Minimal projections in spaces of functions of N variables

Abstract

We will construct a minimal and co-minimal projection from L p ([0,1] n ) onto L p ([0,1] n 1 )+⋯+ L p ([0,1] n k ), where n= n 1+⋯+ n k (see Theorem 2.9). This is a generalization of a result of Cheney, Halton and Light from (Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, Springer, Berlin, 1985; Math. Proc. Cambridge Philos. Soc. 97 (1985) 127; Math. Z. 191 (1986) 633) where they proved the minimality in the case n=2. We provide also some further generalizations (see Theorems 2.10 and 2.11 (Orlicz spaces) and Theorem 2.8). Also a discrete case (Theorem 2.2) is considered. Our approach differs from methods used in [8,13,20].