| Let Ap be the Banach space of all continuous functions on the torus whose Fourier coefficients are in ℓ p. We show that Ap is not an algebra for all 1 < p < p 0, for a certain p0, 1 < p 0 < 2. This is done through a series of attempts which might suggest that the example used is the best one possible. One of the attempts is using the Rudin-Shapiro polynomials and as an aside some new properties of these polynomials are given. We also discuss the space Ap ,infinity: how it relates to Ap and whether or not it is an algebra. Of particular interest is the space A1,infinity which we show is not an algebra, which is a curiosity given that A1 is a well known algebra. We also give examples to show that all of these spaces are indeed different. |
