TY - JOUR
T1 - Asymptotic Bernstein type inequalities and estimation of wavelet coefficients
AU - Spektor, Susanna
AU - ZHUANG, Xiaosheng
PY - 2012
Y1 - 2012
N2 - 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 $C_{k,p}(\psi)$.
AB - 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 $C_{k,p}(\psi)$.
KW - wavelet coefficients
KW - asymptotic estimation
KW - Bernstein type inequalities
KW - Daubechies orthonormal wavelets
KW - semiorthogonal spline wavelets
M3 - 22_Publication in policy or professional journal
VL - 19
SP - 289
EP - 312
JO - Methods and Applications of Analysis
JF - Methods and Applications of Analysis
SN - 1073-2772
IS - 3
ER -