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A G p(L according to algebraic adjoint algebraic multiplicity AnnAnnAn AnRx(Ln approximating the eigenvalues asymptotic formula Aw n)I Banach space boundary conditions CN CN Consider the eigenvalue consider the problem countable set counted according denote diagonal elements dimension dimensional domain eigenvalue problem eigenvalues and eigenvectors equal error estimates Example 7.l f 6 H f G S Gram matrix H H H Hence Hermitian form Hilbert space implies inequality in Lemma inner product integer intersect least eigenvalue Lemma 3.l Let h let Lf Lf,f Mathieu equation matrix mutually disjoint n l<q<n q q null space number of eigenvalues operator defined operator norm ordinary differential operators orthogonal P P P paper positive definite proof of Lemma q eigenvalues RA(T range of P(t result Section self-adjoint operator shows single precision Sobolev space Sturm type Table Theorem 4.4 unique positive solution vector