Lower Bounds for Finite Wavelet and Gabor Systems
Lower Bounds for Finite Wavelet and Gabor Systems
复制标题
有限小波和 Gabor 系统的下界
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
A. Lindner
中科院分区：
文献类型：
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作者：
O. Christensen;A. Lindner
AbstractGiven ψ∈L2(R) and a finite sequence {(ar,λr)}r∈Γ⫅R+XR consisting of distinct points, the corresponding wavelet system is the set of functions
$$left{ {frac{1}{{a_gamma ^{1/2} }}phi (frac{x}{{a_gamma }} - lambda _gamma )gamma varepsilon r}
ight}$$
. We prove that for a dense set of functions ψ∈L2(R) the wavelet system corresponding to any choice of {(ar,λr)}r∈Γis linearly independent, and we derive explicite estimates for the corresponding lower (frame) bounds. In particular, this puts restrictions on the choice of a scaling function in the theory for multiresolution analysis. We also obtain estimates for the lower bound for Gabor systems
$$left{ {e^{2rie_{gamma x} } g(x - lambda _gamma )}
ight}gamma varepsilon r$$
for functions g in a dense subset of L2(R).
AbstractGiven ψ∈L2(R) and a finite sequence {(ar,λr)}r∈Γ⫅R+XR consisting of distinct points, the corresponding wavelet system is the set of functions
$$left{ {frac{1}{{a_gamma ^{1/2} }}phi (frac{x}{{a_gamma }} - lambda _gamma )gamma varepsilon r}
ight}$$
. We prove that for a dense set of functions ψ∈L2(R) the wavelet system corresponding to any choice of {(ar,λr)}r∈Γis linearly independent, and we derive explicite estimates for the corresponding lower (frame) bounds. In particular, this puts restrictions on the choice of a scaling function in the theory for multiresolution analysis. We also obtain estimates for the lower bound for Gabor systems
$$left{ {e^{2rie_{gamma x} } g(x - lambda _gamma )}
ight}gamma varepsilon r$$
for functions g in a dense subset of L2(R).