ISSN:
1572-9265
Keywords:
primary 30E05
;
Moment problem
;
orthogonal rational functions
;
quasi-definite
;
positive-definite
;
Hermitian functional
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract Leta 1,...,a p be distinct points in the finite complex plane ℂ, such that |a j|〉1,j=1,..., p and let $$b_j = 1/\bar \alpha _j ,$$ j=1,..., p. Let μ0, μ π (j) , ν π (j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [−π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ℂ* outside {a 1,...,a p,b 1,...,b p}.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02141919
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