Vol. 24, No. 3, 1968

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On the characterization of measures of the cone dual to a generalized convexity cone

Zvi Ziegler

Vol. 24 (1968), No. 3, 603–626

We consider in this paper the cone C(u0,,un1) of functions which are convex with respect to an Extended Complete Tchebycheffian system {u0(t),u1(t),,un1(t)}. The cone dual to C(u0,,un1) 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.

Mathematical Subject Classification
Primary: 26.52
Secondary: 42.00
Received: 6 August 1966
Revised: 31 January 1967
Published: 1 March 1968
Zvi Ziegler