5.9 From Fourier Series to the Fourier Transform
Every expansion so far has needed a bounded interval: the eigenvalues of a Sturm–Liouville problem on \([a,b]\) are discrete, so the expansion is a sum. A function on the whole line has no such interval, and this section shows what becomes of the series when the interval is allowed to grow without bound. The sum becomes an integral, and the discrete spectrum becomes a continuum.
Take \(f\) defined on \([-L,L]\) with complex coefficients \[c_n = \frac {1}{2L}\int ^{L}_{-L}f(t)e^{-i\omega _n t}\,dt, \qquad \omega _n = \frac {n\pi }{L},\] so that \(f(x)=\sum c_n e^{i\omega _n x}\). The spacing of the frequencies is \(\Delta \omega = \pi /L\). Substituting and writing the sum in terms of \(\Delta \omega \), \[f(x) = \frac {1}{2\pi }\sum ^{\infty }_{n=-\infty } \left [\int ^{L}_{-L}f(t)e^{-i\omega _n t}\,dt\right ]e^{i\omega _n x}\,\Delta \omega .\] As \(L\to \infty \) the spacing \(\Delta \omega \to 0\), the sum becomes a Riemann integral over \(\omega \), and we obtain the following pair.
Definition 5.9.1 (Fourier transform). \[\widehat {f}(\omega ) = \int ^{\infty }_{-\infty }f(t)\,e^{-i\omega t}\,dt, \qquad f(x) = \frac {1}{2\pi }\int ^{\infty }_{-\infty }\widehat {f}(\omega )\,e^{i\omega x}\,d\omega .\]
Note 5.9.2. The passage above is a derivation in outline rather than a proof — the interchange of limit and integral needs justification, and the inversion formula holds under conditions comparable to those of Theorem 5.5.1. What it does show is where the transform comes from. It is not a new idea but the same idea on an unbounded domain: \(\widehat {f}(\omega )\) is the coefficient \(c_n\) with the discrete index \(n\) replaced by a continuous \(\omega \), and the inversion formula is the series with the sum replaced by an integral.
Theorem 5.9.3 (Basic properties). For suitable \(f\) and \(g\):
- (i)
- linearity: \(\widehat {\alpha f+\beta g} = \alpha \widehat {f}+\beta \widehat {g}\);
- (ii)
- shifting: \(\widehat {f(t-a)}(\omega ) = e^{-i\omega a}\widehat {f}(\omega )\);
- (iii)
- scaling: \(\widehat {f(at)}(\omega ) = \frac {1}{|a|}\widehat {f}(\omega /a)\);
- (iv)
- differentiation: \(\widehat {f'}(\omega ) = i\omega \,\widehat {f}(\omega )\);
- (v)
- convolution: \(\widehat {f*g} = \widehat {f}\cdot \widehat {g}\), where \((f*g)(x)=\int ^{\infty }_{-\infty }f(t)g(x-t)\,dt\).
Note 5.9.4. Properties (iv) and (v) are why the transform is used. Differentiation becomes multiplication by \(i\omega \), so a linear differential equation with constant coefficients becomes an algebraic equation; and convolution, which is an integral, becomes an ordinary product. Both replace a hard operation by an easy one, at the cost of having to transform and invert.
Example 5.9.5 (The Gaussian). For \(f(t)=e^{-at^{2}}\) with \(a>0\), completing the square in the exponent and using \(\int ^{\infty }_{-\infty }e^{-au^{2}}du = \sqrt {\pi /a}\) from Section 2.3 gives \[\widehat {f}(\omega ) = \sqrt {\frac {\pi }{a}}\;e^{-\omega ^{2}/(4a)} .\] The Gaussian transforms to a Gaussian. Note the reciprocal widths: a sharply concentrated \(f\) (large \(a\)) has a broad transform, and conversely. This is the uncertainty principle in its analytic form, and it is a theorem about Fourier transforms before it is a statement about physics.
Remark 5.9.6. The reciprocal-width phenomenon has a statistical counterpart worth noting. The characteristic function of a random variable, \(\varphi _X(t)=E\left (e^{itX}\right )\), is the Fourier transform of its density with a sign convention change, and the example above is the statement that the characteristic function of a normal variable is again of normal form. Convolution property (v) is then the statement that the density of a sum of independent random variables is the convolution of their densities, which is why characteristic functions turn sums into products and why they are the standard route to the central limit theorem.
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