MHB Fourier Transform: Calculate $\hat{g}(\omega)$

[evinda]

Hello! (Wave)

I want to calculate the Fourier transform of $g(x)=|x|$.

I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$

Is it right so far?

How can we calculate $\lim_{x \to +\infty} \frac{x \sin{(\omega x)}}{\omega}$ ?

 

[Ackbach]

What's your definition of the Fourier Transform (including constants)?

 

[evinda]

What's your definition of the Fourier Transform (including constants)?

$$\hat{g}(\omega)=\int_{-\infty}^{+\infty} g(x) e^{-i x \omega} dx$$

 

[Opalg]

Hello! (Wave)

I want to calculate the Fourier transform of $g(x)=|x|$.

I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$

Is it right so far?

How can we calculate $\lim_{x \to +\infty} \frac{x \sin{(\omega x)}}{\omega}$ ?

For a function to have a Fourier transform, it has to satisfy some integrability condition. The function $|x|$ is not at all integrable, in fact it tends to infinity as $x\to\infty$. I don't think there is any sense in which this function can have a Fourier transform.

 

[evinda]

For a function to have a Fourier transform, it has to satisfy some integrability condition. The function $|x|$ is not at all integrable, in fact it tends to infinity as $x\to\infty$. I don't think there is any sense in which this function can have a Fourier transform.

Neither using distribution theory?