The generalized Walsh functions
The generalized Walsh functions
复制标题
广义沃尔什函数
DOI：
10.1090/s0002-9947-1950-0042535-2
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发表时间：
1950
影响因子：
1.3
通讯作者：
N. Fine
中科院分区：
文献类型：
--
作者：
N. Fine
1. 介绍。在最近的一篇论文（‘）中，作者讨论了Walsh函数{1t’，(x)}的各种性质，并试图展示它们与指数{exp 2-7rinx}之间的密切类比。这种类比被认为是源于这样一个事实，即每个系统本质上都是某个紧交换群的特征群，并且有可能在两个群之间建立合理的忠实对应关系。人们很自然地要问，这个类比是否可以推广到系统{exp 27riyx}，也就是说，沃尔什函数是否可以嵌入到一个更大的类{，6f (x)}中，以便保留指数的大多数性质，这些性质在分析中是可取的和有用的。这个问题在这里得到了肯定的回答，群论的考虑再次发挥了重要作用。2构造了一个类似于实数的拓扑域a，并证明了a的加性群F的字符是由单个字符和a中的乘法产生的。如果Xl (x)是这个字符，x是任意字符，则存在一个唯一的yEF，使得xGc)= Xj （yt）。对应的X确实是F和它的字符群X之间的同构，由此可以使X成为与的同构域。关系F_X也可以从Fc-GXChar G推导出来，其中G是WF (？2)，但与实数的类比在某种程度上被F的直接乘积分解所掩盖，而对于实数则不存在类似的情况。相关事实是，F在G上存在一个同态a，其核与Char G同构，并且X包含一个与Char G同构的子群X‘，由对应的XCChar G-> X’（?）定义_X （a (Qx)）。在这里我们应该注意到，群F和它的特征群X已经被Paley和Wiener(2)简要地讨论过，然而，没有提到场或与Walsh函数的联系。鉴于Paley对Walsh函数的研究（），他们很可能意识到了这种联系。
1. Introduction. In a recent paper (') the author discussed various prop-erties of the Walsh functions {1t',(x)} and attempted to exhibit the close analogy between them and the exponentials {exp 2-7rinx}. This analogy was seen to stem from the fact that each system is essentially the character group of a certain compact commutative group, and that it is possible to set up a reasonably faithful correspondence between the two groups. It is natural to ask whether the analogy can be extended to the system {exp 27riyx}, that is, whether the Walsh functions can be imbedded in a larger class {, 6f (x)} so as to preserve most of the properties of the exponential which are desirable and useful in analysis. This question is answered in the affirmative here, and again group-theoretic considerations play an important role.In? 2 we construct a topological field a analogous to the reals, and show that the characters of the additive group F of a are generated by means of a single character and the multiplication in a. If Xl (x) is this character, and X an arbitrary character, there is a unique yEF such that xGc)= Xj (yt). The correspondence yx is indeed an isomorphism between F and its char-acter group X. It follows that X may be made into a field isomorphic with. The relation F_X may also be deduced from Fc-GXChar G, where G is the dyadic group defined in WF (? 2), but the analogy with the reals is somewhat obscured by the direct product decomposition of F, the analogue of which does not exist for the reals. The relevant facts are that there is a homomorphism a of F on G, the kernel of which is isomorphic with Char G, and that X contains a subgroup X'isomorphic with Char G, defined by the cor-respondence XCChar G-> X'(?) _X (a (Qx)). We should remark here that the group F and its character group X have been discussed briefly by Paley and Wiener (2), without, however, any mention of the field or of the connection with the Walsh functions. It is quite likely, in view of Paley's work on the Walsh functions (), that they were aware of the connection.