Let $L^1(\mathbb{R})+L^2(\mathbb{R})$ be the space of $h=h_1+h_2$ functions, where $h_1 \in L^1(\mathbb{R})$ and $h_2 \in L^2(\mathbb{R})$.
The FT on $L^1(\mathbb{R})+L^2(\mathbb{R})$ is defined as $\hat{h}_1 + \hat{h}_2$
Consider the Fourier transform from the space $L_1(\mathbb{R})+L_2(\mathbb{R})$ to $C_0(\mathbb{R})$. Is this operator surjective? Namely, for any $f$ in $C_0(\mathbb{R})$ can we find $h\in L^1(\mathbb{R})+L^2(\mathbb{R})$ such that $\hat{h}=f$?
My attempt:
Fourier inversion theorem (assume I can apply it here, but not completely sure if I can)
Ignoring constants, checking the $L^1$ norm
\begin{align} \int_{{\mathbb{R}} }\left|\int_{\mathbb{R}} f(\xi )e^{i \xi x } d \xi\right|dx &\leq \int_{{\mathbb{R}} }\left|\int_{\mathbb{\infty}}^{-\delta_1} f(\xi )e^{i \xi x } d \xi+\int_{\mathbb{-\delta_1}}^{\delta_2} f(\xi )e^{i \xi x } d \xi + \int_{\mathbb{\delta_2}}^{\infty} f(\xi )e^{i \xi x } d \xi\right|dx\\ &\leq \int_{{\mathbb{R}} }\left|C_1\int_{\mathbb{-\infty}}^{-\delta_1} e^{i \xi x } d \xi+\int_{\mathbb{-\delta_1}}^{\delta_2} f(\xi )e^{i \xi x } d \xi + C_2\int_{\mathbb{\delta_2}}^{\infty} e^{i \xi x } d \xi\right|dx\\ &\leq \int_{{\mathbb{R}} }\left|C_1 \left.\frac{-i e^{i \xi x}}{x}\right|_{\xi=-\infty}^{-\delta_1}+\int_{\mathbb{-\delta_1}}^{\delta_2} f(\xi )e^{i \xi x } d \xi + C_2 \left.\frac{-i e^{i \xi x}}{x}\right|_{\delta_2}^{\infty} \right|dx\\ &\leq \int_{{\mathbb{R}} }\left|C_1'\frac{1}{x}\right|+ \left|\int_{\mathbb{-\delta_1}}^{\delta_2} f(\xi )e^{i \xi x } d \xi\right| + \left|C_2'\frac{1}{x}\right|dx\\ \end{align} The deltas are just parts where I cut off the function and bound it by some constant as the function decays from here and will reach a max of $C_1$,$C_2$.
Without even dealing with the middle term, I cannot get the nicer two beside it to converge under this integral or under the $L^2$ norm, my guess is that this statement is not true, but I am also not sure if my bounding was too rough... I think I can deal with the middle term in a similar fashion though as it is a continuous function on a bounded interval.
Naturally I look for fourier inverse of $\frac{1}{x}$ using wolfram alpha, the fourier inverse is $-i \sqrt\frac{\pi}{2} sgn(x)$. https://www.wolframalpha.com/input/?i=-i+sqrt%28%CF%80%2F2%29+sgn%28t%29&assumption=%22ClashPrefs%22+-%3E+%7B%22Math%22%7D Which looks something like a delta function, but clearly the $L^1$ norm and $L^2$ norm of this diverges as it is basically a constant function. This function only behaves badly at $0$. Are functions which do not behave like this at $0$ have similar fourier inverses which make it so they are not in $L^1(\mathbb{R})+L^2(\mathbb{R})$.
This is my first time posting on MSE, please tell me what to edit before flagging or reporting, I am not familiar with the guidelines.
