We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$
For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=\sum_{n=-N}^N(1-\frac{|n|}{N+1})e^{2\pi inx}$$ and we take $f*F(x)$ where $''*''$ denotes the convolution of two functions.
Is there an analogue of this theorem in two dimensions,i.e in $L^1([0,1)^2)$?
If there is,then where can i find o proof of this result?
Thank you in advance.