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Register a new userConstruction of Orthonormal Quasi-Shearlets based on quincunx dilation subsampling
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Rujie Yin
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We consider the construction of orthonormal directional wavelet bases in the multi-resolution analysis (MRA) framework with quincunx dilation downsampling. We show that the Parseval frame property in MRA is equivalent to the identity summation and shift cancellation conditions on M functions, which essentially characterize the scaling (father) function and all directional (mother) wavelets. Based on these two conditions, we further derive sufficient conditions for orthonormal bases and build a family of quasishearlet orthonormal bases, that has the same frequency support as that of the least redundant shearlet system. In addition, we study the limitation of our proposed bases design due to the shift cancellation conditions.
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We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work, We show that the supports of orthonormal wavelets in our framework are discontinuous in the frequency domain, yet this irregularity constraint can be avoided in frames, even with redundancy factor less than 2. In this paper, we focus on the extension of orthonormal wavelets to biorthogonal wavelets and show that the same obstruction of regularity as in orthonormal schemes exists in biorthogonal schemes. In addition, we provide a numerical algorithm for biorthogonal wavelets construction where the dual wavelets can be optimized, though at the cost of deteriorating the primal wavelets due to the intrinsic irregularity of biorthogonal schemes.
In the present work we provide the bounds for Daubechies orthonormal wavelet coefficients for function spaces $mathcal{A}_k^p:={f: |(i omega)^khat{f}(omega)|_p< infty}$, $kinmathbf{N}cup{0}$, $pin(1,infty)$.
In this article, we prove a Strichartz type inequality %associated with Schrodinger equation for a system of orthonormal functions associated with the special Hermite operator $mathcal{L}=-Delta+frac{1}{4}|z|^{2}-i sum_{1}^{n}left(x_{j} frac{partial}{partial y_{j}}-y_{j} frac{partial}{partial x_{j}}right), $ where $Delta$ denotes the Laplacian on $mathbb{C}^{n}$.
The Strichartz inequality for the system of orthonormal functions for the Hermite operator $H=-Delta+|x|^2$ on $mathbb{R}^n$ has been proved in cite{lee}, using the classical Strichartz estimates for the free Schrodinger propagator for orthonormal systems cite{frank, frank1} and the link between the Schrodinger kernel and the Mehler kernel associated with the Hermite semigroup cite{SjT}. In this article, we give an alternative proof of the above result in connection with the restriction theorem with respect to the Hermite transform with an optimal behavior of the constant in the limit of a large number of functions. As an application, we show the well-posedness results in Schatten spaces for the nonlinear Hermite-Hartree equation.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $mathcal{H}_1$ which contains $mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
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