Function Transforms
The Fourier Series considered in the last section was defined for periodic functions, i.e any function \( f \) with \( f (L) = f(-L) \). We could also consider the same construction for any function that is non-zero only in a finite region (we say \( f \) has compact support)- can you think why the construction still works?
Suppose \( f \) is such a function, and that \(L \) is large enough that \([-L, L]\) contains all of the \( x \)-values for which \( f \) is non-zero. The \(n\)-th Fourier series coefficient is then
| \[ c_n(f) = \frac{1}{2L} \int^L_{-L} f(x)e^{\frac{2\pi i n x}{L}}\mathrm{d} x . \] |
In a certain sense the coefficients \( c_n(f) \) define the values of a function defined on a lattice of width and height \( 2L \), \( c_n(f) = \frac{1}{\sqrt{2L}}\hat{f}\left( \frac{n}{L} \right).\) The function \( \hat{f}(k) \) is the Fourier transform of \( f \), defined as follows:
| \[ \hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty} f(x)e^{-2\pi i k x}\mathrm{d}x . \] |
Note
Several different conventions exist for defining the Fourier transform. Some authors omit the factor of \( \frac{1}{\sqrt{2\pi}} \) in front of the integral (we include it so the transform is a unitary operator), whilst others omit the factor of \( 2 \pi \) in the integrand and define
\[\hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty} f(x)e^{- i k x}\mathrm{d}x .\]
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This function can thus be thought of as measuring how many each wave \( e^{-2\pi i k}\) is present in the function \( f \).
When we transform a function of a spatial variable \(x\) the variable \(k\) is the wavenumber \( k = \frac{1}{\lambda} \), where \( \lambda\) is the wavelength, and hence the domain of the transformed function is called the wavenumber domain. Note the definition of the wavenumber here may vary from the one you are familiar with- this is a consequence of the conventions chosen in the definition of the Fourier transform; the factor of \(2 \pi\) in our definition simply switches between waves/metre and radians/metre.
The Fourier transform also has many important applications when applied to a function of a temporal variable \(t\). In this case, the variable \(k\) is that of frequency, and thus the domain of the transformed function is called the frequency domain. Had we chosen the other convention and omitted the factor of \( 2 \pi \) in the integrand of our transform, the transformed function would be a function of angular frequency instead of regular frequency.
Analogously to the Fourier series we then have
| \[ f(x ) = \frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty} \hat{f}(p)e^{2\pi i k x }\mathrm{d} k . \] |
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