On the completeness of the spherical vector wave functions

Kultegin Aydin, A. Hizal

    Research output: Contribution to journalArticle

    19 Citations (Scopus)

    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.

    Original languageEnglish (US)
    Pages (from-to)428-440
    Number of pages13
    JournalJournal of Mathematical Analysis and Applications
    Volume117
    Issue number2
    DOIs
    StatePublished - Aug 1 1986

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    Wave functions
    Wave Function
    Completeness
    Helmholtz equation
    Helmholtz Equation
    Lyapunov
    Exterior Problem
    Mean Square
    Dirichlet Problem
    Linear Combination
    Eigenvalue

    All Science Journal Classification (ASJC) codes

    • Analysis
    • Applied Mathematics

    Cite this

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

    In: Journal of Mathematical Analysis and Applications, Vol. 117, No. 2, 01.08.1986, p. 428-440.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - On the completeness of the spherical vector wave functions

    AU - Aydin, Kultegin

    AU - Hizal, A.

    PY - 1986/8/1

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    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.

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