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Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel

Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel Through the eigenvalue problem we associate to the classical orthogonal polynomials two classes of weighted Riemannian $$1$$ 1 -manifolds having the coordinate $$x$$ x . For the first class the eigenvalues contains $$x$$ x and the metric is fixed as being the Euclidean one while for the second class the eigenvalues are independent of this variable and the metric and weight function are founded. The Hermite polynomials is the only case which generates the same manifold. The geometry of second class of weighted manifolds is studied from several points of view: geodesics, distance and exponential map, harmonic functions and their energy density, volume, zeta function, heat kernel. A partial heat equation is studied for these metrics and for the Poincaré ball model of hyperbolic geometry. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel

Analysis and Mathematical Physics , Volume 5 (4) – Apr 18, 2015

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-015-0102-8
Publisher site
See Article on Publisher Site

Abstract

Through the eigenvalue problem we associate to the classical orthogonal polynomials two classes of weighted Riemannian $$1$$ 1 -manifolds having the coordinate $$x$$ x . For the first class the eigenvalues contains $$x$$ x and the metric is fixed as being the Euclidean one while for the second class the eigenvalues are independent of this variable and the metric and weight function are founded. The Hermite polynomials is the only case which generates the same manifold. The geometry of second class of weighted manifolds is studied from several points of view: geodesics, distance and exponential map, harmonic functions and their energy density, volume, zeta function, heat kernel. A partial heat equation is studied for these metrics and for the Poincaré ball model of hyperbolic geometry.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Apr 18, 2015

References