Source: Real Anal. Exchange
Volume 27, Number 1
In this note we show that a theorem due to L.I. Holder on the absolute
summability of Fourier series by the Bosanquet-Linfoot method $|\alpha,\beta|,$
can be improved upon. Our theorem then provides a better refinement to the
classical theorem of Bosanquet on summability $|C,\gamma|$ of Fourier series.
B. J. Boyer and L. I. Holder, A generalization of absolute Rieszian summability, Proc. Amer. Math. Soc., 14 (1963), 459–464.
Mathematical Reviews (MathSciNet): MR149155
P. Chandra and V. Karanjgaokar, An aspect of local property of the Fourier series, (forthcoming: Indian J. Math.)
G. D. Dikshit, Absolute Nevanlinna summability and Fourier series, J. Math. Anal. Appl., 248 (2000), 482–508.
A. F. Moursund, On a method of summation of Fourier series, Ann. of Math., 33(2) (1932), 773–784.
A. F. Moursund, On a method of summation of Fourier series, Ann. of Math., 34(2) (1933), 778–798.
F. Nevanlinna, Über die Summation der Fourier'schen Reihen und Integrale, Översikt Finska Vetenskapps-Societetens Förhandlinger A, 64(3) (1921–1922).
B. Patra, On the absolute zero order summability of a Fourier series and its allied series, Indian J. Pure and Appl. Math., 13 (1982), 785–794.
Mathematical Reviews (MathSciNet): MR666954
B. Patra, On the absolute $(\alpha,\beta)$ summability of some series associated with a Fourier series and its allied series, Indian J. Pure and Appl. Math., 21 (1990), 530–543.
B. K. Ray and M. Samal, Application of the absolute $N_q$-method to some series and integrals, J. Indian Math. Soc., 44 (1980), 217–236.
Mathematical Reviews (MathSciNet): MR752659
M. Samal, On the absolute $N_q$-summability of some series associated with Fourier series, J. Indian Math. Soc., 50 (1986), 191–209.
Mathematical Reviews (MathSciNet): MR989026