Approximation by pairs of functions.

The problem of approximating a fixed function ⨍ ∈ Ϲ[a,b] by a pair of functions ⨍₁,⨍₂ ∈ Ϲ[a,b] will be explored. A natural error function gf is defined and approximation pairs are chosen from [equation], where IIn represents the set of all algebraic polynomials of degree less than or equal to n. The questions of existence of best approximations, characterization of such approximations, and uniqueness of best approximations are examined. Best approximations will be shown to exist. A partial characterization of these approximations will be developed along with some local uniqueness results. Extensions to approximation by k-tuples, and approximation from Haar spaces are also considered.