We present here a new approach to wavelet transforms based on techniques of SHA. It is a version of HA which operates in spline spaces. SHA enjoys various applications but here we outline a recent application of SHA techniques, namely, accomplishing on the base of periodic splines the wavelet transforms of periodic signals as well as the remarkable informative digital representation of the signals. SHA approach to wavelets yields a tool just as for the constructing a diversity of spline wavelet bases, so for a fast implementation of a decomposition of a function into a fitting wavelet representation and its reconstruction. This approach allows to construct WP bases for refined frequency resolution of signals.