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

Abstract

In their lecture note [1], P. Singh et al. (2022) argue that the Dirichlet conditions (DCs) commonly
stated in engineering texts are incomplete, and propose bounded variation (BV) together with an “extended
Carleson–Hunt theorem” as a more appropriate framework. This paper re-examines these claims in light
of the historical development of Fourier convergence theory and the distinction between Fourier-series and
Fourier-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 Dirichlet
type sufficient conditions for Fourier-series convergence on a finite period. Bounded variation, introduced by
Jordan (1881), constitutes a later and strictly broader criterion rather than a correction of Dirichlet’s original
hypotheses. Moreover, while BV(T) ⊂ Lp(T) holds on any finite period, BV(R) ̸⊂ L1(R); consequently
BV alone does not guarantee the existence of the classical Fourier transform on the real line. Re-analysis
of the seven examples in [1] confirms that they do not expose any deficiency in the classical Dirichlet-type
conditions, 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 finite
periodic domain.