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On the problem of modified moments

Author: Rupert Lasser
Journal: Proc. Amer. Math. Soc. 90 (1984), 360-362
MSC: Primary 42C05; Secondary 44A60
MathSciNet review: 728348
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Abstract: The problem of modified moments is studied. Let $ \left( {{P_n}\left( x \right)} \right)_n^\infty = 0$ be an orthogonal polynomial sequence. Given a sequence $ \left( {{d_n}} \right)_n^\infty = 0$ of real numbers, does there exist a bounded nondecreasing function with infinitely many points of increase such that for every $ n \in {{\mathbf{N}}_0}$, $ {d_n} = \int_{ - \infty }^\infty {{P_n}} (x)d\mu (x)$? Is there any information about the support of $ \mu$? A necessary and sufficient condition for the existence of such a function $ \mu $ is given in terms of the positivity of certain determinants. For certain $ \left( {{P_n}\left( x \right)} \right)_{n = 0}^\infty $ a description of the support of $ \mu $ is established.

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Keywords: Moment problems, orthogonal polynomials
Article copyright: © Copyright 1984 American Mathematical Society