The inverse Fourier transform of a frequency power-law corresponds to a non-integer or fractional derivative. However, the measurements are always band-limited, i.e. for medical ultrasound one only has measurements from say 1 to 20 MHz. The Fourier property, however, is valid for all frequencies from 0 to infinity.
Here we want to develop new operators that correspond better to measurements of the frequency response of real band limited propagation media.
A set of chapters from S. Holm, Waves with Power-Law Attenuation, 2019, will be a special curriculum for the Master degree for this problem.
References:
- S. Holm, Power-laws and fractional calculus, presentation, Dec 2020
- J. Sabatier, Some proposals for a renewal in the field of fractional behaviour studies and modeling, presentation Dec 2020.
- S. Holm, Waves with Power-Law Attenuation, 2019, Springer, chs. 1, 5.1-5.3, and 9.4.
- E. Torgersen, Niels Henrik Abel får enda en fjær i hatten, Titan, februar 2020.