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Unformatted text preview: on L 2 (0 , 2 π ) so we know that it has a complete set oF eigenFunctions, e k , with eigenvalues τ k = 0 . ±rom the discussion above we then know that each e k is actually continuous – since it is Aw with w ∈ L 2 (0 , 2 π ) and hence also twice continuously diﬀerentiable. So indeed, these e k satisFy the eigenvalue problem (with Dirichlet boundary conditions) with eigenvalues (22.3) T k = τ k − 1 + c → ∞ as k → ∞ . The solvability part also Follows much the same way....
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- Spring '09
- Linear Algebra, Eigenvalue, eigenvector and eigenspace, Hilbert space, Eigenfunction, Dirichlet boundary condition