# How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way:

$$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$

Because we can suppose $f$ is written in this form, and then multiply if by $\cos(x)$ and also by $\sin(x)$, integrate in its period and then use some properties of orthogonal integration to zero out every coeficiente except the one you need. Then you isolate $a_n$ and $b_n$ to get:

\begin{align}a_0 &= \frac{1}{L}\int_{-L}^{L}f(t)dt \\a_n &= \frac{1}{L}\int_{-L}^{L}f(t)\cos\left(\frac{n\pi t}{L}\right) dt \\b_n &= \frac{1}{L}\int_{-L}^{L}f(t)\sin\left(\frac{n\pi t}{L}\right) dt\end{align}

This is nice, I understand how and why we can write the coefficients in this way. I've learned all this in a book, but it didn't explain me where the $a_0$ comes from.

However, my real question is: how we go from this, to the Fourier transform? What's so good about Fourier Series that able me to create a transformation? What's the intuition? What is the reason someone, looking for these series, imagine a transformation?

• Why can't you see how they find $a_0$? – Mhenni Benghorbal Sep 14 '14 at 5:38
• See this where they go from Fourier series to Fourier transform, A technique which is used in quantum mechanics. – Mhenni Benghorbal Sep 14 '14 at 5:54
• @Guerlando OCs: $a_0$ and $b_0$ comes from setting $n=0$ in the formulas for $a_n$ and $b_n$. Not just "some function" can be written this way, but any continuous function on a limited interval... – Lehs Sep 14 '14 at 6:01
• @Lehs so why they make people remember the formula for $a_0$? And why they don't make one for $b_0$? – Guerlando OCs Sep 14 '14 at 21:48
• @Guerlando OCs: I don't know, but $b_0=0$. Now, hope someone can explain the transform for us. – Lehs Sep 14 '14 at 22:13

There is a quite nice way of explaining the connection of Fourier transforms and Fourier series using distributions. Let us first write the function $f$ as a complex Fourier series $$f(x)=\sum_{n=-\infty}^\infty c_n e^{in\pi x/L}.$$
The next step is to compute the distributional Fourier transform of a pure exponential of the form $g_n(x)=\exp(in\pi x/L)$, which is given by $\mathcal F(g)=\delta_{n/L}$. The delta distribution $\delta_x$ is explained here. It is loosely speaking just a spike located at $x$ and $0$ elsewhere, I don't want to go into detail on distributional Fourier transforms. Since the Fourier transform is linear we can transform $f$ to find $$\hat f(\xi)= \sum_{n=-\infty}^\infty c_n \delta_{n/L},$$ so $\mathcal F(f)$ is a train of spikes. The spacing between these spikes is $1/L$, which corresponds to the periodicity of the function $f$. Now if we want to Fourier transform a general square integrable function we can not assume any periodicity so we let $L \rightarrow \infty$. By doing so the spacing between the spikes becomes smaller and we get closer and closer to cover the whole real line. I understand that this argument is not air tight but it should give you an intuition of what the connection between Fourier series and Fourier transforms is.