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

Abstract

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.