Dilational Hilbert Scales and Deconvolutional Sharpening
For researchers working on inverse problems, particularly deconvolution, this offers a theoretical framework for sharper error bounds, though the contribution is incremental as it extends existing scale concepts.
This paper introduces dilational Hilbert scales (DHS) as a generalization of ordinary Hilbert scales, providing new interpolatory inequalities that yield improved error estimates for deconvolution problems.
Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms. Using a one parameter family of index functions based on the dilations of a given index function, a new class of scales (dilational Hilbert scales (DHS)) is derived which generates new interpolatory inequalities (dilational interpolatory inequalities (DII)) which have the ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They therefore represent a one-parameter family generalization of OHS, and are a precise and concise subset of VHS approriate for deriving error estimates for deconvolution. The role of the Hilbert scales in deriving error estimates for the approximate solution of inverse problems is discussed along with an application of DHS to deconvolution sharpening.