[Math] Fourier transform of a Schwartz space function and norm

analysisfourier analysis

The Schwartz space (of rapidly decreasing functions) is the set of all $C^{\infty}$ functions $f\colon\mathbb{R}\to\mathbb{C}$ such that
$$
x^jf^{(k)}(x) \to 0
$$
for all integers $j,k\geq0$ as $x \to {\pm\infty}$.

Let $f$ be a rapidly decreasing function in Schwartz space and let $\mathcal{F}(f)$ its Fourier transform. If
$$
\| f \|_2 = \| \mathcal{F}(f) \|_2 \quad,
$$
can we deduce that $\mathcal{F}(f)$ is a Schwartz function?

Best Answer

You don't need their norms to be equal - this is a consequence of the normalization factor chosen for the Fourier transform, and is called the Plancherel Theorem.

To show that the Fourier transform of a Schwartz function is Schwartz, it suffices to show the following:

  • $k^\alpha\hat{f}(k)=i^{-\alpha}\mathcal{F}\left[f^{(\alpha)}\right]$
  • $\frac{d^\alpha}{dk^\alpha}\hat{f}(k)=(-ik)^\alpha\hat{f}(k)$

Combine these two facts with the Riemann-Lebesgue lemma and you've got it.

Note: I'm using the following convention for the Fourier transform:

$$ \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x)e^{-ikx}dx $$

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