Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)at...Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.展开更多
Computer-aided experimental technique was used to study the Stokes deconvolution of X-ray diffraction profile. Considerable difference can be found between the Fourier coefficients obtained from the deconvolutions of ...Computer-aided experimental technique was used to study the Stokes deconvolution of X-ray diffraction profile. Considerable difference can be found between the Fourier coefficients obtained from the deconvolutions of singlet and doublet experimental profiles. Nevertheless, the resultant physical profiles corresponding to singlet and doublet profiles are identical. An approach is proposed to refine the Fourier coefficients, and the refined Fourier coefficients coincide well with that obtained from the deconvolution of singlet experimental profile.展开更多
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.
A free-form lens(FFL)is a special surface that is difficult to be fabricated.FFL are usually fabricated by computer-controlled optical surfacing(CCOS)technique.During CCOS,the material removal amount is determined by ...A free-form lens(FFL)is a special surface that is difficult to be fabricated.FFL are usually fabricated by computer-controlled optical surfacing(CCOS)technique.During CCOS,the material removal amount is determined by the unit removal function(URF)convoluting with the input data――the dwell time.When the removal amount and the URF are known,how to solve the input data involves the special algorithm――deconvolution.Usually,the input data are solved by virtue of low pass filter or iterative methods.However,an approximation solution would destroy the machining stability nec-essary to perform precisely CCOS.In this paper,to solve the input data,a new method that is based on the finite Fourier coefficient is put forward.It can give a continuous and accurate solution for fabricating choice.Several parameters are simulated,which influ-ence the process of CCOS,and evaluating on the effect of the method is also carried out.Experimental results verify that this method is suitable for guiding the manufacturing of high precision FFL.展开更多
Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of t...Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.展开更多
Let f be a Maass eigenform that is a new form of level N onΓ0(N),with Laplace eigenvalue 1/4+νf2.Then,all its Fourier coefficients{λf(n)}∞n=1 are real,and we may normalize so thatλf(1)=1.It is proved,in this pape...Let f be a Maass eigenform that is a new form of level N onΓ0(N),with Laplace eigenvalue 1/4+νf2.Then,all its Fourier coefficients{λf(n)}∞n=1 are real,and we may normalize so thatλf(1)=1.It is proved,in this paper,that the first sign change in the sequence{λf(n)}n∞=1 occurs at some n satisfying n《{(3+|νf|2)N}1/2-δand(n,N)=1.This generalizes the previous results for holomorphic Hecke eigenforms.展开更多
An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu...An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu¨ller space.By a complex analysis approach,we give a characterization for the Fourier coefficients of a Zygmund function by a quadratic form.Some related topics are also discussed,including those analytic functions with quasiconformal deformation extensions.展开更多
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
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
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
The aim of this work is to construct a new quadrature formula for Fourier Chebyshev coef ficients based on the divided differences of the integrand at points 1, 1 and the zeros of the n th Chebyshev polynomial o...The aim of this work is to construct a new quadrature formula for Fourier Chebyshev coef ficients based on the divided differences of the integrand at points 1, 1 and the zeros of the n th Chebyshev polynomial of the second kind. The interesting thing is that this quadrature rule is closely related to the well known Gauss Turn quadrature formula and similar to a recent result of Micchelli and Sharma, extending a particular case due to Micchelli and Rivlin.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
The otolith morphology of two croaker species(C ollichthys lucidus and C ollichthys niveatus) from three areas(Liaodong Bay, LD; Huanghe(Yellow) River estuary, HRE; Jiaozhou Bay, JZ) along the northern Chinese coast w...The otolith morphology of two croaker species(C ollichthys lucidus and C ollichthys niveatus) from three areas(Liaodong Bay, LD; Huanghe(Yellow) River estuary, HRE; Jiaozhou Bay, JZ) along the northern Chinese coast were investigated for species identifi cation and stock discrimination. The otolith contour shape described by elliptic Fourier coefficients(EFC) were analysed using principal components analysis(PCA) and stepwise canonical discriminant analysis(CDA) to identify species and stocks. The two species were well dif ferentiated, with an overall classifi cation success rate of 97.8%. And variations in the otolith shapes were significant enough to discriminate among the three geographical samples of C. lucidus(67.7%) or C. niveatus(65.2%). Relatively high mis-assignment occurred between the geographically adjacent LD and HRE samples, which implied that individual mixing may exist between the two samples. This study yielded information complementary to that derived from genetic studies and provided information for assessing the stock structure of C. lucidus and C. niveatus in the Bohai Sea and the Yellow Sea.展开更多
文摘Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.
文摘Computer-aided experimental technique was used to study the Stokes deconvolution of X-ray diffraction profile. Considerable difference can be found between the Fourier coefficients obtained from the deconvolutions of singlet and doublet experimental profiles. Nevertheless, the resultant physical profiles corresponding to singlet and doublet profiles are identical. An approach is proposed to refine the Fourier coefficients, and the refined Fourier coefficients coincide well with that obtained from the deconvolution of singlet experimental profile.
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This work was supported by the National Natural Science Foundation of China(Grant Nos.50175062,60508001).
文摘A free-form lens(FFL)is a special surface that is difficult to be fabricated.FFL are usually fabricated by computer-controlled optical surfacing(CCOS)technique.During CCOS,the material removal amount is determined by the unit removal function(URF)convoluting with the input data――the dwell time.When the removal amount and the URF are known,how to solve the input data involves the special algorithm――deconvolution.Usually,the input data are solved by virtue of low pass filter or iterative methods.However,an approximation solution would destroy the machining stability nec-essary to perform precisely CCOS.In this paper,to solve the input data,a new method that is based on the finite Fourier coefficient is put forward.It can give a continuous and accurate solution for fabricating choice.Several parameters are simulated,which influ-ence the process of CCOS,and evaluating on the effect of the method is also carried out.Experimental results verify that this method is suitable for guiding the manufacturing of high precision FFL.
基金supported by National Natural Science Foundation of China(Grant No.10971119)Program for Changjiang Scolars and Innovative Research Team in University(Grant No.1264)
文摘Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.
文摘Let f be a Maass eigenform that is a new form of level N onΓ0(N),with Laplace eigenvalue 1/4+νf2.Then,all its Fourier coefficients{λf(n)}∞n=1 are real,and we may normalize so thatλf(1)=1.It is proved,in this paper,that the first sign change in the sequence{λf(n)}n∞=1 occurs at some n satisfying n《{(3+|νf|2)N}1/2-δand(n,N)=1.This generalizes the previous results for holomorphic Hecke eigenforms.
基金supported by National Natural Science Foundation of China (Grant No.11071179)
文摘An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu¨ller space.By a complex analysis approach,we give a characterization for the Fourier coefficients of a Zygmund function by a quadratic form.Some related topics are also discussed,including those analytic functions with quasiconformal deformation extensions.
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
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 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The aim of this work is to construct a new quadrature formula for Fourier Chebyshev coef ficients based on the divided differences of the integrand at points 1, 1 and the zeros of the n th Chebyshev polynomial of the second kind. The interesting thing is that this quadrature rule is closely related to the well known Gauss Turn quadrature formula and similar to a recent result of Micchelli and Sharma, extending a particular case due to Micchelli and Rivlin.
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.
基金Supported by the National Key Basic Research Program of China(973 Program)(No.2015CB453302)the NSFC-Shandong Joint Fund for Marine Science Research Centre(No.U1606404)the Aoshan Science and Technology Innovation Project(No.2015ASKJ02-04)
文摘The otolith morphology of two croaker species(C ollichthys lucidus and C ollichthys niveatus) from three areas(Liaodong Bay, LD; Huanghe(Yellow) River estuary, HRE; Jiaozhou Bay, JZ) along the northern Chinese coast were investigated for species identifi cation and stock discrimination. The otolith contour shape described by elliptic Fourier coefficients(EFC) were analysed using principal components analysis(PCA) and stepwise canonical discriminant analysis(CDA) to identify species and stocks. The two species were well dif ferentiated, with an overall classifi cation success rate of 97.8%. And variations in the otolith shapes were significant enough to discriminate among the three geographical samples of C. lucidus(67.7%) or C. niveatus(65.2%). Relatively high mis-assignment occurred between the geographically adjacent LD and HRE samples, which implied that individual mixing may exist between the two samples. This study yielded information complementary to that derived from genetic studies and provided information for assessing the stock structure of C. lucidus and C. niveatus in the Bohai Sea and the Yellow Sea.
