Nitpick but the finite sum of sines is not an approximation ( except for quantization effects when transforming between time/frequency) A defining characterist

Nitpick but the finite sum of sines is not an approximation ( except for quantization effects when transforming between time/frequency) A defining characteristic of finite length orthogonal transforms is that there exists an inverse transform that preserves energy. What that means is for any particular orthogonal transform, such as the DFT which maps a time domain vector to frequency domain vector, the transformed vector is not an approximation or the time domain. It's the same energy, arranged differently. Nothing is lost in the transform, and it can be transformed back. This is actually why lossy codecs employ lossless transforms - they decorrelate the energy of a signal over a window of time into a representation that has the same energy but compacted into fewer elements of the vector. Some information arranged differently, in a context that makes it more useful. There is no fundamental building block of a signal other than energy. Whatever domain you choose to view that energy on depends on the context for which the signal is meaningful. For our ears that's time. For our brains it's frequency. For any given algorithm or encoding, it could be something totally different (eg the DCT domain, which is like frequency, but isn't the same). Another way to think of it is that frequency and time are two compositions of information, there are infinite such compositions, and infinite ways to map between them. None is any more fundamental than the other, only more meaningful or useful in specific contexts.