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Astron. Astrophys. 321, 1024-1026 (1997)

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3. The Hilbert space of the displacement field

Let [FORMULA] be a function space whose elements are [FORMULA] and in which the inner product is defined as

[EQUATION]

where the integration is over the volume of the star. [FORMULA] on [FORMULA] is self adjoint, [FORMULA]. There follows the eigenvalue problem

[EQUATION]

where [FORMULA] is the eigenfrequency of an oscillation mode, [FORMULA] is its eigendisplacement vector, and n is a collection of three indices, indicating the three wave numbers in, say, [FORMULA] directions of a spherical polar coordinates. Furthermore, [FORMULA] is an orthogonal set and can be normalized to unity,

[EQUATION]

The set [FORMULA] is also complete and may serve as a basis for [FORMULA]. See Dixit et al. (1980). Thus, any [FORMULA] may be expanded in a unique way in terms of [FORMULA]. Numerical values of [FORMULA] for hypothetical or actual star models are abundant in the astronomical literature of seventies and eighties.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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