As a typical family of mono-component signals,the nonlinear Fourier basis {eikθa(t)}k∈Z,defined by the nontangential boundary value of the M¨obius transformation,has attracted much attention in the field of non...As a typical family of mono-component signals,the nonlinear Fourier basis {eikθa(t)}k∈Z,defined by the nontangential boundary value of the M¨obius transformation,has attracted much attention in the field of nonlinear and nonstationary signal processing in recent years.In this paper,we establish the Jackson's and Bernstein's theorems for the approximation of functions in Xp(T),1 p ∞,by the nonlinear Fourier basis.Furthermore,the analogous theorems for the approximation of functions in Hardy spaces by the finite Blaschke products are established.展开更多
We propose a method of complex-amplitude Fourier single-pixel imaging(CFSI)with coherent structured illumination to acquire both the amplitude and phase of an object.In the proposed method,an object is illustrated by ...We propose a method of complex-amplitude Fourier single-pixel imaging(CFSI)with coherent structured illumination to acquire both the amplitude and phase of an object.In the proposed method,an object is illustrated by a series of coherent structured light fields,which are generated by a phase-only spatial light modulator,the complex Fourier spectrum of the object can be acquired sequentially by a single-pixel photodetector.Then the desired complex-amplitude image can be retrieved directly by applying an inverse Fourier transform.We experimentally implemented this CFSI with several different types of objects.The experimental results show that the proposed method provides a promising complex-amplitude imaging approach with high quality and a stable configuration.Thus,it might find broad applications in optical metrology and biomedical science.展开更多
Let g（x） ∈L<sup>2</sup>（R） and （ω） be the Fourier transform of g（x）.Define g<sub>mn</sub>（x）=e<sup>imx</sup>g（x- 2πn）.In this paper we shall give a sufficient and nec...Let g（x） ∈L<sup>2</sup>（R） and （ω） be the Fourier transform of g（x）.Define g<sub>mn</sub>（x）=e<sup>imx</sup>g（x- 2πn）.In this paper we shall give a sufficient and necessary condition under which {g<sub>mn</sub>（x）} constitutes an orthonormal basis of L<sup>2</sup>（R） for compactly supported g（x） or （ω）.展开更多
A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems.The method first discretizes the entire domain into a set of overlapping small subdomains,and then in ea...A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems.The method first discretizes the entire domain into a set of overlapping small subdomains,and then in each of the subdomains,the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method.The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations.The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries.Preliminary numerical experiments involving Poisson,Helmholtz,and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.展开更多
We investigate two classes of orthonormal bases for L^2（[0, 1）^n）. The exponential parts of those bases are multi-knot piecewise linear functions which are called spectral sequences. We characterize the multi-knot ...We investigate two classes of orthonormal bases for L^2（[0, 1）^n）. The exponential parts of those bases are multi-knot piecewise linear functions which are called spectral sequences. We characterize the multi-knot piecewise linear spectral sequences and give an application of the first class of piecewise linear spectral sequences.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.11071261,60873088,10911120394)
文摘As a typical family of mono-component signals,the nonlinear Fourier basis {eikθa(t)}k∈Z,defined by the nontangential boundary value of the M¨obius transformation,has attracted much attention in the field of nonlinear and nonstationary signal processing in recent years.In this paper,we establish the Jackson's and Bernstein's theorems for the approximation of functions in Xp(T),1 p ∞,by the nonlinear Fourier basis.Furthermore,the analogous theorems for the approximation of functions in Hardy spaces by the finite Blaschke products are established.
基金Project supported by the Natural Science Foundation of Hebei Province,China(Grant Nos.A2022201039 and F2019201446)the MultiYear Research Grant of University of Macao,China(Grant No.MYRG2020-00082-IAPME)+2 种基金the Science and Technology Development Fund from Macao SAR(FDCT),China(Grant No.0062/2020/AMJ)the Advanced Talents Incubation Program of the Hebei University(Grant No.8012605)the National Natural Science Foundation of China(Grant Nos.11204062,61774053,and 11674273)。
文摘We propose a method of complex-amplitude Fourier single-pixel imaging(CFSI)with coherent structured illumination to acquire both the amplitude and phase of an object.In the proposed method,an object is illustrated by a series of coherent structured light fields,which are generated by a phase-only spatial light modulator,the complex Fourier spectrum of the object can be acquired sequentially by a single-pixel photodetector.Then the desired complex-amplitude image can be retrieved directly by applying an inverse Fourier transform.We experimentally implemented this CFSI with several different types of objects.The experimental results show that the proposed method provides a promising complex-amplitude imaging approach with high quality and a stable configuration.Thus,it might find broad applications in optical metrology and biomedical science.
基金This work is supported by the National Natural Science Foundation of China(No.19801005)the Project of New Stars of Science and Technology of Beijing a Grant of Young Fellow of Educational Ministry.
文摘Let g（x） ∈L<sup>2</sup>（R） and （ω） be the Fourier transform of g（x）.Define g<sub>mn</sub>（x）=e<sup>imx</sup>g（x- 2πn）.In this paper we shall give a sufficient and necessary condition under which {g<sub>mn</sub>（x）} constitutes an orthonormal basis of L<sup>2</sup>（R） for compactly supported g（x） or （ω）.
基金The work described in this paper was supported by the National Natural Science Foundation of China (Nos.11872220,12111530006)the Natural Science Foundation of Shandong Province of China (No.ZR2021JQ02)the Russian Foundation for Basic Research(No.21-51-53014).
文摘A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems.The method first discretizes the entire domain into a set of overlapping small subdomains,and then in each of the subdomains,the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method.The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations.The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries.Preliminary numerical experiments involving Poisson,Helmholtz,and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.
基金Supported in part by National Natural Science Foundation of China (Grant No.10631080)Natural Science Foundation of Beijing (Grant No.1092004)
文摘We investigate two classes of orthonormal bases for L^2（[0, 1）^n）. The exponential parts of those bases are multi-knot piecewise linear functions which are called spectral sequences. We characterize the multi-knot piecewise linear spectral sequences and give an application of the first class of piecewise linear spectral sequences.
