Estimation of wavelet coefficients on some classes of functions

Abstract

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