IJPAM: Volume 73, No. 4 (2011)

A STUDY OF DIFFERENTIAL OPERATORS FOR
PARTICULAR COMPLETE ORTHONORMAL
SYSTEMS ON A 3D BALL

M. Akram$^1$, I. Amina$^2$, V. Michel$^3$
$^{1,2}$Department of Mathematics
GC University
Katchery Road, Lahore, 54000, PAKISTAN
$^3$Department of Mathematics
University of Siegen
Siegen, 57068, GERMANY


Abstract. In this article, we introduce a class of differential operators for two complete orthonormal systems in $\mbox{L}^{2}(\cal B)$, where $\cal B$ is a ball in $\R^{3}$, such that these orthonormal systems are eigenfunctions. We study further properties of these operators. It turns out, for instance, that the Sobolev norm, which is used in geomathematics for a spline interpolation and approximation method on ${\cal B}$, can be interpreted as the ${\mathrm{L}}^2({\cal B})$-norm of the image of a (pseudo-)differential operator. This result justifies an analogy of the splines on the ball to their counterparts on the real line and the sphere.

Received: November 15, 2011

AMS Subject Classification: 33C45, 33C50, 33C55, 33E30, 46E35

Key Words and Phrases: ball, complete orthonormal system, differential operators, eigenfunctions, orthogonal polynomials, Sobolev space, spline

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 73
Issue: 4