每日一言 - 5月31日言

今日一言, 来自施瓦茨关于傅里叶级数与傅里叶积分的评价:

The theory of Fourier series and integrals has always had major difficulties and necessitated a large mathematical apparatus in dealing with questions of convergence. It engendered the development of methods of summation, although these did not lead to a completely satisfactory solution of the problem…. For the Fourier transform, the introduction of distributions (hence the space $\mathscr{S}$) is inevitable either in an explicit or hidden form…. As a result one may obtain all that is desired from the point of view of the continuity and inversion of the Fourier transform.

傅里叶级数和傅里叶积分理论一直存在一些主要困难, 需要用到大量的数学工具以处理收敛性问题. 这一理论促进了求和方法的发展, 尽管这些方法并没有完全令人满意地解决这个问题… 而对于傅里叶变换, 我们不可避免的要引入分布(因此空间 $\mathscr{S}$), 无论是显式的还是隐藏的形式… 而由此带来的结果就是, 人们可以从傅里叶变换的连续性和反演的角度获得他们所需的一切.

*L. Schwartz*, 1950