Some Classes of Rational Functions and Related Banach Spaces

Throughout this paper, d and r will be positive integers and 0 < δ < 1. These numbers will be arbitrary with the given properties unless they are further specified. For a matrix or vector X let X ′ denote its transpose. Thus if x = (x1, ..., xd) ′ ∈ R is a column vector and A is a d× d matrix, then x′Ax gives the quadratic form defined by the matrix A, ∑n i,j=1 Aijxixj . Let Pd denote the set of symmetric d × d positive definite matrices. Let ‖A‖ be the usual operator norm of a matrix A, ‖A‖ := sup{|Ax| : |x| = 1}, where | · | is the usual Euclidean norm on R. Recall that N is the set of nonnegative integers. Let MMr := MMr,d be the set of monic monomials from R into R of degree r, namely the set of all functions g(x) = Πi=1x ni i with ni ∈ N and ∑d i=1 ni = r. Let Wδ := Wδ,d := {C ∈ Pd : ‖C‖ < 1/δ, ‖C−1‖ < 1/δ}. Let