For a continuous complex-valued continuous function $f$ on the unit circle $\mathbb{T}$, we have that $f\ast K_n$ converges uniformly to $f$, where $K_n$ are the Fejér kernels defined by taking Césaro means of Dirichlet kernels: $$ K_n(x) = \frac{1}{1+n}\sum_{j=0}^n D_j(x). $$ Recall that $D_j(x) = \sum_{k=-j}^j e^{ikx}. $ I would like to see a version of this result for functions on the torus $\mathbb{T}^N$, but I couldn't find it in the literature. So I ask for an indication in this forum.

I tried to use a similar idea to obtain the result, defining (for $N=2$) $$ D'_j(x_1,x_2) = \sum_{-j\leq k_1,k_2 \leq j } e^{i(k_1x_1+ k_2x_2)}, $$ $$ K'_n(x_1,x_2) = \frac{1}{1+n}\sum_{j=0}^n D'_j(x_1,x_2), $$ but couldn't advance. So here are a couple of questions: given a continuous function $f$ on $\mathbb{T}^2$ does $f\ast K'_n$ converge uniformly to $f$? If not is there another family of functions to put in place of $K'_n$ in order to have uniform convergence?

I am actually interested only in pointwise convergence, but I'm convinced that there must be results for uniform convergence.


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