  Since its appearing in 1996, the Stockwell transform (S-transform) has been
applied to medical imaging, geophysics and signal processing in general. In
this paper, we prove that the system of functions (so-called DOST basis) is
indeed an orthonormal basis of L^2([0,1]), which is time-frequency localized,
in the sense of Donoho-Stark Theorem (1989). Our approach provides a unified
setting in which to study the Stockwell transform (associated to different
admissible windows) and its orthogonal decomposition. Finally, we introduce a
fast -- O(N log N) -- algorithm to compute the Stockwell coefficients for an
admissible window. Our algorithm extends the one proposed by Y. Wang and J.
Orchard (2009).
