Unifor Convergence of certain class trigonoetrical series, Proc. London Math. Soc., s2-15 (1917), Received: Noveber 1, 2015; Published: February 9, 2016