Time-frequency localisation of Shannon wavelet packets

It is well known that the 2π minimally supported frequency scaling function φ^α(x) satisfying φ̂^α(ω) = χ_[-α,2π-α](ω), 0 < α < 2π, is not time localised. The Shannon wavelet packets φ_n^α(x) are generated from φ^α(x) via an arbitrary pair of low-pass and high-pass filters (h,g), which is associated with an orthogonal multiresolution analysis. The authors prove that φ_n^α(x) is time localised if α and n satisfy certain conditions. They also show that the decay properties of φ_n^α(x) depend on the multiplicity of the zero ω = π of the symbol m₀(ω) of the low-pass filter h.