A Rebuttal to “Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations" by P. Singh et al.

In their lecture note [1], P. Singh et al. (2022) argue that the Dirichlet conditions (DCs) commonlystated in engineering texts are incomplete, and propose bounded variation (BV) together with an “extendedCarleson–Hunt theorem” as a more appropriate framework. This paper re-examines these claims in lightof the historical development of Fourier convergence theory and the distinction between Fourier-series andFourier-transform settings. The standard textbook conditions—absolute integrability, finitely many extrema,and finitely many finite discontinuities—are shown to be a faithful modern formulation of classical Dirichlettype sufficient conditions for Fourier-series convergence on a finite period. Bounded variation, introduced byJordan (1881), constitutes a later and strictly broader criterion rather than a correction of Dirichlet’s originalhypotheses. Moreover, while BV(T) ⊂ Lp(T) holds on any finite period, BV(R) ̸⊂ L1(R); consequentlyBV alone does not guarantee the existence of the classical Fourier transform on the real line. Re-analysisof the seven examples in [1] confirms that they do not expose any deficiency in the classical Dirichlet-typeconditions, but rather illustrate the need to distinguish domains, function spaces, and modes of convergence.No replacement of the standard Dirichlet-type conditions is required for Fourier-series convergence on a finiteperiodic domain.