An analogue spectrum analyzer uses basically a variable bandpass filter which mid-frequency is automatically tuned (shifted, swept) through the range of frequencies of which the spectrum is to be measured.
A digital spectrum analyzer uses fast Fourier transform (FFT), a mathematical processes transforms a waveform into its compositions of frequencies in the spectrum. As a result, computer programs can compute such transforms, and makes audio processing easier. FFTs have applications in much wider fields.
In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find.
In Hilbert spaces in general, the statement of the spectral theorem for compact self-adjoint operators is virtually the same as in the finite-dimensional case.
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