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Jan 24 at 18:12 comment added Tom Copeland Googling 'Fejer kernel Dirac delta" brings up refs with details on the relations (you note) among Cesaro summation, Fourier series, and the Dirac delta. E.g., ocw.mit.edu/courses/… and victorchen.org/2019/10/21/summability-tauberian-fourier-series .
Jan 23 at 6:01 comment added Piero D'Ancona Periodic distributions and Fourier series with polynomially growing coefficients are exactly the same objects. Convergence of the series is in $S'$ (tempered distributions), or equivalently in the space of periodic distributions
Jan 23 at 5:38 history asked Caleb Briggs CC BY-SA 4.0