On $L^{1}$ convergence of certain cosine sums
HTML articles powered by AMS MathViewer
- by John W. Garrett and Časlav V. Stanojević PDF
- Proc. Amer. Math. Soc. 54 (1976), 101-105 Request permission
Abstract:
Rees and Stanojević introduced a new class of modified cosine sums $\{ {g_n}(x) = \tfrac {1} {2}\sum \nolimits _{k = 0}^n {\Delta a(k) + \sum \nolimits _{k = 1}^n {\sum \nolimits _{j = k}^n {\Delta a(j)\cos kx\} } } }$ and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series $f$ with coefficients of bounded variation, a Rees-Stanojević cosine sum ${g_n}$ can be associated such that ${g_n}$ converges to $f$ pointwise, and a necessary and sufficient condition for ${L^1}$ convergence of ${g_n}$ to $f$ is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions.References
- N. K. Bari, Trigonometricheskie ryady, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). With the editorial collaboration of P. L. Ul′janov. MR 0126115
- Charles S. Rees and Caslav V. Stanojevic, Necessary and sufficient conditions for integrability of certain cosine sums, J. Math. Anal. Appl. 43 (1973), 579–586. MR 322432, DOI 10.1016/0022-247X(73)90278-3
- Č. V. Stanojević, On integrability of certain trigonometrical series, Srpska Akad. Nauka. Zb. Rad. 55, Mat. Inst. 6 (1957), 53–57 (Serbo-Croatian, with English summary). MR 0093680
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 101-105
- DOI: https://doi.org/10.1090/S0002-9939-1976-0394002-8
- MathSciNet review: 0394002