Improvement by Iteration for Compact Operator Equations

The equation $y = f + Ky$ is considered in a separable Hilbert space H, with K assumed compact and linear. It is shown that every approximation to y of the form y₁ₙ = Σ ⁿaₙᵢuᵢ (where {uᵢ} is a given complete set in H, and the aₙᵢ,1 i n, are arbitrary numbers) is less accurate than the best approximation of the form y₂ₙ = f + Σ ⁿbₙᵢKuᵢ, if n is sufficiently large. Specifically it is shown that if y₁ₙ is chosen optimally (i.e. if the coefficients aₙᵢ are chosen to minimize \| y - y₁ₙ \|), and if y₂ₙ is chosen to be the first iterate of y₁ₙ, i.e. y₂ₙ = f + Ky₁ₙ, then \| y - y₂ₙ \| α ₙ \| y - y₁ₙ \|, with α ₙ → 0. A similar result is also obtained, provided the homogeneous equation $x = Kx$ has no nontrivial solution, if instead y₁ₙ is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms y₃ₙ,y₄ₙ, … obtained by further iteration is also shown to be valid, if the range of K is dense in H.