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III.—On Fourier Constants

Published online by Cambridge University Press:  15 September 2014

E. T. Copson
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Extract

§ 1. Notation.—Suppose that a0, a1a2,… are real constants, such that, for some fixed p (≥1), the series is convergent. It is a consequence of the generalised Riesz-Fischer Theorem that there exists a function f(t) L1+p over ( – 1 , 1), of which

is the Fourier Series.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 48 , 1929 , pp. 15 - 19
Copyright
Copyright © Royal Society of Edinburgh 1929

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References

page 15 note * Hobson, Functions of a Real Variable (2nd ed.), 2, 599, Theorem II.

page 15 note † That an(l) and bn(l) exist follows from the use of Hölder's inequality for integrals, and the fact that f(t) is L1+p over (0, 1) and therefore over (0, l).

page 15 note ‡ Hobson, loc. cit., 575.

page 16 note * Hobson, loc, cit., 599.

page 16 note † Math. Ann., 97 (1926), 159–209Google Scholar.

page 16 note ‡ Math. Zeit., 25 (1926), 321–347CrossRefGoogle Scholar.

page 17 note * Hobson, loc. cit., 614 (§ 399).

page 18 note * By Minkowski's inequality.

page 18 note † By Hölder's inequality.