Complete Sets of Bessel and Legendre Functions
Complete Sets of Bessel and Legendre Functions
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完整的贝塞尔和勒让德函数集
DOI：
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发表时间：
1947
期刊：
影响因子：
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通讯作者：
H. Pollard
中科院分区：
文献类型：
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作者：
R. Boas;H. Pollard
1. 尽管对指标的积分值和非积分值的勒让德函数PA(X)已经进行了广泛的研究，但对具有非积分指标的勒让德函数集的完备性问题似乎关注甚少。一个例外出现在Hille[3]的工作中，他表明，对于某些使函数正交的指标集，它们在L2（- 1,1）中是完全的。贝塞尔函数I J，(X)的集合也存在类似的情况。X)} -i，其中v是固定的。显然，只有当Xva是边值问题对应的特征值时，才讨论了这种集的完备性；这里这些函数又形成了一个正交集，相对于一个合适的权函数。本文的目的是建立{X，，}具有更一般性质的Legendre和Bessel函数集的完备性判据。在每种情况下，获得两种类型的标准。一类判据指出，如果相关的三角函数集是完备的，则所讨论的集合是完备的。由于后者已被Paley、Wiener和Levinson等人广泛地研究过，因此可以从他们的结果中得出进一步的完备性条件。但是请注意，这些作者所说的“闭包”就是我们所说的“完备性”。另一种类型的判据要求Xa满足不等式，这些不等式本质上表明它们不会增长得太快。一个函数集fn(x)在函数F(A, b)，
1. Although the Legendre functions PA(X) have been extensively studied for both integral and non-integral values of the index, little attention seems to have been paid to the problem of completeness of sets of such functions with non-integral indices. An exception to this occurs in the work of Hille [3], who showed that, for certain sets of indices which make the functions orthogonal, they are complete in L2(-1, 1). A similar situation exists for sets of Bessel functions I J,(X.x)} -i, where v is fixed. The completeness of such sets has apparently been discussed only in the case where the Xva are eigenvalues corresponding to boundary-value problems; here again the functions form an orthogonal set, with respect to a suitable weight function. It is the purpose of this paper to establish completeness criteria for sets of Legendre and Bessel functions where the {X,,} are of a more general character. In each case two types of criteria are obtained. A criterion of one type states that the set in question is complete if an associated set of trigonometric functions is complete. Since the latter have been studied extensively by Paley and Wiener and Levinson [6], further conditions for completeness can be read off from their results. Note, however, that these authors call "closure" what we call "completeness." A criterion of the other type demands that the Xa satisfy inequalities which state essentially that they do not grow too rapidly. A set of functions fn(x) is said to be complete in a class of functions F(a, b),