Are trigonometric polynomials dense in $C_{2\pi}^m$?

Question

Let $m \in N$ be fixed and let $C_{2\pi}^m$ be a class of functions $f : R \rightarrow R$ of class $C^m$ and periodic with a period $2\pi$ with the following metric $$d(f,g)=\sum_{k=0}^m \sup_{\{x \in R \}}|f^{(k)}(x)-g^{(k)}(x)|$$ for $f,g \in C_{2\pi}^{m}$. Is the set of trigonometric polynomials dense in this metric space?

Answer 1

Yes they are, because you can restrict to an interval $[0, 2 \pi]$ of function $f(0) = f(2 \pi)$, and use that they are dense in $C[0,2 \pi]$. Integration $k$-times and observing carefully what gets added will give you an affirmative answer. It is a continuous, surjective operator $C \mapsto C^k$.

Answer 2

I have an another idea using Fourier series. Let $f \in C_{2\pi}^m$ and let $(S_n)_{n=0}^\infty$ be the sequence of partial sum of the Fourier series of $f$. Then it is known that $(S_n^{(k)})_{n=0}^\infty$, for each $k\in \{0,\ldots m\}$, (that is sequuences of $k$-th derivatives) is a sequence of the partial sum for $f^{(k)}$. By continuity and $2\pi$-periodicity of each $f^{(k)}$, for $k=0,\ldots,m$, and the Feier Theorem we have that for each $k\in \{0,\ldots ,m \}$:

$$\left(\frac{S_0(x)+S_1(x)+\ldots S_n(x)}{n+1}\right)^{(k)}=\frac{S_0^{(k)}(x)+S_1^{(k)}(x)+\ldots S_n^{(k)}(x)}{n+1} \rightrightarrows f^{(k)}(x) $$ as $n\rightarrow \infty$, for $x \in R$. Obviously $\left(\frac{S_0+S_1+\ldots S_n}{n+1}\right)_{n=0}^\infty$ is a sequence of trigonometric polynomials.