The “Fourier” Theory of the Cardinal Function

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