We consider in this paper the
cone C(u0,⋯,un−1) of functions which are convex with respect to an Extended
Complete Tchebycheffian system {u0(t),u1(t),⋯,un−1(t)}. The cone dual to
C(u0,⋯,un−1) is examined and necessary conditions as well as sufficient conditions
for a measure to belong to this cone are developed. The merit of these conditions lies
in the fact that they involve only the pattern of sign changes of the measure and
related functions, and thus are easily verifiable. Several applications are given. These
include new inequalities for the Euler-Fourier coefficients of functions belonging to
given convexity cones. Some new inequalities for the Fourier coefficients of
the expansion of a function in a series of orthogonal polynomials are also
obtained.
