Evaluating polynomials over the unit disk and the unit ball
نویسندگان
چکیده
منابع مشابه
Orthogonal Polynomials and Partial Differential Equations on the Unit Ball
Orthogonal polynomials of degree n with respect to the weight function Wμ(x) = (1 − ‖x‖2)μ on the unit ball in R are known to satisfy the partial differential equation [ ∆− 〈x,∇〉 − (2μ+ d)〈x,∇〉 ] P = −n(n+ 2μ+ d)P for μ > −1. The singular case of μ = −1,−2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2,−3, . . . is shown to have comple...
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We establish asymptotic formulas for polynomials that are orthogonal over the unit disk with respect to a weight of the form |w(z)|2, where w(z) is a polynomial without zeros on the unit circle |z| = 1. The formulas put in evidence a strong connection between the behavior of the polynomials and the reproducing kernel of an associated weighted Bergman space, which produces interesting new featur...
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For a class of weight functions invariant under reflection groups on the unit ball, a family of orthogonal polynomials is defined via a Rodrigues type formula using the Dunkl operators. Their properties and their relation with several other bases are explored.
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An explicit family of polynomials on the unit ball B of R is constructed, so that it is an orthonormal family with respect to the inner product 〈f, g〉 = ρ Z Bd ∇f(x) · ∇g(x)dx + L(fg), where ρ > 0, ∇ is the gradient, and L(fg) is either the inner product on the sphere S or f(0)g(0).
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A family of orthonormal polynomials on the unit ball B of R with respect to the inner product
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ژورنال
عنوان ژورنال: Numerical Algorithms
سال: 2013
ISSN: 1017-1398,1572-9265
DOI: 10.1007/s11075-013-9817-5