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I am studying and trying to do something with fourier analysis, though I have little background and do not know much about the subject.

I believe, and please correct me if I am wrong that if your function $f$ is $n$ times differentiable and its fourier expansion is

$f(x)=\sum_{k=1}^\infty a_ke^{2\pi ikx}$

then we have $|a_k|\leq\frac{M}{k^n}$, and this result comes from differentiating $f$ and knowing that the coefficients must go to zero in the derivative as well as in the original function.

My question is, what is the higher dimensional analogue to this? Specifically what are the conditions needed on the function (derivatives, partial derivatives) and what are the upper bound on the coefficient in the case where $f$ is from $\mathbb{R}^2$ to $\mathbb{R}$?

Thanks so much, any help is appreciated. Also any reference to a book/place online where I could find this information would be appreciated as well. I have spent some time searching the web and a couple books on fourier analysis for this information.

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only periodic functions can be expanded by Fourier series. – chaohuang Jul 19 '12 at 4:53

Your statement is correct in the $1$-dimensional case if $f$ has period $1$ and is $k$ times continuously differentiable (i.e. $f^{(k)}$ is continuous). In $n$ dimensions, if $f$ has period $1$ in each coordinate and $k$ times continuously differentiable we have $f(x) = \sum_{\alpha \in {\mathbb Z}^n} a_\alpha e^{2 \pi i \alpha \cdot x}$, and since $D^\beta e^{2 \pi i \alpha . \cdot x} = (2 \pi i \alpha)^\beta e^{2 \pi i \alpha \cdot x}$ (where $(2 \pi i \alpha)^\beta = \prod_j (2 \pi i \alpha_j)^{\beta_j}$) we have that $|\alpha^\beta c_\alpha|$ is bounded for every multi-index $\beta$ with $\sum_{j=1}^n \beta_j \le k$.

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beautiful, thank you – user36119 Jul 23 '12 at 21:18

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