What Does Each Subscript in the Fourier Transform Function Represent?

[R.Harmon]

Homework Statement

Generate the Fourier transform of the function:
$$f(x)=f_{-Na}(x)+f_{-(N-1)a}(x)+\cdots+f_{(N-1)a}(x)+f_{Na}(x)$$

Homework Equations

$$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(x)e^{ikx}dx$$
The following formula may also be used:
$$\frac{1}{x^N}+\frac{1}{x^{N-1}}+\cdots+1+\cdots+x^{N-1}+x^N=x^{-N}(1+x+x^2+x^{2N}) =\frac{x^{-(N+0.5)}-x^{N+0.5}}{x^{-0.5}-x^{0.5}}$$
It also says in the question that this is a mathematical form of a diffraction grating.

The Attempt at a Solution

My problem with this question is not so much doing the Fourier transformation, but more that I don't understant what the subscripts in f(x) mean (for example $$f_{-Na}(x)$$), so I can't even make a start on the question without doing it in some general form, so could anybody possibly point me in the right direction with how to tackle these? Any help is greatly appreciated.

 

[mighty2000]

The subscripts in f(x) indicate different functions that are being added together. For example, f_{-Na}(x) represents one function, f_{-(N-1)a}(x) represents another function, and so on. These functions are being added together to create the overall function f(x).

To generate the Fourier transform of this function, you will need to use the formula given in the Homework Equations section. You will need to apply this formula to each individual function in the summation, and then add all of the resulting Fourier transforms together.

Since the functions in the summation are not specified, you will need to use the general form of the formula. This means that instead of using f(x), you will use a variable such as g(x) or h(x) to represent the individual functions.

Once you have applied the formula to each individual function and added them together, you will have the Fourier transform of the overall function f(x). This will give you a mathematical representation of the diffraction grating described in the question.

I hope this helps to clarify the question and guide you in the right direction for solving it. Good luck!