Piecewise linear spectral sequences

We study a class of orthonormal exponential bases for the space L 2 [0,1] and introduce the concept of spectral sequences. We characterize piecewise linear spectral sequences with the knot at 1/2 and investigate the non-continuity of the piecewise linear spectral sequences. From a special construction of a piecewise constant spectral sequence, the classical Walsh system is recovered.