Nongeometric convergence of best 𝐿_{𝑝}(𝑝≠2) polynomial approximants

For an arbitrary function f f analytic in the disk D : | z | > 1 D:\left | z \right | > 1 and continuous in D ¯ \bar D , we show that geometric convergence in D D of best L p ( 1 ≤ p ≤ ∞ ) {L_p}(1 \leq p \leq \infty ) polynomial approximants to f f on C : | z | = 1 C:\left | z \right | = 1 is assured only when p = 2 p = 2 .