Gegenbauer Expansion Research Articles

Dispersive derivation of the pion distribution amplitude

We derive the dependence of the leading-twist pion light-cone distribution amplitude (LCDA) on a parton momentum fraction $x$ by directly solving the dispersion relations for the moments with inputs from the operator product expansion (OPE) of the corresponding correlation function. It is noticed that these dispersion relations must be organized into those for the Gegenbauer coefficients first in order to avoid the ill-posed problem appearing in the conversion from the moments to the Gegenbauer coefficients. Given the values of various condensates in the OPE, we find solutions for the pion LCDA, which are convergent and stable in the Gegenbauer expansion. Moreover, the solution from summing contributions up to 18 Gegenbauer polynomials is smooth, and can be well approximated by a function proportional to ${x}^{p}(1\ensuremath{-}x{)}^{p}$ with $p\ensuremath{\approx}0.45$ at the scale $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$. Turning off the condensates, we get the asymptotic form for the pion LCDA as expected. We then solve for the pion LCDA at a different scale $\ensuremath{\mu}=1.5\text{ }\text{ }\mathrm{GeV}$ with the condensate inputs at this $\ensuremath{\mu}$, and demonstrate that the result is consistent with the one obtained by evolving the Gegenbauer coefficients from $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$ to 1.5 GeV. That is, our formalism is compatible with the QCD evolution. The strength of the above framework that goes beyond analyses limited to only the first few moments of a LCDA in conventional QCD sum rules is highlighted. The precision of our results can be improved systematically by including higher-order and higher-power terms in the OPE.

Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

On even-dimensional Euclidean space for integer powers of the positive Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on hyperspheres in arbitrary even dimensions.

Positivity of the Veneziano amplitude in D = 4

The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix bootstrap. However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious. In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function. We use this to read off an exact form of this coefficient on the leading Regge trajectory in D = 4. We find that it decays with the spin but always remains positive. Since for large spin these coefficients are expected to be smaller than those on the subleading trajectories, our result indicates the positivity of the full Veneziano amplitude in D = 4.

Analysis of multivariate Gegenbauer approximation in the hypercube

In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the d-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an lq ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parametrized PDEs.

Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities

In this paper, new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of Jacobi and Gegenbauer polynomial series are presented. For a class of functions with interior singularities, the decay of the coefficient is of the same asymptotic order for arbitrary $$\alpha ,\,\beta >-1$$, which confirms that the decay of the coefficients in the Jacobi polynomial series without normalization is a factor of $$ \sqrt{n}$$ slower compared with the Chebyshev expansion. While for functions with boundary singularities, the decay depends on $$\alpha $$ and $$\beta $$ with $$\alpha ,\,\beta >-1$$. For Gegenbauer expansion, it is related to the parameter $$\lambda $$ whatever f with interior or boundary singularities. All of these asymptotic analysis are optimal. Moreover, under the optimal asymptotic analysis, it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates, i.e., the same as the best polynomial approximation in the sense of absolute maximum error norm. Numerical examples illustrate the perfect coincidence with the estimates.

An Axisymmetric Torsion Problem of an Elastic Layer on a Rigid Circular Base

A solution is presented to a doubly mixed boundary value problem of the torsion of an elastic layer, partially resting on a rigid circular base by a circular rigid punch attached to its surface. This problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer expansion formula of the Bessel function we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. Both the two contact stresses under the punch and on the lower face of the layer are expressed as appropriate Chebyshev series. The effects of the radius of the disc with the rigid base and the layer thickness on the displacements, contact stresses as well as the shear stress and the stress singularity factor are discussed. A numerical application is also considered with some concluding results.

Accurate computation of screened Coulomb potential integrals in numerical Hartree–Fock programs

The frequently-used Zk and Yk functions in the framework of numerical Hartree–Fock method are formulated for atomic systems embedded in weakly-coupled plasmas modeled by Debye–Hückel potential. The Gegenbauer expansion is used to decompose the screened interelectronic Coulomb potential. Two numerical methods, the integration and differentiation ones, are developed to accurately and efficiently calculate the Zk and Yk integrals with each one showing special advantages in the implementation of numerical Hartree–Fock programs. Comparing to the analytical results specialized for Slater-type radial functions, it is generally concluded that the integration method possesses more accuracy and robustness than the differentiation one. Calculations on the two-electron C4+ ion in three different screening models are performed to demonstrate the applicability and stability of present methods in a wider range of screening parameters than existing Hartree–Fock calculations. With benchmark Hartree–Fock energies obtained in this work and accurate non-relativistic energies calculated based on Hylleraas-CI basis functions, it is found that the correlation energies are smallest in the fully screening model.

Double Gegenbauer expansion of |s–t|α

ABSTRACTMotivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function in terms of the ultraspherical polynomials and . Generalization, specialization, and limits of the expansion are also discussed.

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature.

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators (-Δ±β 2) in d-dimensional, R-radius hyperbolic and hyperspherical geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of (-Δ±β 2) on , (-Δ±β 2) on , and present two candidate fundamental solutions for (-Δ - β 2) on . Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

In this paper, we study optimal estimates and comparison of the coefficients in the Gegenbauer series expansion. We propose an alternative derivation of the contour integral representation of the Gegenbauer expansion coefficients which was recently derived by Cantero and Iserles [SIAM J. Numer. Anal., 50 (2012), pp. 307--327]. With this representation, we show that optimal estimates for the Gegenbauer expansion coefficients can be derived, which in particular includes Legendre coefficients as a special case. Further, we apply these estimates to establish some rigorous and computable bounds for the truncated Gegenbauer series. In addition, we compare the decay rates of the Chebyshev and Legendre coefficients. For functions whose singularity is outside or at the endpoints of the expansion interval, asymptotic behavior of the ratio of the $n$th Legendre coefficient to the $n$th Chebyshev coefficient is given, which provides us an illuminating insight for the comparison of spectral methods based on Legendre a...

Reconstruction of sparse Legendre and Gegenbauer expansions

We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony-like methods. The method proposed is robust with respect to noisy sampled data. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order.

Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

Computation of two-electron screened Coulomb potential integrals in Hylleraas basis sets

The Gegenbauer expansion and Taylor expansion methods are developed to accurately and efficiently calculate the two-electron screened Coulomb potential integrals in Hylleraas basis sets. The combination of these two methods covers the entire parameter space of the integrals, including arbitrary total angular momenta, two-electron configurations, powers of inter-electronic coordinate, and complex screening parameters. Numerical examples are given and comparisons with other computational methods in some restricted situations are made. The present methods can be easily applied to calculate the bound and resonant states of two-electron atoms or exotic three-body systems embedded in the screening environment by using the Hylleraas or Hylleraas-CI basis functions.

Poisson Wavelets on $$n$$ n -Dimensional Spheres

In this paper, Poisson wavelets on $n$-dimensional spheres, derived from Poisson kernel, are introduced and characterized. We compute their Gegenbauer expansion with respect to the origin of the sphere, as well as with respect to the field source. Further, we give recursive formulae for their explicit representations and we show how the wavelets are localized in space. Also their Euclidean limit is calculated explicitly and its space localization is described. We show that Poisson wavelets can be treated as wavelets derived from approximate identities and we give two inversion formulae.

Calculation of screened Coulomb potential matrices and its application to He bound and resonant states

We present two analytical methods, Taylor expansion and Gegenbauer expansion, to efficiently and accurately calculate the two-electron screened Coulomb potential matrix elements with Slater-type configuration-interaction basis functions. The former permits great advantages in fast computation of the potential matrices at small screening parameters and the latter allows accurate calculation of the matrices at all screening parameters. The bound and resonant states of a He atom embedded in the screening environment are calculated by employing the variational and complex-scaling methods, respectively, and the results are compared with other theoretical predictions. The expectation values of some physical quantities for He ground state are compared with the recent calculation of Ancarani and Rodriguez [Phys. Rev. A 89, 012507 (2014)] and extended to stronger screening environment. The energies and widths for the doubly excited resonant states are in good agreement with previous calculations, while the interelectronic angle $arccos\ensuremath{\langle}cos({\ensuremath{\theta}}_{12})\ensuremath{\rangle}$ show significant discrepancies with the Feshbach projection calculation of Ord\'o\~nez-Lasso et al. [Phys. Rev. A 88, 012702 (2013)]. The expectation values of $\ensuremath{\langle}{\mathbf{p}}_{\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{p}}_{\mathbf{2}}\ensuremath{\rangle}$ are also calculated for the resonant states investigated here. We conclude that the present methods in the framework of complex scaling enable us to get reliable energy, width, and other physical quantities of the resonant states in a variety of screening conditions.

Mode solver based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections

We present a modal method for the computation of eigenmodes of cylindrical structures with arbitrary cross sections. These modes are found as eigenvectors of a matrix eigenvalue equation that is obtained by introducing a new coordinate system that takes into account the profile of the cross section. We show that the use of Hertz potentials is suitable for the derivation of this eigenvalue equation and that the modal method based on Gegenbauer expansion (MMGE) is an efficient tool for the numerical solution of this equation. Results are successfully compared for both perfectly conducting and dielectric structures. A complex coordinate version of the MMGE is introduced to solve the dielectric case.

Strictly and non-strictly positive definite functions on spheres

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Hardy spaces for Dunkl–Gegenbauer expansions

We develop the harmonic analysis associated with the Dunkl–Gegenbauer expansions, which is in terms of the system {ϕnλ(eiθ),e−iθϕn−1λ(e−iθ)}, orthogonal with respect to the weight |sinθ|2λ on the circumference S1=∂D, where ϕnλ(eiθ)=Pnλ(cosθ)+i2λn+2λPn−1λ+1(cosθ)sinθ, and Pnλʼs are the Gegenbauer polynomials. This system is connected with the operators Tzf(z)=∂zf+λ[f(z)−f(z¯)]/(z−z¯) and Tz¯f(z)=∂z¯f−λ[f(z)−f(z¯)]/(z−z¯). The theory of the Hardy spaces Hλp(D) for p⩾p0:=2λ/(2λ+1) is studied, which extends the theory of Muckenhoupt and Stein from the upper half-disk to the whole disk. As a corollary, a remarkable generalization of Rieszʼs theorem is obtained. Under certain sharp estimates on the mean growth of Hλp(D) functions, a variety of inequalities is proved for all p>p0. The paper concludes with the Lp–Lq boundedness and the boundedness on weighted Morrey spaces of the associated Riesz potential Iλδf, by means of two different fractional maximal functions, and also the Hλp(D)–Hλq(D) boundedness of Iλδf for p0<p<(2λ+1)/δ and q−1=p−1−δ/(2λ+1).

Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method

This paper reports a novel direct Gegenbauer (ultraspherical) transcription method (GTM) for solving continuous-time optimal control (OC) problems (CTOCPs) with linear/nonlinear dynamics and path constraints. In (Elgindy et al. 2012) [1], we presented a GTM for solving nonlinear CTOCPs directly for the state and the control variables, and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment. This article extends the GTM to deal further with problems including higher-order time derivatives of the states by solving the CTOCP directly for the control u(t) and the highest-order time derivative x(N)(t),N∈Z+. The state vector and its derivatives up to the (N−1)th-order derivative can then be stably recovered by successive integration. Moreover, we present our solution method for solving linear–quadratic regulator (LQR) problems as we aim to cover a wider collection of CTOCPs with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature. The proposed numerical scheme fully parameterizes the state and the control variables using Gegenbauer expansion series. For problems with various order time derivatives of the state variables arising in the cost function, dynamical system, or path/terminal constraints, the GTM seeks to fully parameterize the control variables and the highest-order time derivatives of the state variables. The time horizon is mapped onto the closed interval [0,1]. The dynamical system characterized by differential equations is transformed into its integral formulation through direct integration. The resulting problem on the finite interval is then transcribed into a nonlinear programming (NLP) problem through collocation at the Gegenbauer–Gauss (GG) points. The integral operations are approximated by optimal Gegenbauer quadratures in a certain optimality sense. The reduced NLP problem is solved in the Gegenbauer spectral space, and the state and the control variables are approximated on the entire finite horizon. The proposed method achieves discrete solutions exhibiting exponential convergence using relatively small-scale number of collocation points. The advantages of the proposed direct GTM over other traditional discretization methods are shown through four well-studied OC test examples. The present work is a major breakthrough in the area of computational OC theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points, states and controls expansion terms. Moreover, the GTM produces very small-scale NLP problems, which can be solved very quickly using the modern NLP software.

Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation

In this paper we present an extension of the modal method by Gegenbauer expansion (MMGE) [J. Opt. Soc. Am. A28, 2006 (2011)], [Progress Electromagn. Res.133, 17 (2013)] to the study of nonperiodic problems. The nonperiodicity is introduced through the perfectly matched layers (PMLs) concept, which can be introduced in an equivalent way either by a change of coordinates or by the use of a uniaxial anisotropic medium. These PMLs can generate strong irregularities of the electromagnetic fields that can significantly alter the convergence and stability of the numerical scheme. This is the case, e.g., for the famous Fourier modal method, especially when using complex stretching coordinates. In this work, it will be shown that the MMGE equipped with PMLs is a robust approach because of its natural immunity against spurious modes.