A Constructive Framework for Generalized Fourier Transforms via Truncate-and-Generalized Limits

Abstract

This paper presents a truncate-and-generalized-limit (t.g.l.) formulation of the Fourier transform, providing a unified constructive framework for functions beyond the classical L^1(R) setting, including non-decaying, oscillatory, and locally singular functions. Generalized Fourier inversion is constructed through ordered truncations and successive generalized limit operations in Fourier-dual domains. A characteristic feature is an inherent asymmetry between forward and inverse transforms: the forward transform is a first-order generalized-limit family not requiring pointwise convergence in the frequency domain, while the inverse transform requires frequency-domain truncation to generate meaningful reciprocal-domain localization through Dirichlet-type oscillatory kernels. Generalized spectral meaning emerges through second-order generalized limits, via pairing operations between the first-order transform family and admissible auxiliary functions on the reciprocal domain. The formulation provides a constructive operational framework for generalized Fourier analysis, finite-band signal synthesis, and asymptotic signal reconstruction beyond the classical L^1 framework, while preserving the infinite reciprocal Fourier domains and orthogonal exponential basis structure. The paper clarifies the distinction between the t.g.l. approach and distribution theory, revealing an explicit constructive realization of a dual-domain structure that remains largely implicit in both classical and distributional Fourier formulations for non-L^1 functions. Several examples indicate that the framework provides a constructive extension of classical Fourier analysis beyond the conventional L^1 setting.