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Reciprocal Convergence Classes for Fourier Series and Integrals

Published online by Cambridge University Press:  20 November 2018

A. P. Guinand
A. P. Guinand*
Affiliation:
University of Saskatchewan
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The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L2(0, ∞) states that (see (7) for references)

converges in mean square to a function g(x) which also belongs to L2(0, ∞), and furthermore

Some years ago in a series of papers (1; 2; 3) on summation formulae I showed that a similar symmetrical theory for narrower classes of functions and ordinary convergence of the integrals can also be developed.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 19 - 36
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Guinand, A. P., Summation formulae and self-reciprocal functions, I, II, Quart. J. Math. (1), 9 (1938), 53–67 and (1), 10 (1939), 104-118.Google Scholar
2. Guinand, A. P., On Poisson's summation formula, Ann. Math. (2), 42 (1941), 591–603.Google Scholar
3. Guinand, A. P., General transformations and the Parseval theorem, Quart. J. Math. (1), 12 (1941), 51–56.Google Scholar
4. Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities (Cambridge, 1934) 239–246.Google Scholar
5. Miller, J. B., A symmetrical convergence theory for general transforms, Proc. London Math. Soc. (3), 8 (1958), 224–241.Google Scholar
6. Titchmarsh, E. C., The Theory of Functions (Oxford, 1939).Google Scholar
7. Titchmarsh, E. C., An introduction to the theory of Fourier integrals (Oxford, 1948).Google Scholar