  We study the effects of adding a local perturbation in a pattern forming
system, taking as an example the Ginzburg-Landau equation with a small
localized inhomogeneity in two dimensions. Measuring the response through the
linearization at a periodic pattern, one finds an unbounded linear operator
that is not Fredholm due to continuous spectrum in typical translation
invariant or weighted spaces. We show that Kondratiev spaces, which encode
algebraic localization that increases with each derivative, provide an
effective means to circumvent this difficulty. We establish Fredholm properties
in such spaces and use the result to construct deformed periodic patterns using
the Implicit Function Theorem. We find a logarithmic phase correction which
vanishes for a particular spatial shift only, which we interpret as a
phase-selection mechanism through the inhomogeneity.
