Müntz-type theorem on the segments emerging from the origin

A necessary and sufficient condition is obtained for the linear span of a system of monomials {zλ:λ∈Λ} to be dense in the space of all continuous functions defined on the line segments emerging from the origin, where Λ is a set of nonnegative integers. The result is a generalization of the Müntz theorem to the segments emerging from the origin and an extension of the Mergelyan theorem to lacunary polynomials.