On the convergence of Fourier representations and Schwartz distributions

While Fourier theory remains a cornerstone for analyzing and interpreting the spectral content of diverse signals, its limitations often surface in practical applications. Notably, popular signals like sinusoids, Dirac deltas, signums, and unit steps lack convergent Fourier representations within the conventional framework. This discrepancy necessitates distribution theory to build and interpret suitable representations for such signals. However, existing literature in signal processing and communication engineering often glosses over these intricacies, leaving researchers grappling with obscure concepts regarding the very existence of Fourier representations for non-conforming signals. This work bridges this critical gap by offering a comprehensive exploration of the conditions guaranteeing the existence of Fourier representations. We introduce a novel linear space – the Gauss–Schwartz (GS) function space – and its corresponding class of tempered superexponential (TSE) distributions. We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space over which the FT can be defined through duality. This theoretical advancement signifies a twofold contribution. Firstly, it clarifies the existence and interpretation of Fourier representations for prevalent signals that evade conventional analysis. Secondly, the introduction of the GS-TSE framework expands the reach of the FT to encompass a broader spectrum of functions, enabling novel applications in diverse fields. Ultimately, this work paves the way for a more robust and accessible understanding of Fourier analysis, empowering researchers and practitioners to leverage its full potential in exploring and processing a wider range of signals.