The Angular Function

If a complex valued function on the unit disk has invariance in form, then it is separable into radial and angular components. Furthermore, the angular component is of a particular form, and readily visualized. 1 Introduction This paper is motivated by a study of orthogonal functions on the unit disk having invariance in form [1]. To summarize Section 2 of this work, a function f : D 2 \Gamma! C is said to be invariant in form if f(ae; ` + OE) = f(ae; `)\Psi(OE) in radial-polar coordinates. Under a differentiability assumption, f must be of the form f(ae; `) = L(ae)\Psi(`) for some L : R \Gamma! C and \Psi : R \Gamma! C . Furthermore, if full rotations by 2ß are to be indistinguishable then \Psi(`) = exp(iq`) for some q 2 Z. We can thus think of the angular function \Psi as being an envelope modulating a radial signal L. In particular, we may assert that L is real valued, examples of where this is done being the Zernike, pseudo-Zernike and Orthogonal Fourier-Mellon basis fun..