Estimation of wavelet coefficients on some classes of functions

Let $\Psi_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$W_{2,p}^k=\{f \in L_2(R):\|(I \omega)^k\hat f(\omega)\|_p\leq 1\}, \, k \in N.$$ We prove that $$\lim_{m \to \infty}\, \sup\left\{\frac{|(\Psi_m^D)|}{\|(\hat \Psi_m^D)\|_q}: f \in W_{2, p'}^k\right\}=\frac{\frac{(2\pi)^{1/p-1/2}}{\pi^k}\left(\frac{1-2^{1-pk}}{pk-1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$