SOME EXTENSIONS OF PALEY-WIENNER THEOREM

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Left-inverses of fractional Laplacian and sparse stochastic processes

The fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ ^d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I _γ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I _γ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I _γ f is not p-integrable. We then introduce the new unique left-inverse I _{γ, p} of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} with the property that I _{γ, p} is dilation-invariant (but not translation-invariant) and that I _{γ, p} f is p-integrable for any Schwartz function f. We finally apply that linear operator I _{γ, p} with p = 1 to solve the stochastic partial differential equation (-\triangle)^g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w.     

Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

One-pot photochemical synthesis of ultrathin Au nanocrystals on co-reduced graphene oxide and its application

In this study, we have developed a simple and green method for one-pot synthesis of ultrathin gold nanocrystals attached to graphene through photo irradiation. High-yield ultrathin Au nanocrystals are distributed on the reduced graphene oxide, immediately followed by the deoxygenation of graphene oxide in the absence of chemical reductants and surfactants. The procedure has been thoroughly completed and the products have been analyzed by transmission electron microscopy (TEM), atomic force microscopy (AFM), X-ray photoelectron spectroscopy (XPS), etc. The results show that both graphene oxide and photo irradiation play essential roles in the formation of ultrathin Au nanocrystals. The nanohybrids also display excellent electro-catalytical performance to methanol oxidation. This study not only has potential in the applications of bio-sensing and fuel cells, but also provides new procedures for the preparation of metal/graphene nanomaterials.     

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

Behaviour of an oscillatory singular integral on weighted local Hardy spaces

The boundedness on weighted local Hardy spacesh _w ^1,p of the oscillatory singular integral

Two problems about singular integral operator

In this paper we consider L² bounds and weak type (1.1) bounds for a class of singular integral operators introduced by R. Fefferman and Namazi. Then, we solve two problems proposed by Shi.     

奇异积分中的一些问题

设n≥2,Ω为R~n中单位球面S~(n-1)上的可积函数且Ω在S~(n-1)上的平均值为零,即∫_S~(n1)~Ω(x)dσ(x)=0.其中dσ为S~(n-1)上的体积元.定义奇异积分算子T_0,和相应的极大算子T~*,其中h∈L~∞(R~+).关于算子T和T~*已有许多研究([1]-[6]等).在1986年,Namazi利用Fourier变换的Hausdorff-Young不等式证明了     

A new kind of spaces of Triebel-Lizorkin type

In this note, we define a new kind of spaces TF _p ^aq of Triebel-Lizorkin type and characterize it via sequence space l^p (l^q). Also a Fourier multiplier theorem is established in this new spaces TF _p ^aq . The author is partially supported by the National Natural Science of China and the Zhejiang Provincal Science Foundation.