Abstract
The existence of sine and cosine series as a Fourier series, their \(L^1\)-convergence seems to be one of the prominent question in theory of convergence of trigonometric series in \(L^1\)-metric norm. In the literature, till now most of the authors have studied the \(L^1\)-convergence of cosine trigonometric series. However, very few of them have studied the \(L^1\)-convergence of trigonometric sine series. In this paper, new modified cosine and sine sums of Fourier series are introduced and a criterion for the summability and \(L^1\)-convergence of these modified sums is obtained. Also, necessary and sufficient condition for the \(L^1\)-convergence of cosine and sine series is deduced as corollaries. Further an application is given to illustrate the main result.
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This work is the part of UGC sponsored Major Research Project entitled “On \(L^1\)-convergence of trigonometric series with special coefficients”.
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Chouhan, S.K., Kaur, J. & Bhatia, S.S. Convergence and Summability of Fourier Sine and Cosine Series with Its Applications. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 89, 141–148 (2019). https://doi.org/10.1007/s40010-017-0471-5
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DOI: https://doi.org/10.1007/s40010-017-0471-5