## Dilational interpolatory inequalities

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- by Markus Hegland and Robert S. Anderssen PDF
- Math. Comp.
**80**(2011), 1019-1036

## 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.## References

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## Additional Information

**Markus Hegland**- Affiliation: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia
- Email: markus.hegland@anu.edu.au
**Robert S. Anderssen**- Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia
- Email: bob.anderssen@csiro.au
- Received by editor(s): March 18, 2008
- Received by editor(s) in revised form: March 9, 2010
- Published electronically: October 5, 2010
- © Copyright 2010 CSIRO, Mathematics, Informatics and Statistics
- Journal: Math. Comp.
**80**(2011), 1019-1036 - MSC (2010): Primary 65J20
- DOI: https://doi.org/10.1090/S0025-5718-2010-02431-7
- MathSciNet review: 2772107