Rotation invariant moment problems
Rotation invariant moment problems
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旋转不变矩问题
DOI：
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
M. Thill
中科院分区：
文献类型：
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作者：
C. Berg;M. Thill
Marcel Riesz（参见[14]）的一个重要定理指出，当/x是实线上的确定测度时，多项式在L2（/x）中是密集的。在不确定情况下，Riesz还描述了L2（#）中多项式密集的测度/x。这就是所谓的涅万林纳极限措施，在涅万林纳1985年提出。当/x是R d， d>上的确定测度时，似乎不知道多项式在L2(g)中是否密集，参见Fuglede >的说说性论文，以及研究问题书[8，第529页]，其中Devinatz将问题作为问题1提出，并将其归因于物理学家John Challifour（1978）。在本文中，我们将否定地解决这个问题。在R d， d bb0 1上存在旋转不变测度#，它们是确定的，但其多项式在L2(p)中不是稠密的。这些测度/x必然是下列非常特殊的形式
An important theorem of Marcel Riesz, cf. [14], states that the polynomials are dense in L2(/x), when/x is a determinate measure on the real line. In the indeterminate case Riesz also characterized the measures/x for which the polynomials are dense in L2(#). They are the so-called Nevanlinna extremal measures, introduced in Nevanlinna [11]. It does not seem to be known whether the polynomials are dense in L2(g), when/x is a determinate measure on R d, d> 1, cf. the expository paper by Fuglede [7], as well as the research problems book [8, p. 529], where Devinatz poses the problem as question 1 and ascribes it to the physicist John Challifour (1978). In this paper we shall settle the question in the negative. There exist rotation invariant measures # on R d, d> 1, which are determinate but for which the polynomials are not dense in L2(p). Such measures/x are necessarily of the following very special form