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The “Fourier” Theory of the Cardinal Function

Abstract

The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that

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Proc. Camb. Phil. Soc., 23 (1926), 373.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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