When a signal undergoes time scaling, the Fourier series coefficients remain the same, but the representation of the Fourier series changes due to an alteration in the fundamental frequency.

Function symmetries, even, odd, or half-wave odd, are also essential for understanding the Fourier series.

A function is considered even if it remains the same when its input is negated. If a function is even, the Fourier series is simplified as specific components vanish.

A function is deemed odd if it changes sign when its input is negated. If a function is odd, again, the Fourier series simplifies as other components disappear.

A function exhibits half-wave symmetry if it looks identical to its original form when shifted by half a period and inverted. Interestingly, the Fourier series of a half-wave symmetric function contains only odd harmonics.

In music production and multimedia, time scaling adjusts the speed of audio signals, allowing for pitch correction and playback control.

In image processing, even-odd symmetry aids in efficient image reconstruction and compression, leading to optimal storage and improved visualization.