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 ${y_{1n}} = {\Sigma ^n}{a_{ni}}{u_i}$ (where {${u_i}$} is a given complete set in H, and the ${a_{ni}},1 \leqslant i \leqslant n$, are arbitrary numbers) is less accurate than the best approximation of the form ${y_{2n}} = f + {\Sigma ^n}{b_{ni}}K{u_i}$, if n is sufficiently large. Specifically it is shown that if ${y_{1n}}$ is chosen optimally (i.e. if the coefficients ${a_{ni}}$ are chosen to minimize $\left \| {y - {y_{1n}}} \right \|$), and if ${y_{2n}}$ is chosen to be the first iterate of ${y_{1n}}$, i.e. ${y_{2n}} = f + K{y_{1n}}$, then $\left \| {y - {y_{2n}}} \right \| \leqslant {\alpha _n}\left \| {y - {y_{1n}}} \right \|$, with ${\alpha _n} \to 0$. A similar result is also obtained, provided the homogeneous equation $x = Kx$ has no nontrivial solution, if instead ${y_{1n}}$ is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms ${y_{3n}},{y_{4n}}, \ldots$ obtained by further iteration is also shown to be valid, if the range of K is dense in H.