A continued fraction analysis of periodic wavelet coefficients

We define and prove the existence of crossings of wavelet coefficients translated by integer multiples of the numerator of a continued fraction convergent of the ratio of the sampling interval to the period of the wavelet coefficients. Crossings are found to be translation invariant $\pm 1$. Intervals between crossings are analyzed for wavelets with $n$ vanishing moments. These wavelets act as multiscale differential operators. These crossings reveal different locations in the period where there is equality in the $n$th derivative of an averaging of the signal. These results will be employed in the estimation of frequency components in future publications.