We study two classes of orthonormal bases for
![$$
L^{2} {\left[ {0,1} \right]}
$$](/content/mp8p7334375681r5/10444_2005_9006_Article_IEq1.gif)
in this paper. The exponential parts of these bases are multi-knot piecewise linear functions. These bases are called spectral
sequences. Characterizations of these multi-knot piecewise linear functions are provided. We also consider an opposite problem
for single-knot piecewise linear spectral sequences, where the piecewise linear functions are defined on
and
![$$
{\left[ {\theta ,1} \right]}
$$](/content/mp8p7334375681r5/10444_2005_9006_Article_IEq3.gif)
. We show that such spectral sequences do not exist except for
.
*Supported by the Technology and Research project 2002YF015 of the Ministry of Railway of China and by the Natural Science
Foundation of China under grant 10371122.
**Supported by the Presidential Foundation of Graduate School of the Chinese Academy of Sciences (yzjj200505).