Spherical functions on the Grassmann manifold and generalized Jacobi polynomials — part 1

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α,β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the gener...

Generating Functions of Jacobi Polynomials

Multiplicative renormalization method (MRM) for deriving generating functions of orthogonal polynomials is introduced by Asai–Kubo– Kuo. They and Namli gave complete lists of MRM-applicable measures for MRM-factors h(x) = ex and (1 − x)−κ. In this paper, MRM-factors h(x) for which the beta distribution B(p, q) over [0, 1] is MRM-applicable are determined. In other words, all generating function...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials

Product Formula for Jacobi Polynomials, Spherical Harmonics and Generalized Bessel Function of Dihedral Type

Abstract. We work out the expression of the generalized Bessel function of B2-type derived in [4]. This is done using Dijskma and Koornwinder’s product formula for Jacobi polynomials and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step toward the explici...

Classifying Handwritten Digits on the Grassmann Manifold

Empirical studies have shown that the collection of handwritten digits when acquired under a uniform condition forms a differentiable manifold which can be well approximated with linear structures. That is, each point on the manifold is associated with a geometry that parameterizes linear structures. Because of this, the problem of comparing a pair of digits can be turned into the problem of ca...