Estimation of wavelet coefficients on some classes of functions

Let ${\mathrm{\Psi }}_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$W_{2,p}^k=\{f\in L_2(R):\mathrm{\parallel }(I\omega {)}^k\widehat{f}(\omega ){\mathrm{\parallel }}_p\le 1\},k\in N\mathrm{.}$$ We prove that $$\underset{m\to \mathrm{\infty }}{\lim }\sup \{\frac{\mathrm{\mid }({\mathrm{\Psi }}_m^D)\mathrm{\mid }}{\mathrm{\parallel }({\widehat{\mathrm{\Psi }}}_m^D){\mathrm{\parallel }}_q}:f\in W_{2,p^{\mathrm{\prime }}}^k\}=\frac{\frac{(2\pi {)}^{1\mathrm{/}p-1\mathrm{/}2}}{{\pi }^k}{\left(\frac{1-2^{1-pk}}{pk-1}\right)}^{1\mathrm{/}p}}{(2\pi {)}^{1\mathrm{/}q-1\mathrm{/}2}}\mathrm{.}$$