Hilbert points in hardy spaces

A Hilbert point in H-p(T-d), for d >= 1 and 1 <= p <= infinity, is a nontrivial function phi in H-p(T-d) such that parallel to phi parallel to(Hp(Td)) <= parallel to phi + f parallel to(Hp(Td)) whenever f is in H-p(T-d) and orthogonal to phi in the usual L-2 sense. When p not equal 2, phi is a Hilbert point in H-p(T) if and only if phi is a nonzero multiple of an inner function. An inner function on T-d is a Hilbert point in any of the spaces H-p(T-d), but there are other Hilbert points as well when d >= 2. The case of 1-homogeneous polynomials is studied in depth and, as a byproduct, a new proof is given for the sharp Khinchin inequality for Steinhaus variables in the range 2 < p < infinity. Briefly, the dynamics of a certain nonlinear projection operator is treated. This operator characterizes Hilbert points as its fixed points. An example is exhibited of a function parallel to phi that is a Hilbert point in H-p(T-3) for p = 2, 4, but not for any other p; this is verified rigorously for p > 4 but only numerically for 1 <= p < 4.