A discrete Fourier transform will produce frequency samples sufficient to recreate the original time sampled signal from sinusoidal waveforms.
That works great as long as the original signal is actually composed of sinusoidal waveforms. Let's assume that we know a signal is composed of non-sinusoidal waveforms, under which assumption its decomposition may be simpler, such as triangular waveforms. Does a discrete Fourier style transform exist which will decompose the sampled signal into triangular, rectangular or other waveforms? Does such a transform exist for any arbitrary waveform?
It's desirable here to have a similarly "clean" transform using the assumption that the signal is composed of sawtooth waveforms. So my question, does a transform exist that would decompose the signal into some non-sinusoidal waveforms?