an orthogonality theorem of dines related to moment problems and linear programming英文.pdf

JOURNAL OF COMBINATORIAL THEORY 2, 1-26 (1967)
An Orthogonality Theorem of Dines Related to Moment
Problems and Linear Programming*
R. J. DurriN
Carnegie Institute of Technology,
Pittsburgh, Pennsylvania
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 se-
quence 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 non-
negative moments relative to the given sequence of functions. (13') Every linear com-
bination of the sequence having non-negative coefficients either changes sign or van-
ishes 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 objec-
tive 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