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}. $

$\textbf{Question.}$ Is $\{E_m\}_{m\in \mathbf{Z}^2}$ dense in $C^k(Q)$ for $k\geq 0$.

$\textbf{Ideas.}$ 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 https://math.stackexchange.com/questions/381321/are-trigonometric-functions-dense-in-cks1?rq=1. 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.

$\textbf{Secondary Question.}$ 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 $i+j \leq k$.

$\textit{Claim:}$ $\delta^{(i+j)}_{x_l,\xi_l}$ 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.

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.

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}. $

$\textbf{Question.}$ Is $\{E_m\}_{m\in \mathbf{Z}^2}$ dense in $C^k(Q)$ for $k\geq 0$.

$\textbf{Ideas.}$ 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 https://math.stackexchange.com/questions/381321/are-trigonometric-functions-dense-in-cks1?rq=1. 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.

$\textbf{Secondary Question.}$ 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 $i+j \leq k$.

$\textit{Claim:}$ $\delta^{(i+j)}_{x_l,\xi_l}$ 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.

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}. $

$\textbf{Question.}$ Is $\{E_m\}_{m\in \mathbf{Z}^2}$ dense in $C^k(Q)$ for $k\geq 0$.

$\textbf{Ideas.}$ 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 https://math.stackexchange.com/questions/381321/are-trigonometric-functions-dense-in-cks1?rq=1. 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.

$\textbf{Secondary Question.}$ 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 $i+j \leq k$.

$\textit{Claim:}$ $\delta^{(i+j)}_{x_l,\xi_l}$ 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.