Improvement by iteration for compact operator equations
Ian H. Sloan · Mathematics of Computation · 1976
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