Reflection coefficient (Schur parameter) representation for convex compact sets in the plane

We obtain a one-to-one relation between the shape and orientation of a convex compact planar set and a complex-valued reflection coefficient (Schur (1917) parameter) sequence, such that (1) the reflection coefficient magnitudes are less than or equal to one, (2) if any reflection coefficient has a magnitude equal to one, then all subsequent reflection coefficients are equal to zero, and (3) the first reflection coefficient is equal to zero. Three additional independent parameters specify the position of the set in the plane, and the size of the set (specifically its circumference). For a finite duration reflection coefficient sequence, if the last nonzero reflection coefficient has a magnitude that is less than one, then the boundary of the set is an infinitely differentiable convex curve. The boundary is a convex polygon if and only if the magnitude of the last reflection coefficient is equal to one; the number of sides of the polygon is equal to the index of the last reflection coefficient. Almost all planar convex compact sets have reflection coefficient sequences of infinite duration. Such sets can be accurately approximated with convex compact sets that are generated from relatively small numbers of reflection coefficients.

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