Imposing the constraints

Abstract

Let Ωn# denote the parental function space for Ωn consisting of all spectral distribution functions F on the torus T° = [-Π, Π] endowed with the Lebesgue σ-algebra T which satisfy the following constraints: (2.1) F is real-valued and absolutely continuous with respect to Lebesgue measure (dω) on T° (2.2) F′ = f, which exists by (2.1), is a strictly positive, even, continuous density on T° which is of bounded variation (2.3) 0 < δ < f(ω) < Γ < ∞ for all ω e [0, π] (δ and Γ fixed and independent of f and n) (2.4) f has an absolutely convergent Fourier (cosine) series on [0, π].