Improvement by iteration for compact operator equations

Abstract

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