On the Order of Regularization Methods

Abstract

Let A: L₂(Ω) → H, Ω ⊆ ℝⁿ, H a Hilbert space, be a linear bounded injective operator. With H_q we denote either the Sobolev space H^q(Ω) or the Sobolev space ${\text{H}}_{\text{O}}^{\text{q}}\left( \Omega \right)$ of distributions in H^q(ℝⁿ) with support in Ω. The norm in H_q is denoted by ∥ · ∥_q. We assume that there is a c > O such that for all f ∈ L₂(Ω)

$$c{\left\| f \right\|_{ - a}}{\mkern 1mu} {\mkern 1mu} \left\| {Af} \right\|{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{1}{c}{\mkern 1mu} {\mkern 1mu} {\left\| f \right\|_{ - a}}$$

((1.1))

for some a > O, i.e. the norms ∥f∥_−a, ∥Af∥ are equivalent. It follows that the equation Af = g is ill-posed.