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On the Convergence of Fourier Representations and Schwartz Distributions
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  • Pushpendra Singh ,
  • Amit Singhal ,
  • Binish Fatimah ,
  • Anubha Gupta ,
  • Shiv Dutt Joshi
Pushpendra Singh
School of Engineering, National Institute of Technology Hamirpur

Corresponding Author:[email protected]

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Amit Singhal
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Binish Fatimah
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Anubha Gupta
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Shiv Dutt Joshi
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Fourier theory is a popular tool for analyzing various signals and interpreting their spectral contents. It finds applications in a wide variety of subject areas. However, some of the popular signals, such as sinusoidal signals, Dirac delta, signum, and unit step, fail to have a convergent Fourier representation (FR) in the conventional sense. Hence, it becomes imperative to utilize the distribution theory to understand and build a suitable representation for these signals. The signal processing and communication engineering literature does not explain these concepts clearly. As a result, many of the concepts of FR for signals that do not conform to the conventional derivations remain obscure to researchers. We attempt to bridge this gap and provide a comprehensive explanation regarding the existence of FR. Further, we have proposed a new linear space of Gauss–Schwartz (GS) functions and corresponding tempered superexponential (TSE) distributions. It is shown that the Fourier transform (FT) is an isomorphism on the GS space of test functions and hence by duality is an isomorphism on TSE distributions. The space GS is smallest in the sense that its dual space, a set of TSE distributions, is the largest linear space over which FT can be defined by duality.