Fourier transform of the linear function

• Irid
In summary, the conversation discusses the meaning of the Fourier transform of a linear function and whether the expression \frac{\delta(k)}{ik} makes sense. It is concluded that the expression is undefined and a handwaving way to understand it is through the use of the derivative and the Dirac delta function. The conversation also addresses the presence of a minus sign in the expression and explains that it is necessary to obtain the correct result for the original integral.
Irid
Hello,
I was wondering if one can give meaning to the Fourier transform of the linear function:

$$\int_{-\infty}^{+\infty} x e^{ikx}\, dx$$

I found that it is $$\frac{\delta(k)}{ik}$$, does this make sense?

This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
$$\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).$$

1 person
vanhees71 said:
This expression doesn't make sense since it's intrinsically undefined. A handwaving way is
$$\int_{\mathbb{R}} \mathrm{d} x x \exp(\mathrm{i} k x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \int_{\mathbb{R}} \mathrm{d} x \exp(\mathrm{i} k x)=-2 \pi \mathrm{i} \frac{\mathrm{d}}{\mathrm{d} k} \delta(k).$$
Hmm.. seems to make sense. Why is there a minus sign popping up?

d/dk(exp(ikx)) = ixexp(ikx). you need -i to get 1 for the original integral.

1 person

What is the Fourier transform of a linear function?

The Fourier transform of a linear function is another linear function. It is given by the equation F(f(x)) = F(x), where f(x) is the original linear function and F(x) is the Fourier transform of that function.

How does the Fourier transform of a linear function affect the frequency domain?

The Fourier transform of a linear function shifts the original function from the time domain to the frequency domain. This means that the Fourier transform allows us to analyze the different frequencies present in the original linear function.

What is the relationship between the Fourier transform of a linear function and the Fourier series?

The Fourier transform of a linear function is essentially the continuous version of the Fourier series. While the Fourier series decomposes a periodic function into a sum of sinusoidal functions, the Fourier transform extends this concept to non-periodic functions.

How is the Fourier transform of a linear function calculated?

The Fourier transform of a linear function is calculated using the Fourier transform integral, which is given by the equation F(x) = ∫f(x)e-2πixξdx. This integral can be solved using techniques such as integration by parts or the use of Fourier transform tables.

What is the significance of the Fourier transform of a linear function in signal processing?

The Fourier transform of a linear function is a crucial tool in signal processing as it allows us to analyze the frequency content of a signal. This is useful in applications such as filtering, noise reduction, and compression of signals. It also provides a way to represent signals in the frequency domain, which can make certain calculations and manipulations easier.

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