On the minimal property of the Fourier projection

Let C be the space of real 27r-periodic continuous functions normed with the supremum norm. Let Pn denote the subspace of trigono-metric polynomials of degree ^n. It is known [l] that the Fourier projection F of C onto P » is minimal; i.e., if A is a projection of C onto Pn then \\F\ \ Û\\A\\. We prove that F is the only minimal projection of C onto P n. The proof is constructed by verifying the assertions listed below. Details will appear elsewhere. ASSERTION. If there exists a minimal projection different from F, then there exist minimal projections L and H, different from F such tha