AI 中文总结
提出傅里叶变换的截断和广义极限公式，为经典\(L^1(R)\)之外函数提供统一构造框架，通过特定操作构建广义傅里叶反演，有不对称性，还阐述了广义谱意义及与分布理论区别。
AI 中文摘要
本文提出了傅里叶变换的截断和广义极限（t.g.l.）公式，为超出经典\(L^1(R)\)设置的函数提供了一个统一的构造框架，包括非衰减、振荡和局部奇异函数。广义傅里叶反演是通过在傅里叶对偶域中的有序截断和连续广义极限运算来构造的。一个特征是正向和反向变换之间存在固有的不对称性：正向变换是一个一阶广义极限族，在频域中不需要逐点收敛，而反向变换需要频域截断，以通过狄利克雷型振荡核在倒数域中产生有意义的局部化。广义谱意义通过二阶广义极限出现，通过一阶变换族与倒数域上的可允许辅助函数之间的配对运算。该公式为广义傅里叶分析、有限带信号合成和超出经典\(L^1\)框架的渐近信号重建提供了一个构造性操作框架，同时保留了无限倒数傅里叶域和正交指数基结构。本文阐明了t.g.l.方法与分布理论之间的区别，揭示了双域结构的明确构造实现，这在非\(L^1\)函数的经典和分布傅里叶公式中在很大程度上仍然是隐含的。几个例子表明，该框架提供了超出传统\(L^1\)设置的经典傅里叶分析的构造性扩展。
英文摘要
This paper introduces a constructive definition of generalized Fourier transforms based entirely on ordinary truncated Fourier integrals and ordered dual-domain limits, within the framework of improper Riemann integration and classical analysis. The proposed truncate-and-generalized-limit (t.g.l.) formulation does not require test-function spaces, Lebesgue measure theory, or duality pairings in its proofs: the forward and inverse transforms are defined directly through finite-domain truncation of the target function, followed by successive ordered limits in the time and frequency domains. As consequences of this constructive definition, the formulation provides a unified treatment of non-decaying, oscillatory, and locally singular functions beyond the classical L1(R) setting; reveals an inherent asymmetry between the forward transform, interpreted as a first-order generalized-limit family, and the inverse transform, which requires frequency-domain truncation to generate pointwise reconstruction through Dirichlet-type oscillatory localisation; and clarifies the distinction between the t.g.l. approach and distribution theory, where generalized Fourier transforms are introduced through duality pairings rather than constructed from ordinary integrals. The inversion formula is established rigorously for two concrete admissible classes using only the classical Dirichlet convergence theorem. Several examples confirm that the framework covers constants, polynomials, periodic functions, singular kernels, and chirp signals within a single constructive scheme.
CommentsSubmitted to the Journal of Fourier Analysis and Applications v2: Added Section 6.7 discussing the suitability of improper Riemann integration for oscillatory Fourier integrals and its relation to the Dirichlet kernel. Added reference to Feichtinger and Jakobsen (2020). Submitted to the Journal of Fourier Analysis and Applications
