Convergence Rates for Approximate Eigenvalues of Compact Integral Operators

Let K be an integral operator and \ Kₙ \ a sequence of numerical integral operators approximating K. Let λ ₀ ≠ 0 be an eigenvalue of K of multiplicity m and index ν, and let σ ₙ be the eigenvalues of Kₙ within some small fixed neighborhood of λ ₀. Then for some $c > 0$ and all sufficiently large n, \[ | {λ - λ _0 } | cmax \{ {\| {Kφ _i - K_n φ _i } \|^{{1 / ν }} |1 i m} \} \] for all λ ∈ σ ₙ. The set \ φ ₁ , ⋯ ,φ ₘ \ is a basis for null (λ ₀ - K)^ν.