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For example in the 1-dimensional case, it is known that if f satisfies the α-Hölder condition, then $|f(x)-(S_Nf)(x)|\le K \frac{\ln N}{N^\alpha}$ where $S_N f$ is the n-term partial sum of the Fourier series of $f$.

Is there some similar result for multi-dimensional case? Thanks a lot.

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There are a book

L. Zhizhiashvili, Trigonometric Fourier series and their conjugates, Kluwer, Dordrecht, 1996,

and a survey

Alimov, Sh.A.; Ashurov, R.R.; Pulatov, A.K. Multiple Fourier series and Fourier integrals. In: Commutative harmonic analysis. IV: Harmonic analysis in ${\Bbb R}^n$. Encycl. Math. Sci. 42, 1-95 (Springer, 1992)

containing, in particular, conditions of the uniform convergence of multiple Fourier series similar to the classical Dini-Lipschitz condition.

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