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Mathematics of Computation

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Dilational interpolatory inequalities

Authors: Markus Hegland and Robert S. Anderssen
Journal: Math. Comp. 80 (2011), 1019-1036
MSC (2010): Primary 65J20
Published electronically: October 5, 2010
MathSciNet review: 2772107
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Abstract: Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms and, thereby, associated scales of interpolatory inequalities. Using one-parameter families of index functions based on the dilations of given index functions, new classes of interpolatory inequalities, dilational interpolatory inequalities (DII), are constructed. They have ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They represent a precise and concise subset of variable Hilbert scales interpolatory inequalities appropriate for deriving error estimates for peak sharpening deconvolution. Only for Gaussian and Lorentzian deconvolution do the DIIs take the standard form of OHS interpolatory inequalities. For other types of deconvolution, such as a Voigt, which is the convolution of a Gaussian with a Lorentzian, the DIIs yield a new class of interpolatory inequality. An analysis of deconvolution peak sharpening is used to illustrate the role of DIIs in deriving appropriate error estimates.

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Additional Information

Markus Hegland
Affiliation: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia

Robert S. Anderssen
Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia

Received by editor(s): March 18, 2008
Received by editor(s) in revised form: March 9, 2010
Published electronically: October 5, 2010
Article copyright: © Copyright 2010 CSIRO, Mathematics, Informatics and Statistics