2
$\begingroup$

How do I go about proving that $f \in L^p(\mathbb{R})$ for $1\leq p \leq \infty $ if $f, \hat{f} \in L^1(\mathbb{R})$ ? I feel like I will have to somehow use the Fourier inversion theorem with some other inequality like Holder's inequality but I am stuck. Any help is highly appreciated.

$\endgroup$
  • $\begingroup$ If $f \in L^1$, then $\hat{f} \in L^{\infty}$. Interpolate. $\endgroup$ – user296602 Oct 3 '17 at 2:35
  • $\begingroup$ And the Fourier coefficients refers to the Fourier series not to the Fourier transform. $\endgroup$ – reuns Oct 3 '17 at 3:05
3
$\begingroup$

Since $\hat{f}\in L^1$ then $f$ is uniformly continuous. So you can set $$f = f_++f_- \qquad\qquad f_+(x) = f(x) 1_{|f(x)| \ge 1} \qquad\qquad f_-(x) = f(x) 1_{|f(x)| < 1}$$ and $f_+$ is compactly supported.

Finally since $f \in L^1$ $$\|f\|_{L^p}^p \le \|f_-\|_{L^1}^p+\|f_+\|_{L^p}^p < \infty$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.