Dilational Hilbert Scales and Deconvolutional Sharpening

TL;DR

The paper introduces dilational Hilbert scales (DHS), a new family of scales that generalize ordinary Hilbert scales and are useful for deriving error estimates in inverse problems, with an application to deconvolution sharpening.

Contribution

It develops dilational Hilbert scales (DHS) based on dilations of index functions, providing a new framework for error estimates in inverse problems.

Findings

DHS generalize ordinary Hilbert scales.

DHS lead to new dilational interpolatory inequalities.

Application demonstrated in deconvolution sharpening.

Abstract

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.