An orthogonality theorem of dines related to moment problems and linear programming

Abstract

It was found by Lloyd Dines that the following two properties of a finite sequence of functions are equivalent: (A) There exists a positive function that is orthogonal to each member of the given sequence. (B) Every linear combination of the given sequence either changes sign or vanishes identically. This paper gives a proof of the above equivalence by a variational method. Thereby the orthogonal function is given explicitly as the positive part of a linear combination of the sequence of functions. Moreover, this proof leads to various generalizations. In particular, it is shown that the following properties are equivalent: (A′) There exists a positive function with nonnegative moments relative to the given sequence of functions. (B′) Every linear combination of the sequence having non-negative coefficients either changes sign or vanishes identically. Next it is shown possible to transform these properties A′ and B′ into statements about dual linear programs A″ and B″. Program A″ concerns the moments of the sequence of functions with respect to a positive function. The objective of Program A″ is to maximize the moment of the first function subject to the constraint that the moments of the remaining functions equal or exceed preassigned values. The dual program B″ is a minimization problem stated in terms of linear combinations of the sequence. From the equivalence theorem it is shown that the maximum of the primal program is equal to the minimum of the dual program. This last theorem on infinite programs is not merely an analogy with the classical duality theorem of finite programming, because by a suitable refinement it is seen to include the classical theorem as a special case. Various other ramifications, counter examples, and applications related to the Dines theory are developed.