By Houben H., Rigassi V., Garnier P.
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Optical spectra are line and/or band spectra, depending upon the state of the radiating matter, whether atoms or molecules. Mechanical spectra may be line spectra or continuous spectra, and this depends partly on the source: a periodic source gives a line spectrum and a random process gives a continuous spectrum, partly and essentially on the method of analysis: the Fourier series method yields line spectra and the Fourier transform method continuous spectra. (5) Precision. Optical spectra, obtained under well-controlled laboratory conditions, represent a higher degree of precision than mechanical spectra.
Thus, the error depends only on (n + 1/2) f. g. corre- 32 F O U R I E R SERIES A N D F O U R I E R T R A N S F O R M S TABLE I Convergence in approximating the saw-tooth function by a Fourier series expansion N u m b e r of harmonics Independent variable η ] (n Error + l/2)t fn(t) ~ Finite, arbitrary 0, In, ... 0, (η + 1/2)2π, ... e. with increasing number of terms. Only t will be smaller for increasing «, keeping (n + 1 /2) t and the error constant. For the saw-tooth curve, this constant error amounts to about 9 % of the step in f(t).
According to this formula, an extension (contraction) of the time scale by a factor a implies a contraction (extension) of the frequency scale by the same factor, while the area under the spectral curve is preserved. 3. 4 Time-shifting theorem If / ( / ) <-» F(co), then f(t ± a) <-> e ^ ( ω ) , where β is a real constant. Proof. The Fourier inversion formula  gives immediately: ± i f l W an ± a) = 00 J F(co) e dco = - i - J F(a>) e i a , ( i ± f l ) ± i t o a e itot dω that is: f(t ± a)~e ° F(a>) ±ia<  Obviously, when a time function is shifted by a constant time segment, its amplitude spectrum remains unchanged while its phase spectrum changes by an amount determined by the time shift.