Maximum subspace of Lp space which has Fourier transform.

Question

What is the maximum subspace of Lp (p>2) space which has Fourier transform? Is it Schwartz class or bigger than it?

Answer 1

The largest subspace is $L^1 \cap L^p$ since we need $\int_{\mathbb{R}^n} f(x)e^{-2\pi ix \xi}dx$ to exist for each $\xi \in \mathbb{R}^n$. If $f \in L^1$, then the integral exists, and we need $f \in L^1$, which can be seen by taking $\xi = 0$ for example. Since you wanted it defined on $L^p$ as well, we need $f$ to be in $L^p$ as well; hence $L^1 \cap L^p$.

The issue for defining it directly on $L^2$ is that $\int_{\mathbb{R}^n} f(x)e^{-2\pi i \xi x}dx$ might not converge if we only know $f \in L^2$, similar to how $\int_1^\infty (\frac{1}{x})^2dx$ exists but not $\int_1^\infty \frac{1}{x}dx$ does not. We can define the fourier transform of $f \in L^2$ as the limit of the fourier transform of Schwarz functions approximating $f$ (note schwarz functions are in $L^1$ so taking the fourier transform of them makes sense). This implicitly uses the fact that the set of Schwarz functions is dense in $L^2$.