This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite...

The conventional dyadic multiresolution analysis constructs a succession of frequency intervals in the form of (<i>π</i> / 2^<i> j</i>, <i>π</i> / 2^{<i> j</i> - 1}); <i>j</i> = 1, 2, . . . , <i>n</i> of which the bandwidths are halved repeatedly in the descent from high frequencies to low frequencies. Whereas this scheme...