A Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients

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1 It. J. Ope Problems Comput. Sci. Math., Vol., No., Jue 009 A Note O L 1 -Covergece of the Sie ad Cosie Trigoometric Series with Semi-Covex Coefficiets Xhevat Z. Krasiqi Faculty of Educatio, Uiversity of Prishtia Rr. Agim Ramadai, Prishtië 10000, Republic of Kosova address: xheki00@hotmail.com Abstract We itroduce ew modified cosie ad sie sums ad study their L 1 -covergece to the sie ad cosie trigoometric series respectively ad deduce a result of Bala ad Ram [1] as a corollary. Keywords: L 1 -covergece, ew modified sie ad cosie sums, semicovex sequeces. 000 Mathematics Subject Classificatio: 4A0, 4A3. 1 Itroductio ad Prelimiaries Let ad a 0 + a k cos kx 1) a k si kx ) be cosie ad sie trigoometric series respectively with their partial sums deoted by Sx) c = a 0 + a k cos kx, Sx) s = a k si kx ad let fx) = lim Sx), c gx) = lim Sx). s For coveiece, i the followig of this paper we shall assume that a 1 = a 0 = 0.

2 3 Xhevat Z. Krasiqi A sequece a ) is said to be semi-covex if a 0, ad a 1 + a <, 3) =1 where a k = a k ) = a k a k+1 + a k+. R. Bala ad B. Ram [1] have proved that for series 1) with semi-covex ull coefficiets the followig theorem holds true. Theorem A. If a ) is a semi-covex ull sequece, the for the covergece of the cosie series 1) i the metric L 1, it is ecessary ad sufficiet that a 1 log = o1),. Garret ad Staojević [] have itroduced modified cosie sums f x) = 1 a k + j=k a j cos kx. Garret ad Staojević [3], Ram [4], Sigh ad Sharma [5], ad Kaur ad Bhatia [6], [7] studied the L 1 -covergece of this cosie sum uder differet sets of coditios o the coefficiets a. Kumari ad Ram [8] itroduced ew modified cosie ad sie sums as h x) = a 0 + g x) = j=k j=k aj j ) k cos kx ) aj k si kx j ad have studied their L 1 -covergece uder the coditio that the coefficiets a belog to differet classes of sequeces. Likewise, they deduced some results about L 1 -covergece of cosie ad sie series as corollaries. N. Hooda, B. Ram ad S. S. Bhatia [9] itroduced ew modified cosie sums as ) ) r x) = 1 a 1 + a k + a k+1 + a j cos kx ad studied the L 1 -covergece of these cosie sums. Later o, K. Kaur [10] itroduced ew modified sie sums as K x) = 1 j=k j=k a j 1 a j+1 ) si kx,

3 A Note O L 1 -Covergece of the Sie ad Cosie 33 ad studied the L 1 -covergece of this modified sie sum with semi-covex coefficiets provig the followig theorem. Theorem B. Let a ) be a semi-covex ull sequece, the K x) coverges to fx) i L 1 -orm. We itroduce here ew modified cosie ad sie sums as H x) j=k [a j 1 a j+1 ) cos jx] G x) = 1 j=k [a j 1 a j+1 ) si jx]. The mai goal of the preset work is to study the L 1 -covergece of these ew modified sie ad cosie sums with semi-covex coefficiets ad to deduce Theorem A as a corollary. As usually with D x) ad D x) we shall deote the Dirichlet ad its cojugate kerels respectively, defied by D x) = 1 + cos kx, D x) = si kx. Everywhere i this paper the costats i the O-expressios deote positive costats ad they may be differet i differet relatios. Mai Results Regardig the series ) we prove the followig result: Theorem 1. Let a ) be a semi-covex ull sequece, the H x) coverges to gx) i L 1 -orm.

4 34 Xhevat Z. Krasiqi Proof. We have H x) j=k [a j 1 a j+1 ) cos jx] [a k 1 a k+1 ) cos kx a a + ) cos + 1)x] a k 1 a k+1 ) cos kx + + 1) a a + ) a k 1 + a k ) cos kx + + 1) a a + ) cos + 1)x cos + 1)x. 4) Applyig Abel s trasformatio i 4) see [11], p. 17), we have { H x) 1 ) a k 1 + a k D k x) + 1 ) + a 1 + a ) D x) + 1 ) } { + + 1) a a + ) a k 1 + a k ) Dk x) cos + 1)x a 1 a +1 ) D x) } a k 1 a k+1 ) + a 1 + a cos + 1)x + + 1) a a + ) ) a k 1 + a k Dk x) + + 1) a a + ) cos + 1)x. 5)

5 A Note O L 1 -Covergece of the Sie ad Cosie 35 O the other side we have S s x) = 1 si x a k si kx si x a k [cosk + 1)x cosk 1)x] a k 1 a k+1 ) cos kx cos x a +1 a cos + 1)x a k 1 + a k ) cos kx cos x a +1 a cos + 1)x. 6) Applyig Abel s trasformatio to the equality 6) we get { Sx) s 1 ) a k 1 + a k D k x) + 1 ) + a 1 + a ) D x) + 1 ) } cos x a +1 a cos + 1)x { ) a k 1 + a k Dk x) a k 1 a k+1 ) } + a + a +1 ) D x) + a 1 + a cos x a +1 a cos + 1)x ) a k 1 + a k Dk x) a a + ) D x) a cos x +1 a cos + 1)x. 7)

6 36 Xhevat Z. Krasiqi Sice a ) is semi-covex sequece, the from 3) we have + 1) a a + ) = + 1) a k a k+ ) k= = + 1) a k 1 a k+1 ) k=+1 k a k 1 + a k = o1),. 8) k=+1 Usig 8) ad whe we pass o limit as to 5) ad 7) we get gx) = lim S s x) = lim H x) ) a k 1 + a k Dk x). 9) Applyig well-kow iequality D k x) 1/+k, k = 1,,..., ad relatios 5), 8) ad 9) we have π π gx) H x) dx = 1 ) = a k 1 + a k Dk x) k=+1 cos + 1)x + + 1) a a + ) dx ) = O k a k 1 + a k k=+1 +O + 1) a a + ) ) = o1),, which fully proves the Theorem 1. Corollary 1. Let a ) be a semi-covex ull sequece, the the ecessary ad sufficiet coditio for L 1 -covergece of the series ) is a log = o1),. Proof. Let S s x) gx) = o1),. We would show that a log =

7 A Note O L 1 -Covergece of the Sie ad Cosie 37 o1),. Ideed, we have gx) H x) + S s x) gx) + + 1) a cos + 1)x a + ) + a a + ) D x) a cos x +1 + a cos + 1)x a +1 cos x cos + 1)x + π dx = a +1 D x) 1 π ctg x dx = a +1 D x) dx ctg x ) dx π 0 = O a +1 log ), 10) see [11], p. 116). Sice gx) H x) = o1) by Theorem 1, S s x) gx) = o1) by assumptio of Corollary, third ad fourth term at the left side of relatio 10) ted to 0 by 8), cosequetly a log = o1),. Coversely, let a log = o1),. The by 8) S s x) gx) H x) gx) + H x) S s x) = H x) gx) cos + 1)x + + 1) a a + ) + a a + ) D x) + a cos x +1 + a cos + 1)x = o1) + O + 1) a a + ) + O a a + ) + a cos x +1 + a cos + 1)x = o1) + a cos x +1 + a cos + 1)x. 11) But a cos x +1 π + a cos + 1)x π dx a cos x + cos + 1)x π dx = a 1 ctg x D x) dx = O a log ) = o1), π therefore from 11) we get S s x) gx) = o1),, which implies the coclusio of the Corollary.

8 38 Xhevat Z. Krasiqi Regardig the series 1) the followig statemets are true: Theorem. Let a ) be a semi-covex ull sequece, the G x) coverges to fx) i L 1 -orm. Corollary. Let a ) be a semi-covex ull sequece, the the ecessary ad sufficiet coditio for L 1 -covergece of the series 1) is a log = o1),. The proofs of Theorem ad Corollary are similar to the proofs of Theorem 1 ad Corollary, therefore we omit them. Remark 1. Kaur K. [10] itroduced the modified sie sums K x) ad studied L 1 -covergece of cosie series 1). Here we shall itroduce the modified cosie sums F x), similar with K x), as F x) j=k a j 1 a j+1 ) cos kx, where a 1 = a 0 = 0. By a similar argumet, we ca show that the Theorem B also holds for the series ), whe istead of K x) we use F x). More precisely, the followig statemet holds. Theorem 3. Let a ) be a semi-covex ull sequece, the F x) coverges to gx) i L 1 -orm. Remark. The Corollary 1 is a cosequece of Theorem 3, as well. 3 Ope Problem I [6] Kaur K. ad Bhatia S. S. have defied a ew class of sequeces, so-colled geeralized semi-covex sequeces, as follows: A ull sequece a ) is said to be geeralized semi-covex if α α+1 a 1 + α+1 a <, for α > 0, a 0 = 0). =1 Sice for α = 1 a geeralized semi-covex sequece reduces to a semicovex sequece, the the followig problem arises :

9 A Note O L 1 -Covergece of the Sie ad Cosie 39 If we use a geeralized semi-covex sequece istead of a semi-covex sequece, how to defie the ew modified cosie ad sie sums H x) ad G x) so that our results i this paper ca be as corollaries of ew results? Refereces [1] Bala R. ad Ram B., Trigoometric series with semi-covex coefficiets, Tamkag J. Math. Vol. 18, No. 1, 1987) pp [] Garret J. W. ad Staojević Č. V., O itegrability ad L1 -covergece of certai cosie sums, Notices, Amer. Math. Soc. Vol., 1975) A-166. [3] Garret J. W. ad Staojević Č. V., O L1 -covergece of certai cosie sums, Proc. Amer. Math. Soc. No. 54, 1976) pp [4] Ram B., Covergece of certai cosie sums i the metric space L, Proc. Amer. Math. Soc. No. 66, 1977) pp [5] Sigh N. ad Sharma K. M., Covergece of certai cosie sums i the metric space L, Proc. Amer. Math. Soc. No. 75, 1978) pp [6] Kaur K. ad Bhatia S. S., Itegrability ad L 1 -covergece of Ress- Staojević sums with geeralized semi-covex coefficiets, IJMMS 30:11 00) pp [7] Kaur J. ad Bhatia S. S., Itegrability ad L 1 -covergece of Certai Cosie Sums, KYUNGP. Math. J. No. 47, 007), pp [8] Kumari S. ad Ram, B. L 1 -covergece of certai trigoometric sums, Idia J. Pure Appl. Math. Vol. 0, No. 9, 1989) pp [9] Hooda N., Ram B. ad Bhatia S. S., O L 1 -covergece of a modified cosie sum, Soochow J. of Math. Vol. 8, No. 3, July 00), pp [10] Kaur K., O L 1 -covergece of a modified sie sums, A electroic J. of Geogr. ad Math. Vol. 14, Issue 1, 003). [11] Bary K. N., Trigoometric series, Moscow, 1961)i Russia).