Minimal Multi-Convex Projections Onto Subspaces of Incomplete Algebraic Polynomials

Let X = (C ^N [0, 1], ‖·‖), where N ≥ 3 and let V be a linear subspace of Π _N , where Π _N denotes the space of algebraic polynomials of degree less than or equal to N. Denote by 𝒫 _S = 𝒫 _S (X, V) = {P: X → V | P-linear and bounded P| _V  = id _V , PS ? S}, where S denotes a cone of multi-convex functions. In [25 G. Lewicki and M. Prophet ( 2006 ). Minimal shape-preserving projections onto Π _n : generalizations and extensions . Numer. Func. Anal. Optim. 27 ( 7–8 ): 847 – 873 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 26 G. Lewicki and M. Prophet ( 2007 ). Minimal multi-convex projections . Studia Mathematica 178 ( 2 ): 99 – 124 . [Google Scholar]], the multi-convex projections were defined and it was shown the explicite formula for projection with minimal norm in 𝒫 _S for V = Π _N . In this article we present a generalization of these results in the case of V being certain, proper subspaces of Π _N .