On the minimal property of the Fourier projection

E. W. Cheney, C. R. Hobby, P. D. Morris, F. Schurer, D. E. Wulbert
1969 Bulletin of the American Mathematical Society  
Let C be the space of real 27r-periodic continuous functions normed with the supremum norm. Let P n denote the subspace of trigonometric 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 P n 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
more » ... on different from F, then there exist minimal projections L and H, different from F such that $L+$H=F. The proof of this assertion utilizes Berman's equation,
doi:10.1090/s0002-9904-1969-12141-5 fatcat:vk5wmck7tbgwjahoj3x5sgwjpu