Orthogonal Polynomials with Symmetry of Order Three

1984 ◽  
Vol 36 (4) ◽  
pp. 685-717 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Sphere ◽  
Jacobi Polynomials ◽  
Orthogonal Basis ◽  
Lower Degree ◽  
Surface Measure ◽  
Rotation Invariant ◽  
General Measure ◽  
Invariant Surface ◽  
Image Position

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measurecorresponds to the measureon the triangle(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

scholarly journals On weighted estimates for Stein's maximal function

1996 ◽  
Vol 54 (1) ◽  
pp. 35-39 ◽  
Author(s):  
Hendra Gunawan
Keyword(s):  
Maximal Function ◽  
Unit Sphere ◽  
Surface Measure ◽  
Weighted Estimates ◽  
Mellin Transformation ◽  
Transformation Approach ◽  
Image Position

Let φ denote the normalised surface measure on the unit sphere Sn−1. We shall be interested in the weighted Lp estimate for Stein's maximal function Mφf, namelywhere w is an Ap weight, especially for 1 < p ≤ 2. Using the Mellin transformation approach, we prove that the estimate holds for every weight wδ where w ∈ Ap and 0 ≤ δ < (p(n − 1) − n)/(n(p − 1)), for n ≥ 3 and n/(n − 1) < p ≤ 2.

A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Function ◽  
Jacobi Polynomials ◽  
Defining Relation ◽  
Linearization Problem ◽  
Image Position ◽  
Linearization Coefficients ◽  
Negative Representation

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients

1971 ◽  
Vol 70 (2) ◽  
pp. 243-255 ◽  
Author(s):  
Richard Askey ◽  
George Gasper
Keyword(s):  
Orthogonal Polynomials ◽  
Harmonic Analysis ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position ◽  
Polynomial Expansions

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the typesand their dualshold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).

scholarly journals Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

2018 ◽  
Vol 33 (32) ◽  
pp. 1850187 ◽  
Author(s):  
I. A. Assi ◽  
H. Bahlouli ◽  
A. Hamdan
Keyword(s):  
Wave Function ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Basis Functions ◽  
Exact Solvability ◽  
Analytical Properties ◽  
Expansion Coefficients ◽  
Potential Parameters ◽  
Square Integrable Basis

This work aims at introducing two new solvable 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wave function is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. The expansion coefficients are then written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of which are yet to be derived. Moreover, we have computed the numerical eigenenergies for both potentials by considering specific choices of the potential parameters.

On (L1)* for General Measure Spaces

1963 ◽  
Vol 6 (2) ◽  
pp. 211-229 ◽  
Author(s):  
H. W. Ellis ◽  
D. O. Snow
Keyword(s):  
Measurable Function ◽  
Measure Space ◽  
Finite Measure ◽  
Signed Measure ◽  
General Measure ◽  
Measure Spaces ◽  
Arbitrary Measure ◽  
Image Position ◽  
Nikodym Theorem

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

On Sieved Orthogonal Polynomials II: Random Walk Polynomials

1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Orthogonal Polynomials ◽  
Transition Probabilities ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Markov Process ◽  
Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

Linearization of the Product of Jacobi Polynomials. II

1970 ◽  
Vol 22 (3) ◽  
pp. 582-593 ◽  
Author(s):  
George Gasper
Keyword(s):  
Harmonic Analysis ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position

Let [3, p. 170, (16)](1.1)denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by(1.2)where Rn(α, β)(x) = Pn(α, β)(x)/Pn(α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.Question 1. For which (α, β) do we have(1.3)Question 2. For which (α, β) do we have(1.4)where G depends only on (α, β)?

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