Improvement by iteration for compact operator equations

摘要: The equation y f + Ky is considered in a separable Hilbert space H, with K assumed compact and linear. It shown that every approximation to of the form Yln = Enaniui (where {ui} given complete set an1, 1 < i n, are arbitrary numbers) less accurate than best Y2n ynbnjKui, if n sufficiently large. Specifically it chosen optimally (i.e. coefficients ani minimize Ily 11), be first iterate Ylnl i.e. Kyln then ll an II, an0. A similar result also obtained, provided homogeneous x Kx has no nontrivial solution, instead approximate solution by Galerkin or Galerkin-Petrov method. generalization forms Y3n' Y4n' . obtained further iteration valid, range dense H.