approximating differentiable functions with double trigonometric polynomials

Question

Let $Q = [0,1]^2$. For sake of notation, let $$ f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi). $$ Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if $$ \|f\|_{C^k} = \sum_{i+j \leq k}\sup_{(x,\xi)\in Q}|f^{(i,j)}(x,\xi)| < \infty. $$ Additionally, let $m = (m_1,m_2) \in \mathbf{Z}^2$. Define $$ E_m = E_m(x,\xi) = e^{2\pi i m_1 x}e^{2\pi i m_2 \xi}. $$ Question. Is $\{E_m\}_{m\in \mathbf{Z}^2}$ dense in $C^k(Q)$ for $k\geq 0$.

Ideas. The question can be rephrased as follows. For any $f\in C^k(Q)$ and $\epsilon > 0$ does there exist some double trigonometric polynomial $$ p_k(x,\xi) = \sum_{m\in F}c_m E_m,\quad F \subseteq \mathbf{Z},\; |F| <\infty. $$ such that $$ \|f-p_k\|_{C^k} < \epsilon. $$ I am confident that for $k=0$ this result is already been established elsewhere. I've tried a similar technique in this Q&A. However, the multi-index derivatives make it complicated to carry over. I've also thought about some potential "convolution with some kernel" argument, but I haven't given it much time. Any advice would be appreciated or a counterexample if this does not hold.

Secondary Question. I actually want to prove or dispove the following (I think the aforementioned claim helps). Let $$ \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x_l,\xi_l) = \langle f, \delta^{(i,j)}_{x_l,\xi_l}\rangle. $$ Morever, let $C^k(Q)$ be the set of $k$ continuously differentiable periodic functions. Hence, $\delta^{(i+j)}_{x_l,\xi_l}\in C^k(Q)'$ for $i+j \leq k$.

Claim: $\delta^{(i+j)}_{x_l,\xi_l}$ is uniquely determined by its Fourier coefficients.

I believe this is true when $k=0$, however, it seems less trivial for arbitrary $k\geq 1$. Any assistance would be greatly appreciated.

Answer 1

In my opinion this is a question of MSE level.

For the first question the answer is yes. Because finite trigonometric polynomials is dense in $C^k(\Bbb T^n)$. This is essentially another Weierstrass approximation theorem.

Note that it is possible that for a $f(x,y)\in C^k(\Bbb T^2)$ neither $\sum_{|j|+|k|\le N}\hat f(j,k)e^{2\pi i(jx+ky)}$ or $\sum_{|j|^2+|k|^2\le N^2}\hat f(j,k)e^{2\pi i(jx+ky)}$ converges to $f$ in $C^k$. It is fail even in 1-dimension. You can read Stein and Shakarchi's Fourier Analysis: An Introduction for a proof on $k=0$. The larger $k$ can be modified similarly.

Just a remark the lim $N\to\infty$ do not converge in $C^k(\Bbb T^2)$. But if you only consider a suitable subsequence $N_l\to\infty$ it may still converges. And when we just prove Weierstrass approximation we don't even need the $p_k$ to be the partial Fourier series.

For your second question we prefer to use the notation $\partial^\alpha f(x_0)=(-1)^{|\alpha|}\langle f,\partial^\alpha\delta_{x_0}\rangle$. Here $\alpha=(\alpha_1,\dots,\alpha_n)$, $|\alpha|=\alpha_1+\dots+\alpha_n$ and $f\in C^k(\Bbb T^n)$ (yes your formula misses a $\pm$ sign).

$\partial^\alpha\delta_{x_0}$ is a distribution, whose Fourier expansion converges also in the space of distributions. I don't have an immediate reference on hand. You can read Folland's Real Analysis Chapter 9.2 for a brief discussion.