This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The pr...This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.展开更多
Due to the disturbances arising from the coherence of reflected waves and from echo noise,problems such as limitations,instability and poor accuracy exist with the current quantitative analysis methods.According to th...Due to the disturbances arising from the coherence of reflected waves and from echo noise,problems such as limitations,instability and poor accuracy exist with the current quantitative analysis methods.According to the intrinsic features of GPR signals and wavelet time–frequency analysis,an optimal wavelet basis named GPR3.3 wavelet is constructed via an improved biorthogonal wavelet construction method to quantitatively analyse the GPR signal.A new quantitative analysis method based on the biorthogonal wavelet(the QAGBW method)is proposed and applied in the analysis of analogue and measured signals.The results show that compared with the Bayesian frequency-domain blind deconvolution and with existing wavelet bases,the QAGBW method based on optimal wavelet can limit the disturbance from factors such as the coherence of reflected waves and echo noise,improve the quantitative analytical precision of the GPR signal,and match the minimum thickness for quantitative analysis with the vertical resolution of GPR detection.展开更多
Based on the approach of biorthogonal basis, we carry out the quasinormal modes （QNMs） expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Her...Based on the approach of biorthogonal basis, we carry out the quasinormal modes （QNMs） expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the dosed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H . The time-independent perturbation theory for the complex frequencies can be also developed.展开更多
In this paper,a new method is presented for designing M-band biorthogonal symmetric wavelets.The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with ...In this paper,a new method is presented for designing M-band biorthogonal symmetric wavelets.The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated.The closed-form solution is given and a design example is presented.展开更多
The seismic wave consists of many seismic phases, which contain rich geophysical information from the hypocenter, medium of seismic wave passing through and so on. It is very important to detect and pick these seismic...The seismic wave consists of many seismic phases, which contain rich geophysical information from the hypocenter, medium of seismic wave passing through and so on. It is very important to detect and pick these seismic phases for understanding the mechanism of earthquake, the Earth structure and property of seismic waves. In order to reduce or avoid the loss resulted from the earthquake, one of the important goals of seismic event detecting is to obtain its related information before and after it occurs. Because of the particularity of P wave and S wave the seismic event detecting focuses on distinguishing P and S waves and picking their onset time, it has been becoming one of the research hotspots for many geophysicists to pick the P and S wave arrival accurately and effectively.展开更多
A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine trans form (BLDCT). It uses a redundant library of biorthogonal local d...A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine trans form (BLDCT). It uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional trans form domain excision algorithms is that the basis functions are not fixed but ca n be adapted to the time-frequency structure of the interfering signal. It is w e ll suited to transform domain compression and suppression of various types of in terference. Compared to the discrete wavelet transform (DWT) that provides logar ithmic division of the frequency bands, the adaptive BLDCT can provide more flex ible frequency resolution. Thus it is more insensitive to variations of jamming frequency. Simulation results demonstrate the improved bit error rate (BER) perf ormance and the increased robustness of the receiver.展开更多
Traditional lapped transform domain excision techniques obtain good performance at the expense of increased processing delay. Extension of transform domain filtering techniques to the lapped biorthogonal transform dom...Traditional lapped transform domain excision techniques obtain good performance at the expense of increased processing delay. Extension of transform domain filtering techniques to the lapped biorthogonal transform domain can help solve the problem. By incorporating biorthogonality into the lapped transforms, more flexibility is obtained in the design of windows. Thus transform bases with better stopband attenuation can be generated by designing windows, but not by increasing the overlapping factor. In this paper, a new modulated lapped biorthogonal transform (MLBT) with optimized windows is introduced for efficient compression of multi-tone interfering signal energy. The bit error rate (BER) performance of the receiver employing the proposed MLBT excision technique is analyzed and compared with that of the lapped transform domain excision-based receivers. Simulation results demonstrate the improved performance and increased robustness of the proposed technique.展开更多
In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identificat...In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.展开更多
The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavel...The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.展开更多
Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function...Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to construct biorthogonal scramble modulation (BOSM) wireless transmission systems. In a BOSM system, the transmitted signal has significant security performance. Several different BOSM and orthogonal systems are compared on aspects of BER performance and spectrum efficiency, simulation results show that the BOSM systems based on Chebyshev polynomial and Legendre polynomial are better than BOSM system based on Fourier series, also better than orthogonal MCM and OFDM systems.展开更多
In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the metho...In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.展开更多
The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationsh...The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationship for couple stress theory comparing to classical elasticity. A new state vector is presented for three dimensional couple stress problems of prismatic structures. A new biorthogonality relation- ship of couple stress is discovered. The dual partial differential equations of couple stress problem are derived by the new state vector. By two important identical equations the new biorthogonality rela- tionship is proved based on the method of separation of variables. The symplectic orthogonality rela- tionship to three dimensional couple stress theory may be decomposed into two independently and symmetrically orthogonality relationships. The new biorthogonality relationship includes the symplec- tic orthogonality relationship. The biorthogonality relationship of couple stress may also be degener- ated into the theory of elasticity. The new state vector and biorthogonality relationship provide theo- retic foundation for the research on the schemes of separation of variables and eigenfunction expan- sion of couple stress theory.展开更多
Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal mu...Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ? = (? 1, …, ? r)T is in $(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $ is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M ?n = 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.展开更多
A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of th...A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of the eigenfunctions are presen! Based on the biorthogonal relationships, a novel complete biorthogonal expansion of the Mindlin plate bending problems x~ two opposite sides simply supported is proposed through the products of operator matrices. The exact solutions to deflections bending moments for the Mindlin plate with fully simply supported sides are obtained. A numerical example is illustrated to ve~ the accuracy and validity of the expansion method.展开更多
We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian tra...We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of the ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.展开更多
This paper is concerned with seeking the general solutions of matrix equation M（ξ）M＊ （ξ） = Is for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.
Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limi...Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.展开更多
In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functions and These functions are useful in the theory of biorthogonal wavelet.
In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite...In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.展开更多
文摘This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.
基金Projects(51678071,51278071)supported by the National Natural Science Foundation of ChinaProjects(14KC06,CX2015BS02)supported by Changsha University of Science&Technology,China
文摘Due to the disturbances arising from the coherence of reflected waves and from echo noise,problems such as limitations,instability and poor accuracy exist with the current quantitative analysis methods.According to the intrinsic features of GPR signals and wavelet time–frequency analysis,an optimal wavelet basis named GPR3.3 wavelet is constructed via an improved biorthogonal wavelet construction method to quantitatively analyse the GPR signal.A new quantitative analysis method based on the biorthogonal wavelet(the QAGBW method)is proposed and applied in the analysis of analogue and measured signals.The results show that compared with the Bayesian frequency-domain blind deconvolution and with existing wavelet bases,the QAGBW method based on optimal wavelet can limit the disturbance from factors such as the coherence of reflected waves and echo noise,improve the quantitative analytical precision of the GPR signal,and match the minimum thickness for quantitative analysis with the vertical resolution of GPR detection.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10647108, 10547101, and 10604002the National Fundamental Research Program of China under Grant No. 2006CB921200
文摘Based on the approach of biorthogonal basis, we carry out the quasinormal modes （QNMs） expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the dosed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H . The time-independent perturbation theory for the complex frequencies can be also developed.
文摘In this paper,a new method is presented for designing M-band biorthogonal symmetric wavelets.The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated.The closed-form solution is given and a design example is presented.
文摘The seismic wave consists of many seismic phases, which contain rich geophysical information from the hypocenter, medium of seismic wave passing through and so on. It is very important to detect and pick these seismic phases for understanding the mechanism of earthquake, the Earth structure and property of seismic waves. In order to reduce or avoid the loss resulted from the earthquake, one of the important goals of seismic event detecting is to obtain its related information before and after it occurs. Because of the particularity of P wave and S wave the seismic event detecting focuses on distinguishing P and S waves and picking their onset time, it has been becoming one of the research hotspots for many geophysicists to pick the P and S wave arrival accurately and effectively.
基金Project supported by the National Natural Science Foundation of China(Grant No.6017201860372007)
文摘A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine trans form (BLDCT). It uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional trans form domain excision algorithms is that the basis functions are not fixed but ca n be adapted to the time-frequency structure of the interfering signal. It is w e ll suited to transform domain compression and suppression of various types of in terference. Compared to the discrete wavelet transform (DWT) that provides logar ithmic division of the frequency bands, the adaptive BLDCT can provide more flex ible frequency resolution. Thus it is more insensitive to variations of jamming frequency. Simulation results demonstrate the improved bit error rate (BER) perf ormance and the increased robustness of the receiver.
文摘Traditional lapped transform domain excision techniques obtain good performance at the expense of increased processing delay. Extension of transform domain filtering techniques to the lapped biorthogonal transform domain can help solve the problem. By incorporating biorthogonality into the lapped transforms, more flexibility is obtained in the design of windows. Thus transform bases with better stopband attenuation can be generated by designing windows, but not by increasing the overlapping factor. In this paper, a new modulated lapped biorthogonal transform (MLBT) with optimized windows is introduced for efficient compression of multi-tone interfering signal energy. The bit error rate (BER) performance of the receiver employing the proposed MLBT excision technique is analyzed and compared with that of the lapped transform domain excision-based receivers. Simulation results demonstrate the improved performance and increased robustness of the proposed technique.
文摘In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.
基金Supported by Natural Science Foundation of Henan Province（0511013500）
文摘The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.
文摘Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to construct biorthogonal scramble modulation (BOSM) wireless transmission systems. In a BOSM system, the transmitted signal has significant security performance. Several different BOSM and orthogonal systems are compared on aspects of BER performance and spectrum efficiency, simulation results show that the BOSM systems based on Chebyshev polynomial and Legendre polynomial are better than BOSM system based on Fourier series, also better than orthogonal MCM and OFDM systems.
文摘In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.
基金Supported by the National Natural Science Foundation of China (Grant No. 50778071)the Hunan Provincial Natural Science Foundation of China (Grant No. 08JJ3011)the Research Committee of City University of Hong Kong (Grant No. 7002315)
文摘The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationship for couple stress theory comparing to classical elasticity. A new state vector is presented for three dimensional couple stress problems of prismatic structures. A new biorthogonality relation- ship of couple stress is discovered. The dual partial differential equations of couple stress problem are derived by the new state vector. By two important identical equations the new biorthogonality rela- tionship is proved based on the method of separation of variables. The symplectic orthogonality rela- tionship to three dimensional couple stress theory may be decomposed into two independently and symmetrically orthogonality relationships. The new biorthogonality relationship includes the symplec- tic orthogonality relationship. The biorthogonality relationship of couple stress may also be degener- ated into the theory of elasticity. The new state vector and biorthogonality relationship provide theo- retic foundation for the research on the schemes of separation of variables and eigenfunction expan- sion of couple stress theory.
基金This work was partially supported by the National Natural Science Foundation of China(Grant Nos.10071071 and 10471123)the Mathematical Tianyuan Foundation of the National Natural Science Foundation of China NSF(Grant No.10526036)China Postdoctoral Science Foundation(Grant No.20060391063)
文摘Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ? = (? 1, …, ? r)T is in $(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $ is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M ?n = 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.
基金supported by the National Natural Science Foundation of China (Grant No. 10962004)the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2012MS0105)
文摘A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of the eigenfunctions are presen! Based on the biorthogonal relationships, a novel complete biorthogonal expansion of the Mindlin plate bending problems x~ two opposite sides simply supported is proposed through the products of operator matrices. The exact solutions to deflections bending moments for the Mindlin plate with fully simply supported sides are obtained. A numerical example is illustrated to ve~ the accuracy and validity of the expansion method.
基金G.S.is appreciative of support from the NSFC under the Grant Nos.11704186 and 11874220S.P.K is appreciative of support by the National Natural Science Foundation of China under Grant Nos.11674026,11974053,and 12174030.
文摘We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of the ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.
基金supported in part Professor Yuesheng Xu under the program of"One Hundred Outstanding Young Chinese Scientists" of the Chinese Academy of Sciencesthe Graduate Innovation Foundation of the Chinese Academy of Sciences
文摘This paper is concerned with seeking the general solutions of matrix equation M（ξ）M＊ （ξ） = Is for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.
文摘Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.
文摘In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functions and These functions are useful in the theory of biorthogonal wavelet.
文摘In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.
