The interesting thing (provided you assume nice functions that are absolutely integrable) is that both the representations are entire. That is, you could use either the Fourier kernel exp(jwt) or the Laplace kernel exp(st) for unique linear decompositions with arbitrarily bounded LMS error. In some sense it's like how a function can be expanded either as a Fourier series (sine and cosine) or as a Chebyshev series (T
n) with the latter converging much faster within a certain zone (what would be called the 'passband' in Chebyshev filters). Also gives some idea of why wavelet representations are more dense than Fourier or Laplace, and why the Karhunen-Loeve expansion of an AWGN contains the least number of eigenvalues.
Basis functions bear out George Orwell's inunction that "All animals are equal, but some animals are more equal than others"
M.G.Rajan
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