The Schwartz space on [itex]\mathbb{R}^d[/itex] is defined to be(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

S(\mathbb{R}^d) := \{f\in C^{\infty}(\mathbb{R}^d,\mathbb{C})\;|\; \|f\|_{S,N}<\infty\;\forall N\in\{0,1,2,3,\ldots\}\}

[/tex]

where

[tex]

\|f\|_{S,N} := \underset{|\alpha|,|\beta|\leq N}{\textrm{max}}\;\underset{x\in\mathbb{R}^d}{\textrm{sup}}\; |x^{\alpha}\partial^{\beta}f(x)|.

[/tex]

Alpha and beta are multi-indexes. It turns out, that when Fourier transform is defined on this space, with the integral formula, one obtains a continuous mapping [itex]\mathcal{F}:S(\mathbb{R}^d)\to S(\mathbb{R}^d)[/itex]. Since the Schwartz space is dense in [itex]L^p(\mathbb{R}^d)[/itex], [itex]1\leq p < \infty[/itex], it is possible to obtain a continuous extension of the Fourier transform onto the [itex]L^p(\mathbb{R}^d)[/itex] too. I have not seen explicit counter examples yet, but I've heard that one cannot define the Fourier transform directly with the integral formula in [itex]L^p[/itex] when [itex]p>1[/itex].

My question deals with the range of the Fourier transform. Am I correct to guess, that we have

[tex]

\mathcal{F}(L^p)=L^q

[/tex]

with [itex]1/p+1/q=1[/itex]? It is a known result, that [itex]\mathcal{F}(L^2)=L^2[/itex]. It is also easy to show for example that [itex]\|\mathcal{F}f\|_{\infty}\leq \|f\|_1[/itex], but I'm not sure how to show equality. [itex]\mathcal{F}(L^p)=L^q[/itex] would seem plausible result, but I couldn't find it with a quick skim over the Rudin's Fourier transform chapter at least.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier transform, domains, ranges, L^p-spaces

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**