Refinement of Fourier Coefficients from the Stokes Deconvoluted Profile

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.

STUDY ON FREQUENCY ESTIMATION BASED ON WEIGHTED LEAST SQUARE METHOD WITH THREE FOURIER COEFFICIENTS

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.

Order of Magnitude of Multiple Fourier Coefficients

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.

The Cancellation of Fourier Coefficients of Cusp Forms over Different Sparse Sequences

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.

On the Third and Fourth Power Moments of Fourier Coefficients of Cusp Forms

The asymptotic formulae for the third and fourth power moments of Fourier coefficients of cusp forms are proved in this paper.

Quadratic forms connected with Fourier coefficients of Maass cusp forms

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

ON THE FOURIER-VILENKIN COEFFICIENTS

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.

Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions

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.

Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL_m(Z)

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.

Fourier coefficients of Zygmund functions and analytic functions with quasiconformal deformation extensions

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.

On the Distribution and Moments of the Fourier Coefficients of Cusp Forms

Two theorems about the distribution and moments of the Fourier coefficients of cusp forms are proved in this paper.

On the Fourth Power Moment of Fourier Coefficients of Cusp Form

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.

Quadrature formulas for Fourier-Chebyshev coefficients

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.

A Biproportional Construction Algorithm for Correctly Calculating Fourier Series of Aperiodic Non-Sinusoidal Signal

The Fourier series (FS) applies to a periodic non-sinusoidal function satisfying the Dirichlet conditions, whereas the being-processed function in practical applications is usually an aperiodic non-sinusoidal signal. When is aperiodic, its calculated FS is not correct, which is still a challenging problem. To overcome the problem, we derive a direct calculation algorithm, a constant iteration algorithm, and an optimal iteration algorithm. The direct calculation algorithm correctly calculates its Fourier coefficients (FCs) when is periodic and satisfies the Dirichlet conditions. Both the constant iteration algorithm and the optimal iteration algorithm provide an idea of determining the states of . From the idea, we obtain an algorithm for determining the states of based on the optimal iteration algorithm. In the algorithm, the variable iteration step is introduced; thus, we present an algorithm for determining the states of based on the variable iteration step. The presented algorithm accurately determines the states of . On the basis of these algorithms, we build a biproportional construction theory. The theory consists of a first and a second proportional construction theory. The former correctly calculates the FCs of at the present sampling time

Recurrent formula of Bernoulli numbers and the relationships among the coefficients of beam,Bernoulli numbers and Euler numbers

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.

Otolith shape analysis for stock discrimination of two Collichthys genus croaker(Pieces:Sciaenidae,)from the northern Chinese coast

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.

On Kadison’s Similarity Problem for Homomorphisms of the Algebra of Complex Polynomials

We prove that every homomorphism of the algebra Pn 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 Pn generate completely positive maps over the algebras C(Tn)and M2(C(Tn)).

On Von Neumann’s Inequality for Matrices of Complex Polynomials

We prove that every matrix F∈Mk (Pn) is associated with the smallest positive integer d (F)≠1 such that d (F)‖F‖∞ is always bigger than the sum of the operator norms of the Fourier coefficients of F. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality holds up to the constant 2n for matrices of the algebra Mk (Pn).

A Note on the Solution of Water Wave Scattering Problem Involving Small Deformation on a Porous Channel-Bed

The solution of water wave scattering problem involving small deformation on a porous bed in a channel, where the upper surface is bounded above by an infinitely extent rigid horizontal surface, is studied here within the framework of linearized water wave theory. In such a situation, there exists only one mode of waves propagating on the porous surface. A simplified perturbation analysis, involving a small parameter ε （≤1） , which measures the smallness of the deformation, is employed to reduce the governing Boundary Value Problem （BVP） to a simpler BVP for the first-order correction of the potential function. The first-order potential function and, hence, the first-order reflection and transmission coefficients are obtained by the method based on Fourier transform technique as well as Green＇s integral theorem with the introduction of appropriate Green＇s function. Two special examples of bottom deformation： the exponentially damped deformation and the sinusoidal ripple bed, are considered to validate the results. For the particular example of a patch of sinusoidal ripples, the resonant interaction between the bed and the upper surface of the fluid is attained in the neighborhood of a singularity, when the ripples wavenumbers of the bottom deformation become approximately twice the components of the incident field wavenumber along the positive x -direction. Also, the main advantage of the present study is that the results for the values of reflection and transmission coefficients are found to satisfy the energy-balance relation almost accurately.

Segmentation of Images from Fourier Spectral Data

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.