Plancherel theorem

The Plancherel theorem is a result in harmonic analysis that establishes a deep connection between functions and their frequency components. Essentially, it states that you can study a function's behavior by analyzing its Fourier transform, which breaks the function down into simpler parts based on frequency. Importantly, the theorem guarantees that the total "energy" or information of the original function is preserved in its frequency representation. This means you can transition between the time domain (the original function) and the frequency domain (the transformed function) without losing any important details, allowing for powerful analysis and applications in various fields.