Some extremal problems in Lp(w)

Fix a positive integer n and 1 < p < infinity. We provide expressions for the weighted L-p distance [GRAPHICS] where d lambda is normalized Lebesgue measure on the unit circle, w is a nonnegative integrable function, and f ranges over the trigonometric polynomials with frequencies in S-1 = {...,-3,-2,-1} boolean OR {1,2,3,...,n}, S2 = {...,-3,-2,-1}\{-n}, or S3 = {...,-3,-2,-1} boolean OR {n}. These distances are related to other extremal problems, and are shown to be positive if and only if log w is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with w.