In this paper,a sinusoidal signal frequency estimation algorithm is proposed by weighted least square method.Based on the idea of Provencher,three biggest Fourier coefficients in the maximum periodogram are considered...In this paper,a sinusoidal signal frequency estimation algorithm is proposed by weighted least square method.Based on the idea of Provencher,three biggest Fourier coefficients in the maximum periodogram are considered,the Fourier coefficients can be written as three equations about the amplitude,phase,and frequency,and the frequency is estimated by solving equations.Because of the error of measurement,weighted least square method is used to solve the frequency equation and get the signal frequency.It is shown that the proposed estimator can approach the Cramer-Rao Bound(CRB)with a low Signal-to-Noise Ratio(SNR)threshold and has a higher accuracy.展开更多
The order of magnitude of multiple Fourier coefficients of complex valued functions of generalized bounded variations like (∧^1,.. .,∧^N)BV^(p) and r-BV, over [0,2π]^ N, are estimated.
Abstract Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ Sk(F), In this paper, we established nontrivial estimates for ∑n≤x λf(n^i)λf(n^j),where 1≤ij≤4.
For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of
In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems: for any ε∈ (0, 1), there exists a measurable set E C [0, 1) of measure bigger than 1 - s such that ...In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems: for any ε∈ (0, 1), there exists a measurable set E C [0, 1) of measure bigger than 1 - s such that for any function f ∈ LI[0, 1), it is possible to find a function g ∈ L^1 [0, 1) coinciding with f on E and the absolute values of non zero Fourier coefficients of g with respect to the Vilenkin system are monotonically decreasing.展开更多
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic for...Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.展开更多
Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) =∑n≤xa(n).In this paper, we establish an asymptotic formula of tile fourth power mome...Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) =∑n≤xa(n).In this paper, we establish an asymptotic formula of tile fourth power moment of A(x) and prove that ∫1^TA^4(x)dx=3/64κπ^4s4;2(a^~)T^2κ+O(T^2a-δ4+t) with δ4 = 1/8, which improves the previous result.展开更多
Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying...Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.展开更多
We prove that every homomorphism of the algebra P<sub><em>n</em></sub> into the algebra of operators on a Hilbert space is completely bounded. We show that the contractive homomorphism introduc...We prove that every homomorphism of the algebra P<sub><em>n</em></sub> into the algebra of operators on a Hilbert space is completely bounded. We show that the contractive homomorphism introduced by Parrott, which is not completely contractive, is completely bounded (similar to a completely contractive homomorphism). We also show that homomorphisms of the algebra <span style="white-space:normal;">P</span><sub style="white-space:normal;"><em>n</em></sub> generate completely positive maps over the algebras <em>C</em>(T<sup><em>n</em></sup>)and <em>M</em><sub>2</sub>(<em>C</em>(T<sup><em>n</em></sup>)).展开更多
We prove that every matrix </span><i><span style="font-family:"">F</span></i><span style="font-size:6.5pt;line-height:102%;font-family:宋体;">∈</span>...We prove that every matrix </span><i><span style="font-family:"">F</span></i><span style="font-size:6.5pt;line-height:102%;font-family:宋体;">∈</span><i><span style="font-family:"">M</span></i><sub><span style="font-family:"">k </span></sub><span style="font-family:"">(P<sub>n</sub>)</span><span style="font-family:""> is associated </span><span style="font-family:"">with</span><span style="font-family:""> </span><span style="font-family:"">the</span><span style="font-family:""> smallest positive integer </span><i><span style="font-family:"">d</span></i><span style="font-family:""> (<i>F</i>)</span><span style="font-size:8.0pt;line-height:102%;font-family:宋体;">≠</span><span style="font-family:"">1</span><span style="font-family:""> such that </span><i><span style="font-family:"">d </span></i><span style="font-family:"">(<i>F</i>)</span><span style="font-family:宋体;">‖</span><i><span style="font-family:"">F</span></i><span style="font-family:宋体;">‖</span><sub><span style="font-size:9px;line-height:102%;font-family:宋体;">∞</span></sub><span style="font-family:""> </span><span style="font-family:"">is always bigger than the sum of the operator norms of the Fourier coefficients of <i>F</i>. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality hold</span><span style="font-family:"">s</span><span style="font-family:""> up to the constant </span><span style="font-family:"">2<sup>n </sup></span><span style="font-family:"">for matrices of the algebra</span><span style="font-family:""> <i>M</i><sub>k </sub>(P<sub>n</sub>).</span><span style="font-family:""></span> </p> <br /> <span style="font-family:;" "=""></span>展开更多
This paper designs a segmentation method for an image based on its Fourier spectral data.An edge map is generated directly from the Fourier coefficients without first reconstructing the image in pixelated form.Consequ...This paper designs a segmentation method for an image based on its Fourier spectral data.An edge map is generated directly from the Fourier coefficients without first reconstructing the image in pixelated form.Consequently the internal boundaries of the edge map are not blurred by any(filtered)Fourier reconstruction.The edge map is then processed with an edge linking segmentation algorithm.We include examples from magnetic resonance imaging(MRI).Our results illustrate some potential benefits of using high order methods in imaging.展开更多
Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x...Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x) = Σx〈n〈2x Af(n)e(α√n).展开更多
Let L(s, sym2f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2, Z), with tf(n, 1) denoting the nthcoefficient of the Dirichlet series for it. It is proved that, for...Let L(s, sym2f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2, Z), with tf(n, 1) denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧(n)tf(n,1)e(nα)〈〈N exp (-c√logN,where ∧(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov's three primes theorem associated to the coefficients of Rankin-Selberg L-functions.展开更多
The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with dist...The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison's ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.展开更多
We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponent...We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg(n)e(αnβ), when β = 1/2 and |α| is close to 2√ q C Z+, where Ag(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 (Z). The similar natures of the divisor function 7(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.展开更多
In this paper we analyze a long standing problem of the appearance of spurious,non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures.The theory results in a s...In this paper we analyze a long standing problem of the appearance of spurious,non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures.The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem.We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coeffi-cients to small k components would lead to the appearance of non-physical solutions.We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution.This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures.Finally,based on the above requirement of small k,we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.展开更多
文摘In this paper,a sinusoidal signal frequency estimation algorithm is proposed by weighted least square method.Based on the idea of Provencher,three biggest Fourier coefficients in the maximum periodogram are considered,the Fourier coefficients can be written as three equations about the amplitude,phase,and frequency,and the frequency is estimated by solving equations.Because of the error of measurement,weighted least square method is used to solve the frequency equation and get the signal frequency.It is shown that the proposed estimator can approach the Cramer-Rao Bound(CRB)with a low Signal-to-Noise Ratio(SNR)threshold and has a higher accuracy.
文摘The order of magnitude of multiple Fourier coefficients of complex valued functions of generalized bounded variations like (∧^1,.. .,∧^N)BV^(p) and r-BV, over [0,2π]^ N, are estimated.
基金Supported by National Natural Science Foundation of China(Grant No.11101249)
文摘Abstract Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ Sk(F), In this paper, we established nontrivial estimates for ∑n≤x λf(n^i)λf(n^j),where 1≤ij≤4.
基金Acknowledgements This work was supported in part by the Natural Science Foundation of Jiangxi Province (Nos. 2012ZBAB211001, 20132BAB2010031).
文摘For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of
基金supported by State Committee Science MES RA,in frame of the research project N SCS 13-1A313
文摘In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems: for any ε∈ (0, 1), there exists a measurable set E C [0, 1) of measure bigger than 1 - s such that for any function f ∈ LI[0, 1), it is possible to find a function g ∈ L^1 [0, 1) coinciding with f on E and the absolute values of non zero Fourier coefficients of g with respect to the Vilenkin system are monotonically decreasing.
基金This work was supported in part by the Natural Science Foundation of Shandong Province (No. ZR2015AM016).
文摘Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.
文摘Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) =∑n≤xa(n).In this paper, we establish an asymptotic formula of tile fourth power moment of A(x) and prove that ∫1^TA^4(x)dx=3/64κπ^4s4;2(a^~)T^2κ+O(T^2a-δ4+t) with δ4 = 1/8, which improves the previous result.
基金Supported by the National Natural Science Foundation of China(51276017)
文摘Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.
文摘We prove that every homomorphism of the algebra P<sub><em>n</em></sub> into the algebra of operators on a Hilbert space is completely bounded. We show that the contractive homomorphism introduced by Parrott, which is not completely contractive, is completely bounded (similar to a completely contractive homomorphism). We also show that homomorphisms of the algebra <span style="white-space:normal;">P</span><sub style="white-space:normal;"><em>n</em></sub> generate completely positive maps over the algebras <em>C</em>(T<sup><em>n</em></sup>)and <em>M</em><sub>2</sub>(<em>C</em>(T<sup><em>n</em></sup>)).
文摘We prove that every matrix </span><i><span style="font-family:"">F</span></i><span style="font-size:6.5pt;line-height:102%;font-family:宋体;">∈</span><i><span style="font-family:"">M</span></i><sub><span style="font-family:"">k </span></sub><span style="font-family:"">(P<sub>n</sub>)</span><span style="font-family:""> is associated </span><span style="font-family:"">with</span><span style="font-family:""> </span><span style="font-family:"">the</span><span style="font-family:""> smallest positive integer </span><i><span style="font-family:"">d</span></i><span style="font-family:""> (<i>F</i>)</span><span style="font-size:8.0pt;line-height:102%;font-family:宋体;">≠</span><span style="font-family:"">1</span><span style="font-family:""> such that </span><i><span style="font-family:"">d </span></i><span style="font-family:"">(<i>F</i>)</span><span style="font-family:宋体;">‖</span><i><span style="font-family:"">F</span></i><span style="font-family:宋体;">‖</span><sub><span style="font-size:9px;line-height:102%;font-family:宋体;">∞</span></sub><span style="font-family:""> </span><span style="font-family:"">is always bigger than the sum of the operator norms of the Fourier coefficients of <i>F</i>. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality hold</span><span style="font-family:"">s</span><span style="font-family:""> up to the constant </span><span style="font-family:"">2<sup>n </sup></span><span style="font-family:"">for matrices of the algebra</span><span style="font-family:""> <i>M</i><sub>k </sub>(P<sub>n</sub>).</span><span style="font-family:""></span> </p> <br /> <span style="font-family:;" "=""></span>
基金This work was partially supported by NSF grants CNS 0324957,DMS 0510813DMS 0652833NIH grant EB 02553301(AG)The first author would also like to thank the ICOSAHOM committee for the invitation to speak at this conference.
文摘This paper designs a segmentation method for an image based on its Fourier spectral data.An edge map is generated directly from the Fourier coefficients without first reconstructing the image in pixelated form.Consequently the internal boundaries of the edge map are not blurred by any(filtered)Fourier reconstruction.The edge map is then processed with an edge linking segmentation algorithm.We include examples from magnetic resonance imaging(MRI).Our results illustrate some potential benefits of using high order methods in imaging.
基金This work is supported by the National Natural Science Foundation of China (Grant No. 10701048)
文摘Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x) = Σx〈n〈2x Af(n)e(α√n).
文摘Let L(s, sym2f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2, Z), with tf(n, 1) denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧(n)tf(n,1)e(nα)〈〈N exp (-c√logN,where ∧(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov's three primes theorem associated to the coefficients of Rankin-Selberg L-functions.
基金the Dirección General de Investigación of the Spanish Ministry for ScienceTechnology through the project BFM2003-05640/MATE and from the Departament d'Universitats,Recerca i Societat de la Informac
文摘The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison's ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11101239, 10971119), the Program for Changjiang Scholars and Innovative Research Team in University (IRT1264), and the Independent Innovation Foundation of Shandong University (Grant No. 2012ZRYQ005).
文摘We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg(n)e(αnβ), when β = 1/2 and |α| is close to 2√ q C Z+, where Ag(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 (Z). The similar natures of the divisor function 7(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.
文摘In this paper we analyze a long standing problem of the appearance of spurious,non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures.The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem.We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coeffi-cients to small k components would lead to the appearance of non-physical solutions.We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution.This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures.Finally,based on the above requirement of small k,we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.