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Based on the variable Hilbert scale interpolation inequality bounds for theerror of regularisation methods are derived under range inclusions. In thiscontext, new formulae for the modulus of continuity of the inverse of boundedoperators with non-closed range are given. Even if one can show the equivalenceof this approach to the version used previously in the literature, the newformulae and corresponding conditions are simpler than the former ones. Severalexamples from image processing and spectral enhancement illustrate how the newerror bounds can be applied.

Errors of regularisation under range inclusions using variable Hilbert scales