Time-frequency localisation of Shannon wavelet packets

It is well known that the 2π minimally supported frequency scaling function φ^α(x) satisfying φˆ^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 are 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.