Definability and Strength of Fourier Series Transform Obtained by Employing Parametric Integration

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Abstract

Regardless of the term coinage, this article is a story of adding variables that act in a similar way to the “s” in the Laplace transform. This idea is similar to the avoiding of points in improper integration, but instead of taking limits, the endpoints of each maximum integrable interval are made to be variables, but they still preserve the orders. The result is here called a parametric integral. It is applied here to Fourier series transform as an example. Therefore, some functions that formerly had no Fourier series expansion may no longer exist if one employs parametric integration. The resulting transform has advantages in terms of definability and strength. In definability because the transform exists for more functions than even for the well-known Fourier transform. It gives hope for solving more problems that may be of certain practical interests, especially when endpoint variables in the solution are replaced by respective limiting processes. The possibility that a solution still contains non-eliminable variables may lead to interesting non-standard analysis with the parameters as new “numbers”, some of which are given here. The parameters may also serve in determining the partial inverses of the transform. Keywords: Fourier series expansion, Fourier series transform, half-range expansion, Laplace transform, parametric integration. https://doi.org/10.55463/issn.1674-2974.50.8.13
用参数积分法求傅里叶级数变换的可定义性和强度
不管术语是什么,本文是一个关于添加变量的故事,这些变量的作用方式与拉普拉斯变换中的“s”类似。这个想法类似于在反常积分中避免点,但不是取极限,而是使每个最大可积区间的端点成为变量,但它们仍然保持阶数。结果在这里被称为参数积分。这里以傅里叶级数变换为例。因此,如果采用参数积分,以前没有傅里叶级数展开的一些函数可能不再存在。所得到的变换在可定义性和强度方面具有优势。在可定义性上因为这个变换存在于更多的函数中甚至比存在于著名的傅里叶变换中还要多。它为解决更多可能具有一定实际意义的问题提供了希望,特别是当解中的端点变量被各自的极限过程所取代时。一个解仍然包含不可消除变量的可能性可能会导致有趣的非标准分析,将参数作为新的“数字”,其中一些在这里给出。这些参数还可以用于确定变换的部分逆。关键词:傅立叶级数展开,傅立叶级数变换,半范围展开,拉普拉斯变换,参数积分。https://doi.org/10.55463/issn.1674-2974.50.8.13
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