Asymptotic Bernstein type inequalities and estimation of wavelet coefficients

In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p:=\{f:\|(i \omega)^k\hat{f}(\omega)\|_p\le 1\}$, $k\in\N\cup\{0\}$, $p\in(1,\infty)$ using an important quantity $C_{k,p}(\psi):=\sup\{\frac{|\la f,\psi\ra|}{\|\hat{\psi}\|_p}\,:\,{f\in\mathcal{A}_k^{p'}}\}$ with $1/p+1/p'=1$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained an sharp inequality of Bernstein type for splines and a lower bound for the quantity $C_{k,p}(\psi)$ with $\psi$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(\psi)$.