TY - JOUR

T1 - Dilational interpolatory inequalities

AU - Hegland, Markus

AU - Anderssen, Robert S.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78751639086&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2010-02431-7

DO - 10.1090/S0025-5718-2010-02431-7

M3 - Article

SN - 0025-5718

VL - 80

SP - 1019

EP - 1036

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 274

ER -