### Abstract

The set of the regular and radiating spherical vector wave functions (SVWF) is shown to be complete in L_{2}(S) where S is a Lyapunov surface. The completeness fails when k^{2} is an eigenvalue of the Dirichlet problem for the Helmholtz equation (▽^{2} + k^{2})F = 0 in the region D_{i} bounded by S. On the other hand, the set of radiating SVWF is shown to be complete for all values of k^{2}. It is also proved that any vector function, which is continuous in D_{i} + S and satisfies the Helmholtz equation in D_{i}., can be approximated uniformly in D_{i} and in the mean square sense on S by a sequence of linear combinations of the regular SVWF (assuming the set is complete). Similar results are obtained for the exterior problem with the set of radiating SVWF. These results are extended to the set composed of the regular and radiating SVWF on two nonintersecting Lyapunov surfaces, one of which encloses the other.

Original language | English (US) |
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Pages (from-to) | 428-440 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 117 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 1 1986 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*117*(2), 428-440. https://doi.org/10.1016/0022-247X(86)90233-7