Approximations using Hilbert transform of wavelets

Hilbert transform of wavelets has been used to approximate functions in L 2 (R) . It is proved that Hilbert transform of wavelets with many vanishing moments does a good job in approximating smooth functions in L 2 (R) . We also prove that Hlder continuity of a function helps in the decay of wavelet coefficients and thereby helps in approximating it. Finally, we give a result that relates the Hilbert transform of wavelet with dyadic scale differential operator and use it to decrease the wavelet coefficients.