| Research Abstract |
Suppose {c_<-N>, ・・・, c_0, ・・・, c_N} (c_<-k>=c_k^^-) is a positive definite sequence of complex numbers. A finite positive measure mu on the unit circle is called a representing measure of this sequence if. The set of all representing measures, denoted by M(c_0, c_1, ・・・, c_N), is weak ^<**>compact convex set. Our results concern the extreme points of M(c_0, c_1, ・・・, c_N). By introducing new notions of singularly positive definiteness and singular extensions of a given positive definite sequence {c_<-N>, ・・・, c_0, ・・・, c_N}, our results can be summarized as follows.
Theorem A measure mu in M(c_0, c_1, ・・・, c_N) is an extreme point of M(c_0, c_1, ・・・, c_N) if and only if mu is the representing measure of a singularly positive definite sequence {c_0, ・・・, c_N, ・・・, c_M, c_<M+1>) which is a singular extension of {c_0, ・・・, c_N} and N(〕SY.ltoreq.〔)M(〕SY.ltoreq.〔)2N.The representing measure mu of {c_0, ・・・, c_N, ・・・, c_M, c_<M+1>} can be constructed as follows
a) a polynominal has M+1 many simple zeros a_1, a_2, ・・・, a_<M+1> on the unit circle.
b) mu=SIGMA^^<M+1>__W_kdelta_ak#, weights W_k are given by where T_M is the Toeplitz matrix associated with {c_0, ・・・, c_M} and |T_M| is the determinant of T_M.
We have similar results in the case of the power moment problems.
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