Improvement by Iteration for Compact Operator Equations

The equation y = / + Ky is considered in a separable Hubert space H, with K assumed compact and linear. It is shown that every approximation to y of the form yXn = £nanl«. (where {u-} is a given complete set in H, and the an¡, 1 < / < n, are arbitrary numbers) is less accurate than the best approximation of the form y2n = f + ZnbnjKUj, if ii is sufficiently large. Specifically it is shown that if y Xn is chosen optimally (i.e. if the coefficients an¡ are chosen to minimize \\y yXn II), and if y2n is chosen to be the first iterate of yXn, i.e. y2n = f + KyXn, then || v y2„ II < <*„ \\y yXn II, with an -» 0. A similar result is also obtained, provided the homogeneous equation x = Kx has no nontrivial solution, if instead y, 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, y¿\n, ■ ■ ■ obtained by further iteration is also shown to be valid, if the range of K is dense in H.