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- W2057130561 abstract "Let L denote the linear differential operator $Ly = a_0 y^{(n)} + a_1 y^{(n - 1)} + cdots + a_{n - 1} y^{(1)} + a_n y$, where each $a_j $ is a complex-valued function of class $C^{n - j} $ on the closed interval $[a,b]$, and $a_0 (x) ne 0$ for $x in [a,b]$. Let $B_1 ,B_2 , cdots ,B_n $ be linearly independent boundary forms, and suppose that the eigenvalue problem $Ly = lambda y,B_1 y = B_2 y = cdots = B_n y = 0$ is self-adjoins. With each such ,eigenvalue problem there is associated a series expansion whose coefficients are boundary values. For the case in which 0 is not an eigenvalue of the problem, the expansion has the form $f(x) = sum _{k = 0}^infty sum _{j = 1}^infty (B_j L^k f)p_{nk + j} (x)$, which is valid for suitably restricted functions f defined on $[a,b]$. The functions ${ p_k } _{k = 1}^infty $ are developed from solutions of the homogeneous equation $Ly = 0$ and the Green’s function. If the problem has the eigenvalue 0, the representation takes the similar form $f(x) = q_0 (x) + sum _{k = 0}^infty sum _{j = 1}^infty (U_j L^k f)q_{nk + j} (x)$, where $q_0 $ is an eigenfunction corresponding to $lambda = 0$ and where the boundary forms $U_1 ,U_2 , cdots ,U_n $ are linear combinations of $B_1 ,B_2 , cdots ,B_n $. For each expansion, necessary and sufficient conditions for convergence are given in terms of the magnitude of an eigenvalue nearest the origin." @default.
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- W2057130561 date "1976-05-01" @default.
- W2057130561 modified "2023-10-05" @default.
- W2057130561 title "Series Expansions and Linear Differential Operators" @default.
- W2057130561 doi "https://doi.org/10.1137/0507025" @default.
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