Expansion formulas and addition theorems for Gegenbauer functions

Durand, Loyal; Fishbane, Paul M.; Simmons, L. M.

doi:10.1063/1.522831

Cited by 64 publications

(51 citation statements)

References 9 publications

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“…In the framework of the dimensional regularization a natural extension of the Chebyshev polynomial is the Gegenbauer polynomial C λ n (cos θ) (see [26]- [29]) with the index λ related to the transverse space dimension D − 2 as λ = (D − 2)/2 − 1 = D/2 − 2. When λ = 0 we return to the Chebyshev polynomials because…”

Section: The Technique Of Calculationsmentioning

confidence: 99%

NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories

Kotikov¹,

Lipatov²

2000

Nuclear Physics B

295

360

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We study next-to-leading corrections to the integral kernel of the BFKL equation for high energy cross-sections in QCD and in supersymmetric gauge theories. The eigenvalue of the BFKL kernel is calculated in an analytic form as a function of the anomalous dimension γ of the local gauge-invariant operators and their conformal spin n. For the case of an extended N = 4 SUSY the kernel is significantly simplified. In particular, the terms non-analytic in n are canceled. We discuss the relation between the DGLAP and BFKL equations in the N = 4 model.

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Section: The Technique Of Calculationsmentioning

confidence: 99%

NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories

Kotikov¹,

Lipatov²

2000

Nuclear Physics B

295

360

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“…(2.3) and (2.8), and also [11], p.1045, for. 8.936 (1) For the functions P λ α (ch x) and C λ α (ch x) the following formulas (see [9], p. 1939) are valid: 6) where α − n = −1, −2, . .…”

Section: The Properties Of Generalized Gegenbauer Shift and Statementmentioning

confidence: 99%

Gegenbauer Transformations Nikolski-Besov Spaces Generalized by Gegenbauer Operator and Their Approximation Characteristics

Guliyev¹,

Ibrahimov²,

Jafarova³

2017

AAN

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In this paper we consider some problems of the theory of approximation of functions on interval [0, ∞) in the metric of L p,λ with weight sh 2λ x using generalized Gegenbauer shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Gegenbauer shifts. We establish the equivalence of the modulus of smoothness and K-functional, defined in terms of the space of the Sobolev type corresponding to the Gegenbauer differential operator. We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use some functions classes of spectrum. In these classes, we prove analogues of Bernstein's inequality and others for the Gegenbauer differential operator. Our results are analogues of the results for generalized Bessel shifts obtained in the work [30].Keywords: Approximation of functions, generalized Gegenbauer shift, Gegenbauer transformation, Nikol'skii-Besov type spaces, embedding theorems.In the classical theory of approximation of functions on R = (−∞, ∞) the shift operator f (x) → f (x + y), x, y ∈ R. plays a central role. This shift operator is used in the construction of the moduli of continuity and smoothness, which are the basic elements of the direct and inverse theorems of approximation theory. Various generalizations of shift operators enable to state natural analogues of problems in approximation theory. Groups and semigroups of operators on Banach spaces are generalizations of the shift operator. Different problems of approximation theory on Banach spaces with groups and semigroups of operators were considered in [1,3,5,7,44].Generalized shift operator naturally follows from "addition theorem" for eigen functions of the differential operators (for example, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite and other). These operators may not form a group or semigroup, but the generalized moduli of smoothness defined in terms of them can be better adapted to the study of relations between the smoothness properties of functions and the best approximations of these functions in weighted function spaces. Some results on the best approximation of functions using generalized shift operators can be found in [1, 4, 22-26, 29-39, 45]. Note that most of the papers on this topic deal with the approximation of functions by polynomials on a finite segment. For the half-line, most popular examples are the generalized Bessel and Dunkl shifts (see for example [4,23,[30][31][32]). Fourier-Bessel and Fourier-Dunkl harmonic analysis, which deals with Bessel and Dunkl integral transformations and their applications, are closely connected with the generalized Bessel and Dunkl shift. Moreover, generalized Bessel shift is widely used in the potential theory and theory of maximal functions (see, example [5,[11][12][13]). The file of constructions of the theory of generalized shift operators generalize in the theory of transformation of operators (see for example [7]). We reduce only ...

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“…M2P can be expressed as a product of two matrices L2P N ×(p+1) 2 M2L (p+1) 2 ×(p+1) 2 using (5) in a similar manner as in (4). The entries of L2P depend on the observation points, and the entries of M2L depend on the locations of the centers of the source and the observation spheres.…”

Section: Source Spherementioning

confidence: 99%

“…One may be tempted to use the following addition theorem for the Gegenbauer polynomials which is well known [2,4]:…”

Section: Addition Theorems Frommentioning

confidence: 99%

Single Level Multipole Expansions and Operators for Potentials of the Form $r^-\lambda$

Chowdhury¹,

Jandhyala²

2005

SIAM J. Sci. Comput.

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This paper presents the generalized multipole, local, and translation operators for three-dimensional static potentials of the form r −λ , where λ is any real number. Addition theorems are developed using Gegenbauer polynomials. Multipole expansions and error bounds are presented in a manner similar to those for truncated classical multipole expansions. Numerical results showing error behavior versus number of terms, distance, and λ are presented. Introduction.The N-body three dimensional (3D) problem involving the Coulombic r −1 Green's function has been successfully accelerated using the FMM (O(N ) method) by Greengard [1] and other related techniques [5,7,8]. These techniques improve drastically over the classical O(N 2 ) method by efficiently clustering sources and observers in a multilevel manner. Furthermore, the FMM is error-controllable; i.e., the truncation and approximation errors can be predicted in an a priori manner by choosing a specific number of terms in multipole and local expansions.A related area of research has been the development of FMM-like methods based on plane-wave expansions and variations [6] for oscillatory kernels arising in dynamic electromagnetics and acoustics. Apart from the r −1 and dynamic oscillatory kernels, another class includes potential functions of the form r −λ , where λ is a positive integer. For example, the van der Waals forces, Lennard-Jones potentials, and H-bonds have the forms r −6 , r −12 , and r −10 and have important applications in chemistry [10], molecular dynamics [11] and fluid mechanics [12]. Present computational approaches rely heavily on FMM or related methods for Coulombic interactions, but do not have the same approaches for the van der Waals forces, owing to a lack of exact multipole expansions. (Other approaches based on precorrected FFT on a uniform grid are topics of current research.) A generalized FMM technique for nonoscillatory kernels based on singular value decomposition has been presented in [9]. In this paper, to the best of our knowledge, for the first time analytic multipole expansions are developed for potential functions of the form r −λ , for all real λ. In a manner analogous to the classical multipole expansions for electrostatic potentials, these expansions are errorcontrollable and enable efficient clustering of sources and observers. In the multipole method presented in [1] spherical harmonics are used. In this paper the well-known Gegenbauer polynomials are used instead to deduce the necessary addition theorems for source-clustering, observer-clustering, and cluster-cluster interactions. In doing so all the necessary operators for a single level FMM for functions of the form r −λ are obtained.

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Expansion formulas and addition theorems for Gegenbauer functions

Cited by 64 publications

References 9 publications

NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories

NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories

Gegenbauer Transformations Nikolski-Besov Spaces Generalized by Gegenbauer Operator and Their Approximation Characteristics

Single Level Multipole Expansions and Operators for Potentials of the Form $r^-\lambda$

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