Question regarding the indexes $\alpha$ and $\beta$ in the definition of Schwartz space

Question

I was asked to prove that the Fourier transform of any Schwartz function belongs to the Schwartz class, i.e: $$||\xi^{\alpha}\frac{d^{|\beta|}\hat{f}}{d\xi^{\beta}}||<\infty$$ For $\alpha$ and $\beta$ equals a natural number. After applying the properties of the derivative of the Fourier transform and triangular inequality, i got that the expression above should be smaller to: $$ k \int_{-\infty}^{\infty} \frac{d^{|\beta|}x^{\beta}f}{dx^{\alpha}}dx $$ Where k is a constant. However, here I have a doubt about the definition, and is if for f to be Schwartz the condition should be satisfy for any $\alpha$ and $\beta$ belongs to the natural number or only for some specific values. I guess if this hold for any $\alpha$ and $\beta$ the above condition should be trivial, as I can apply chain rule for the derivative and apply multiple integration by parts to see that the integral is finite. Namely f is a Schwartz function if for any $\alpha$ and $\beta$ in the natural numbers the condition expressed above is satisfied.

Answer 1

Suppose $f\in \mathbb{S}$. Then, for all $\alpha\in \mathbb{N}$ and $\beta\in \mathbb{N}$, we have

$$\left|x^\alpha \frac{d^\beta}{dx^\beta}f(x)\right|<\infty$$.

Next, let $F$ be the Fourier Transform of $f$ as given by

$$F(k)=\int_{-\infty}^\infty f(x)e^{ikx}\,dx$$

Since $f\in \mathbb{S}$, we have

$$\begin{align} \left|k^\alpha\frac{d^\beta}{dk^\beta}F(k)\right|&=\left|k^\alpha \frac{d^\beta}{dk^\beta}\int_{-\infty}^\infty f(x)e^{ikx}\,dx\right|\tag1 \end{align}$$

Now, let's integrate by parts, $\alpha$ times, the integral on the right-hand side of $(1)$. Proceeding, we have

$$\begin{align} \left|k^\alpha\frac{d^\beta}{dk^\beta}F(k)\right|&=\left|k^\alpha \frac{d^\beta}{dk^\beta}\int_{-\infty}^\infty (-1)^\alpha \frac{d^\alpha f(x)}{dx^\alpha}\frac{e^{ikx}}{(ik)^\alpha}\,dx\right|\\\\ &=\left| \frac{d^\beta}{dk^\beta}\int_{-\infty}^\infty \frac{d^\alpha f(x)}{dx^\alpha}e^{ikx}\,dx\right|\\\\ &=\left| \int_{-\infty}^\infty \frac{d^\alpha f(x)}{dx^\alpha}\frac{d^\beta e^{ikx}}{dk^\beta}\,dx\right|\\\\ &=\left| \int_{-\infty}^\infty x^\beta\frac{d^\alpha f(x)}{dx^\alpha}e^{ikx}\,dx\right|\\\\ &\le \int_{-\infty}^\infty \left|x^\beta\frac{d^\alpha f(x)}{dx^\alpha}\right|\,dx\\\\ &<\infty \end{align}$$

And we are done!