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ON L1 CONVERGENCE OF MODIFIED COSINE SUM

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Published on August 10, 2007

Author: Ninu07

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ON L1-CONVERGENCE OF MODIFIED COSINE SUM:  ON L1-CONVERGENCE OF MODIFIED COSINE SUM JATINDERDEEP KAUR and S.S. BHATIA School of Mathematics and Computer Application Thapar Institute of Engg. and Technology (Deemed University), Patiala 147004 ABSTRACT:  ABSTRACT We introduce here new modified cosine sum and study its L1-convergence to a cosine trigonometric series. In this paper, we have been able to remove the necessary and sufficient condition for the L1-convergence of cosine series. INTRODUCTION:  INTRODUCTION A sequence is said to be null sequence if Sidon [1939] introduced a class S of coefficient sequences for the series to be a cosine series, satisfying Slide4:  We say that a null sequence belongs to class S if there exists a sequence such that Let the partial sums of the series (1.1) be denoted by and Slide5:  Regarding L1-convergence of (1.1), the following theorem of Teljakovskii [1973] is well known: Theorem A. If the coefficient sequence of the cosine series (1.1) belongs to the class S, then a necessary and sufficient condition for L1-convergence of (1.1) is Rees-Stanojevic [1973] introduced a new modified cosine sums and studied the necessary and sufficient condition for the L1-convergence of (1.1). Slide6:  Concerning the L1-convergence of (1.1) to a cosine trigonometric series, belonging to the class S, Ram [1977] proved the following theorem: Theorem B. If (1.1) belongs to class S. Then In the present paper, we introduce a new modified cosine sum as and study its L1-convergence under class S [1939] of coefficient sequences. Also the L1- convergence of cosine series is deduced as corollary. LEMMA:  LEMMA The following lemma will be used in the proof of the theorem: Lemma (Fomin [1964]) If then where is Dirichlet kernel and C is positive absolute constant. RESULT:  RESULT We prove the following result. THEOREM Let (1.1) belongs to the class S, then Proof. We have Slide9:  Since, and Oliver’s theorem implies that and so Also is finite in . Hence Slide10:  Now, making use of Abel’s transformation and Lemma, we get Slide11:  Under the assumed hypothesis, converges and therefore the first term in (1.6) tends to zero as Moreover, by Zygmund’s theorem ([1], Vol II, p.458). Also since, And for :  And for It follows that Slide13:  Proof. We notice that Since by our theorem and The conclusion of the corollary follows. Corollary If (1.1) belongs to the class S, then Slide14:  REFERENCES [1] N.K. Bary, A treatise on trigonometric series, Vol II, Pergamon Press, London, (1964). [2] G.A. Fomin, On linear methods for summing Fourier series, Mat. Sbornik, 66(107) (1964), 114-152. [3] B. Ram, Convergence of certain cosine sums in the metric space L, Proc. Amer. Math. Soc., 66(2) (1977), 258-260. [4] C.S. Rees and ˇC.V. Stanojevi´c, Necessary and sufficient condition for integrability of certain cosine sums, J. Math. Anal. Appl., 43 (1973), 579-586. [5] S. Sidon, Hinreichende Bedingungen f¨ur den Fourier-Charakter einer trigonometrischen Reihe, J. London Math Soc., 14 (1939), 158-160. [6] S.A. Teljakovski˘ı, A certain sufficient condition of Sidon for the integrability of trigonometric series, Math. Zametki, 14(3) (1973), 317-328. Slide15: 

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