Vanishing moments of wavelet packets and wavelets associated with Riesz projectors

In the present paper, we show that, under some conditions wavelet packet basis of L
²
(ℝ) can be used as a tool for the uniform approximation of a function f ∈ (C
^M
∩ L
²
)(ℝ),M > 0. The necessary and sufficient condition on the wavelet packet transform for estimating the Hölder continuity of a function f has been given. Wavelets associated with Riesz projectors have been proposed and various results related to vanishing moments have been presented. Further, we use such wavelets and prove that Hölder continuity of a function aids in the decay of wavelet coefficients and thus helps in approximating it. Finally, we give some properties of wavelets associated with Riesz projectors.