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The Absolute Summability (A) of Fourier Series

Published online by Cambridge University Press:  20 January 2009

B. N. Prasad
B. N. Prasad
Affiliation:
University of Liverpool.
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In a recent paper Dr J. M. Whittaker has shown that the Fourier series

of a function f(θ) which has a Lebesgue integral in (— π, π), is absolutely summable (A) to sum l, if

exists, where

.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 2 , Issue 3 , January 1931 , pp. 129 - 134
Copyright
Copyright © Edinburgh Mathematical Society 1931

References

page 129 note 1 Proc. Edinburgh Math. Soc. (2), 2 (1930), 1–5.CrossRefGoogle Scholar

page 129 note 2 A series

has been defined to be absolutely summable (A), if

is convergent in ( 0 ≤ x < 1) and if f(x) is of bounded variation in (0, 1).

page 130 note 1 Hobson, E. W., Theory of Functions of a Real Variable, 2 (1926), 629.Google Scholar

page 130 note 2 ibid.. 1 (1927), 630.

page 131 note 1 ibid.. 593.

page 133 note 1 See Hardy, G. H., Messenger of Math., 49 (1919–1920), 150.Google Scholar

page 134 note 1 Abhand. d. Bayer. Akad. (1876), II, 37.Google ScholarSee also Hardy, G. H., Quarterly Journal, 44 (1913), 242–263.Google Scholar