Following in the spirit of Greg Knese's excellent talk from last quarter, I show how
some concepts in complex analysis (three lines theorem and entire
functions) were used to prove a (sharp) basic inequality in classical
Fourier analysis (Hausdorff-Young). I also show how a result of mine from 1974 in operator theory leads to that same inequality, as well as some new (at the time) inequalities on some noncommutative groups including the two dimensional ax+b group. This motivates an introduction to abstract harmonic analysis, which I survey in anticipation of future talks I am planning to give on some 21st century applications of this noncommutative theory to engineering (e.g. robotics, wavelet transforms).