Extrapolation of operator moments, with applications to linear algebra problems
Claude Brezinski, Paraskevi Fika, Marilena Mitrouli · 2010
Let A be a linear self-adjoint operator from H to H, where H is a real infinite dimensional Hilbert space with the inner product (·, ·). For positive powers of A, the Hilbert space H could be infinite dimensional, while, for negative powers it is always assumed to be a finite dimensional, and, in this case, A is also assumed to be invertible. Using the singular value decomposition for a compact linear self-adjoint operator A and its moments, we can define it’s fractional powers by Aνz = ∑ k σν k (z, uk)uk, and its fractional moments by cν(z) = (z, Aνz) = ∑ k σν kα2 k (z), where αk(z) = (z, uk), for ν ∈ Q. We will approximate cq(z) by interpolating or extrapolating, at the point q ∈ Q, the cn(z)’s for different values of the non-negative integer index n by a conveniently chosen function obtained by keeping only one or two terms in the preceding summations. Estimates of the trace of Aq, for q ∈ Q, and of the norm of the error when