# Understanding Fourier Transform Theorem in Folland: Theorem 8.35

While reading Folland's Real Analysis textbook, I came across the Fourier Analysis section and became troubled by Theorem 8.35. It goes as follows:

Focusing on part $$a.$$ of the theorem, I am confused about the sudden change of supposition where we originally considered $$f \in L^1 + L^2$$, but then in $$a.$$ suddenly consider $$f \in L^p \, (1 \leq p < \infty)$$. In particular, I have not seen any mention of the Fourier transform if $$f \in L^p$$ for $$p > 2$$ (could we use Schwartz functions?). Furthermore as he begins the proof he lets $$f = f_1 + f_2$$ where $$f_1 \in L^1$$ and $$f_2 \in L^2$$, which to me signifies that he is still using $$f \in L^1 + L^2$$. The proof of $$a.$$ makes sense to me, but only for $$f \in L^1 + L^2$$, how are we suddenly also considering $$f \in L^p \, (1 \leq p < \infty)$$?

• I think the assumption that $f \in L^1 + L^2$ applies throughout. In part a, $f \in L^p$ is an additional assumption.
– user169852
Nov 3, 2020 at 21:57
• Yes, that's exactly what was throwing me off. I thought Folland was throwing out that assumption. Thank you. Nov 3, 2020 at 22:00

If $$f=f_1+f_2$$ and $$f_j\in L^j,$$ then we know that $$\hat{f_1}\in L^\infty$$ and $$\hat{f_2}\in L^2$$ by the mapping properties of the Fourier transform. In the second sentence of the statement of the theorem, they assume that $$f\in L^1+L^2,$$ so such a decomposition is perfectly kosher. They assumed that $$f\in L^p$$ in part (a) to use theorem 8.14.

• Yes, I am familiar with the mentioned mapping property. I suppose my difficulty is if $f \in L^p$ for $p > 2$, then what exactly is $f^t$ as is defined in the theorem? In particular, what is the $\widehat{f}(\xi)$ term in $f^t$? Nov 3, 2020 at 21:54
• The Fourier transform is linear, so it makes perfect sense to take the Fourier transform of $f_1+f_2$ (which equals $f$). The $L^p$ assumption isn't used to take the Fourier transform.
– cmk
Nov 3, 2020 at 21:56
• If you like, $f\in (L^1+L^2)\cap L^p$ in part a.
– cmk
Nov 3, 2020 at 21:56
• Oh gosh, I get it! I understand what you're saying and understand about the linearity principal. I was assuming we disregarded that $f \in L^1 + L^2$ when we said $f \in L^p$. Thank you. Nov 3, 2020 at 21:58
• Happens to everyone sometimes :)
– cmk
Nov 3, 2020 at 22:03