On the completeness of the spherical vector wave functions

K. Aydin; A. Hizal

doi:10.1016/0022-247X(86)90233-7

On the completeness of the spherical vector wave functions

K. Aydin
, A. Hizal

Research output: Contribution to journal › Article › peer-review

29 Scopus citations

Abstract

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 D_i 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 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	https://doi.org/10.1016/0022-247X(86)90233-7
State	Published - Aug 1 1986

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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

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title = "On the completeness of the spherical vector wave functions",

abstract = "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.",

author = "K. Aydin and A. Hizal",

note = "Funding Information: Partial support for this researchw as provided by National Science Foundation Grant ATM-8003376.T he authors would like to expresst heir appreciationt o T. A. Seliga for his encouragemenatn d support during the preparationo f this paper.",

year = "1986",

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doi = "10.1016/0022-247X(86)90233-7",

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T1 - On the completeness of the spherical vector wave functions

AU - Aydin, K.

AU - Hizal, A.

N1 - Funding Information: Partial support for this researchw as provided by National Science Foundation Grant ATM-8003376.T he authors would like to expresst heir appreciationt o T. A. Seliga for his encouragemenatn d support during the preparationo f this paper.

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.

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

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On the completeness of the spherical vector wave functions

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