### 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) |
---|---|

Pages (from-to) | 428-440 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 117 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 1986 |

### Fingerprint

### 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

}

*Journal of Mathematical Analysis and Applications*, vol. 117, no. 2, pp. 428-440. https://doi.org/10.1016/0022-247X(86)90233-7

**On the completeness of the spherical vector wave functions.** / Aydin, Kultegin; Hizal, A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the completeness of the spherical vector wave functions

AU - Aydin, Kultegin

AU - Hizal, A.

PY - 1986/8/1

Y1 - 1986/8/1

N2 - The set of the regular and radiating spherical vector wave functions (SVWF) is shown to be complete in L2(S) where S is a Lyapunov surface. The completeness fails when k2 is an eigenvalue of the Dirichlet problem for the Helmholtz equation (▽2 + k2)F = 0 in the region Di bounded by S. On the other hand, the set of radiating SVWF is shown to be complete for all values of k2. It is also proved that any vector function, which is continuous in Di + S and satisfies the Helmholtz equation in Di., can be approximated uniformly in Di 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.

AB - The set of the regular and radiating spherical vector wave functions (SVWF) is shown to be complete in L2(S) where S is a Lyapunov surface. The completeness fails when k2 is an eigenvalue of the Dirichlet problem for the Helmholtz equation (▽2 + k2)F = 0 in the region Di bounded by S. On the other hand, the set of radiating SVWF is shown to be complete for all values of k2. It is also proved that any vector function, which is continuous in Di + S and satisfies the Helmholtz equation in Di., can be approximated uniformly in Di 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.

UR - http://www.scopus.com/inward/record.url?scp=0022767757&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0022767757&partnerID=8YFLogxK

U2 - 10.1016/0022-247X(86)90233-7

DO - 10.1016/0022-247X(86)90233-7

M3 - Article

AN - SCOPUS:0022767757

VL - 117

SP - 428

EP - 440

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -