Geodesic
The~rate of Pringsheim convergence of multiple Fourier series of piecewise monotonic functions of many variables in the~space~$L$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 4, pp. 1003-1013.

Voir la notice de l'article provenant de la source Math-Net.Ru

It has been earlier proved by the author that the Fourier series of piecewise monotonic functions of many variables converge in the sense of Pringsheim pointwise and in $C(T^m)$-metric faster than in the case of arbitrary continuous functions. The main result of the paper says that this is not valid for the Pringsheim convergence in $L(T^m)$-metric. 
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     author = {M. I. Dyachenko},
     title = {The~rate of {Pringsheim} convergence of multiple {Fourier} series of piecewise monotonic functions of many variables in the~space~$L$},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1003--1013},
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M. I. Dyachenko. The~rate of Pringsheim convergence of multiple Fourier series of piecewise monotonic functions of many variables in the~space~$L$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 4, pp. 1003-1013. http://geodesic.mathdoc.fr/item/FPM_1999_5_4_a3/