Differentiable Functions Defined in Closed Sets. I
Differentiable Functions Defined in Closed Sets. I
复制标题
闭集中定义的可微函数。
DOI：
10.1007/978-1-4612-2972-8_16
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发表时间：
1934
影响因子：
1.3
通讯作者：
H. Whitney
中科院分区：
文献类型：
--
作者：
H. Whitney
1.引言。在最近的一篇论文中，作者证明了如果定义在n-空间E中的闭集A中的函数j(X)满足某些条件，即如果它是“A中的em类”，则它的定义可以在E上扩展，使得它将有连续的m阶偏导数。在这篇文章中，我们将自己限制在一维的情况下。(关于这种情况下的上述定理，见§4。)设x_0，…，x‘“是A的不同点，若f_p(X)=co+…+c”，x’“是至多m次多项式，使得P(Xi)=j(X)(i=O，…，m)，j(X)在这些点上的第m个差商为~o……j=”‘j(X)=m！c“，本文的主要目的是证明(定义见§2和节3)
1. Introduction. In a recent paper § the author has shown that if a function j (x) defined in a closed set A in n-space E satisfies certain conditions involving Taylor's formula (in finite form), ie if it is" of class em in A," then its definition can be extended over E so that it will have continuous partial derivatives through the mth order. In this paper we restrict ourselves to the one-dimensional case.(For the above theorem in this case, see § 4.) Letxo,..., x'" be distinct pointsofA. IfP (x)= co+...+ c", x'" is the polynomial of degree at most m such that P (Xi)= j (x,)(i= O,..., m), the mth difference quotient ofj (x) at these points iS~ o...... j=~"'j (x)= m! c",. The main object of this paper is to prove (see §§ 2 and 3 for definitions)