  In this paper, we investigate the wavelet coefficients for function spaces
$\mathcal{A}_k^p=\{f: \|(i \omega)^k\hat{f}(\omega)\|_p\leq 1\}, k\in N,
p\in(1,\infty)$ using an important quantity $C_{k,p}(\psi)$. In particular,
Bernstein type inequalities associated with wavelets are established. We
obtained a sharp inequality of Bernstein type for splines, which induces 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 $C_{k,p}(\psi)$.
