This book exploits the classification of a class of linear bounded operators with rank-one
self-commutators in terms of their spectral parameter known as the principal function. The
resulting dictionary between two dimensional planar shapes with a degree of shade and Hilbert
space operators turns out to be illuminating and beneficial for both sides. An exponential
transform essentially a Riesz potential at critical exponent is at the heart of this novel
framework its best rational approximants unveil a new class of complex orthogonal polynomials
whose asymptotic distribution of zeros is thoroughly studied in the text. Connections with
areas of potential theory approximation theory in the complex domain and fluid mechanics are
established. The text is addressed with specific aims at experts and beginners in a wide
range of areas of current interest: potential theory numerical linear algebra operator theory
inverse problems image and signal processing approximation theory mathematical physics.
