Geodesic
Divergence everywhere of the Fourier series of continuous functions of several variables
Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1153-1170 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Fourier series of a function $f$ of $n$ real variables is said to be $\lambda$-convergent at a point $\vec x$ for $\lambda \geqslant 1$ if there exists the limit $$ \lim _{\min \limits _kM_k\to +\infty}S_{\vec M}(\vec x,f) $$ over all indices $\vec M=(M_1,\dots ,M_n)$ such that $1/\lambda \leqslant M_k/M_j\leqslant \lambda$ for all $k$ and $j$. An example of a continuous function of $2m$ variables with modulus of continuity $$ \omega (F,\delta )=\underset {\delta\to +0}O\Bigl (\ln ^{-m}\frac 1\delta \Bigr) $$ is constructed such that the Fourier series of $F$ with respect to the trigonometric system $\lambda$-diverges everywhere for an arbitrary fixed $\lambda >1$. 
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A. N. Bakhvalov. Divergence everywhere of the Fourier series of continuous functions of several variables. Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1153-1170. http://geodesic.mathdoc.fr/item/SM_1997_188_8_a2/

[1] Carleson L., “On convergence and growth of partial sums of Fourier series”, Acta Math., 116 (1966), 135–157 | DOI | MR | Zbl

[2] Tevzadze N. R., “O skhodimosti dvoinogo ryada Fure funktsii, summiruemoi s kvadratom”, Soobsch. AN GSSR, 58:2 (1970), 277–279 | MR | Zbl

[3] Fefferman C., “On the divergence of multiple Fourier series”, Bull. Amer. Math. Soc., 77:2 (1971), 191–195 | DOI | MR | Zbl

[4] Bakhbukh M., Nikishin E. M., “O skhodimosti dvoinykh ryadov Fure ot nepreryvnykh funktsii”, Sib. matem. zhurn., 14:6 (1973), 1189–1199 | MR | Zbl

[5] Dyachenko M. I., “Nekotorye problemy teorii kratnykh trigonometricheskikh ryadov”, UMN, 47:5 (1992), 97–162 | MR | Zbl

[6] Oskolkov K. I., “Otsenka skorosti priblizheniya nepreryvnoi funktsii i ee sopryazhennoi summami Fure na mnozhestve polnoi mery”, Izv. AN SSSR. Ser. matem., 38:6 (1974), 1373–1407 | MR

[7] Zhizhiashvili L. V., “O skhodimosti kratnykh trigonometricheskikh ryadov Fure”, Soobsch. AN GSSR, 80:1 (1975), 17–19 | MR | Zbl