Hey I'm a bit confused about

Usually one shows the fourier density of the functions $\sin(kx)$ in $L^2([0,1])$ using the Fourier transform. This in fact comes from the Stone-Weirstrass theorem however and then uses the density of continuous functions in $L^1([0,1])$. If I apply L^2([0,1])$.

However, the Stone Weirstrass's Weirstrass theorem can be used to show, it tells me for example, that the functions $e^{ikx}$ are indeed dense in the continuous functions C([0,1]) $C([0,1])$ and hence dense in $L^1([0,1])$. Is the difference that the coefficients are not the Fourier ones? ieL^1([0,1])$ as well. there must be other So we obtain (not-necessarily-unique) coefficients $c_k$ so such that one has $f(x) f_k(x) =c_ke^{ikx}$ converge to any given $f \sum c_k e^{ikx}$ converging in L^1([01])$. How should I think about these coefficients? How do they relate to the Fourier series of $L^1([0,1])$?

Thanks!f$ (with basis $e^{ikx})$?

Hey I'm a bit confused about the fourier transform in $L^1([0,1])$. If I apply Stone Weirstrass's theorem, it tells me that $e^ikx$ e^{ikx}$ are indeed dense in the continuous functions C([0,1]) and hence dense in $L^1([0,1])$. Is the difference that the coefficients are not the Fourier ones? ie. there must be other coefficients $c_k$ so that one has $f(x) = \sum c_k e^ikx$.e^{ikx}$ converging in $L^1([0,1])$?

Thanks!

Hey I'm a bit confused about the fourier transform in $L^1$. L^1([0,1])$. If I apply Stone Weirstrass's theorem, it tells me that $e^ikx$ are indeed dense in the continuous functions C([0,1]) and hence dense in $L^1$. L^1([0,1])$. Is the difference that the coefficients are not the Fourier ones? ie. there must be other coefficients so that one has $f(x) = \sum c_k e^ikx$.

Thanks!

Hey I'm a bit confused about the fourier transform in $L^1$. If I apply Stone Weirstrass's theorem, it tells me that $e^ikx$ are indeed dense in the continuous functions and hence dense in $L^1$. Is the difference that the coefficients are not the Fourier ones? ie. there must be other coefficients so that one has $f(x) = \sum c_k e^ikx$.

Thanks!