"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms? I was studying the introduction to wavelets and its benefit over the frequency domain. I said that:


  Fourier analysis can't localize signals both in time and frequency
  domain. Fourier analysis can localize signal in frequency domain very
  well, but not so much in time domain. While wavelet has the advantage
  of localizing signals both in time and frequency domain.


What does this mean? Could you give me any example or link that explains this scenario better?
In the Fourier transform, the basis functions are complex exponentials. These functions are perfectly localized in the frequency domain, i.e., they exist at one frequency, but they have no time localization because of their infinite duration. The localization of a function depends on its spread in time and frequency. A complex exponential has zero spread in frequency, but it has infinite spread in the time domain. Consequently, any signal analyzed by the Fourier transform is only localized in frequency (e.g., frequencies of sinusoidal components can be identified), but not at all in time (e.g., temporal changes of signal properties cannot be localized).