In this dissertation, we first present a unified treatment of compact moment problems, both the truncated and full moment cases. Second, we define the lower and upper functions V±(ð₁,... ð _n) on the convex hull of the curve Γ_n = {(t,.·.,tⁿ): t ∈ [0,1] } for each positive integer n. Explicit formulas of these functions are derived and applied to the study of the subnormal completion problem in operator theory. Last, we show that certain power functions are the building blocks of completely positive functions; by our definition, these functions are the continuous functions on the interval [0, 1] that map each Hausdorff moment sequence of a probability measure into another one.