tailieunhanh - Báo cáo hóa học: " A study on degree of approximation by Karamata summability method"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A study on degree of approximation by Karamata summability method | Nigam and Sharma Journal of Inequalities and Applications 2011 2011 85 http content 2011 1 85 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access A study on degree of approximation by Karamata summability method Hare Krishna Nigam and Kusum Sharma Correspondence kusum31sharma@ Department of Mathematics Faculty of Engineering and Technology Mody Institute of Technology and Science Deemed University Laxmangarh-332311 Sikar Rajasthan India Abstract Vuckovic Maths. Zeitchr. 89 192 1965 and Kathal Riv. Math. Univ. Parma Italy 10 3338 1969 have studied summability of Fourier series by Karamata K1 summability method. In present paper for the first time we study the degree of approximation of function f e Lip a r and f e W Lr t by -summability means of its Fourier series and conjugate of function f e Lip a r and f e w Lr t by Kx-summability means of its conjugate Fourier series and establish four quite new theorems. MsC primary 42B05 42B08 42A42 42A30 42A50. Keywords degree of approximation Lip a r class W Lr Ẹ f class of functions Fourier series conjugate Fourier series KẦ-summability Lebesgue integral 1 Introduction The method K was first introduced by Karamata 1 and Lotosky 2 reintroduced the special case l 1. Only after the study of Agnew 3 an intensive study of these and similar cases took place. Vuckovic 4 applied this method for summability of Fourier series. Kathal 5 extended the result of Vuckovic 4 . Working in the same direction Ojha 6 Tripathi and Lal 7 have studied K-summability of Fourier series under different conditions. The degree of approximation of a function f e Lip a by Cesàro and Norlund means of the Fourier series has been studied by Alexits 8 Sahney and Goel 9 Chandra 10 Qureshi 11 Qureshi and Neha 12 Rhoades 13 etc. But nothing seems to have been done so far in the direction of present work. Therefore in present paper we establish two new theorems on degree

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