Using analyticity to check the consistency 
                       of experimental measurements.  


                                   by

                      David Eyre and Graeme Milton
               Department of Mathematics, University of Utah

                INSCC 110,  3:30pm Monday, March 29, 1999

                                Abstract

This is joint work with Joe Mantese and Roderic Lakes. 
Experimental measurements often give the real and imaginary parts of the
complex dielectric constant or magnetic permeability (or the bulk and
shear moduli of viscoelastic materials) as a function of frequency. 
It is well known that the real and imaginary parts are Hilbert transforms 
of each other, with respect to frequency.  These relations, known as 
the Kramers-Kronig dispersion relations, have a long history dating back 
to 1926 and have proved to be an invaluable tool for interpreting data.  
Such dispersion relations are prevalent throughout physics and derive from 
the causal nature of the response of materials, bodies or
particles to electromagnetic, elastic or other fields.  

In actual experiments one only makes measurements over a
finite range of frequencies. This corresponds to the following
mathematical problem: suppose we know the imaginary part of an analytic
function is positive in the upper half plane, and is known along 
two disjoint intervals along the real axis. What can one say about 
the real part of the function? Our incomplete knowledge of the imaginary 
part of the function means that we can only derive inequalities on 
the real part. These inequalities generalize the Kramers-Kronig 
relations, providing bounds on the combinations of data values 
that are permissible over the measured frequency interval. 
They can provide highly accurate interpolation formulas for the 
real part, given its value at a few selected frequencies and given 
the imaginary part over a range of frequencies.  We illustrate the 
practical utility of the bounds as applied to high frequency 
transmission line measurements of the complex relative magnetic 
permeability of a composite made with equal parts (by volume)  
of barium titanate and a magnesium-copper-zinc ferrite. 
For viscoelasticity we obtain even tighter bounds by assuming 
the complex bulk and shear moduli derive from a relaxation model. 

For more information see:
http://www.math.utah.edu/~eyre/research/kk/node1.html