1. The Limit That Opens the Continuum: Period Stretched to Infinity
Callback — The Fourier Series Starting Point
Recall from the Fourier series block: a periodic function $\tilde{x}(t)$ with period $T$ can be written as a sum of harmonics at frequencies $\omega_k = k\omega_0$, where $\omega_0 = 2\pi/T$ is the fundamental frequency. The Fourier series is:
$$ \tilde{x}(t) = \sum_{k=-\infty}^{\infty} c_k\, e^{ik\omega_0 t}, \qquad c_k = \frac{1}{T}\int_{-T/2}^{T/2} \tilde{x}(t)\,e^{-ik\omega_0 t}\,dt $$Each coefficient $c_k$ lives at a discrete frequency $\omega_k = k\omega_0$. The spectrum is a line spectrum — a set of isolated spikes.
1.1 The Harmonics Crowding into a Continuum
Now ask: what happens when the period $T$ grows without bound?
- The fundamental frequency $\omega_0 = 2\pi/T$ shrinks toward zero.
- The spacing between consecutive harmonics $\Delta\omega = \omega_0 = 2\pi/T$ becomes infinitesimally small.
- The discrete spectral lines crowd together until they merge into a continuous density.
The honest picture: The discrete spectrum's isolated lines don't vanish — they merge into a continuous curve. Each line's height $c_k$ shrinks (it must, because there are more and more of them), but the product $c_k \cdot T$ approaches a well-defined function of the continuous variable $\omega$. That function is the Fourier transform.
1.2 The Derivation Sketched (Motivation, Not Proof)
Define $\hat{f}(\omega_k) = T\,c_k = \displaystyle\int_{-T/2}^{T/2} f(t)\,e^{-i\omega_k t}\,dt$. Then the reconstruction formula becomes: