Visualizing the Fourier transform of some basic functions.
In the periodic case we saw that the Fourier transform passes between $L$-periodic functions $f(x)$ of a real variable $x$
and functions $F(n)$ of integers $n$. In the non-periodic setting, a new version of
the Fourier transform will pass between functions $f(x)$
of a real variable $x$ and functions $F(\lambda)$ of a real variable $\lambda$. Our focus on the 1-dimensional Fourier transform
is sufficient for many signal processing applications and
for PDE's with 1 spatial and 1 time dimension (a good
warm-up before tackling the full complexities of nature).
Let $f:\R \to \C$ be a function for which $|f(x)|$ decays quickly enough as $|x| \to +\infty$ (so
$\int_{\R} |f(x)|\,dx$ converges). The Fourier transform $\widehat{f}:\R \to \C$ is the function defined by
$$\widehat{f}(\lambda) = \int_{\R} f(x) e^{-i\lambda x}\,dx$$
(The integrand involves variables $\lambda$ and $x$, but we "integrate over $x$", leaving a dependence on $\lambda$.)
The Fourier transform can also be reversed: For $f(x)$ as above, if $|\widehat{f}(\lambda)|$ decays quickly enough
as $|\lambda| \to +\infty$
so that $\int_{\R} |\widehat{f}(\lambda)| \,d\lambda$ converges then the following formula
(with suitable interpretation at jump discontinuities of $f(x)$) recovers $f$ from $\widehat{f}$:
$$f(x) = \frac{1}{2\pi} \int_{\R} \widehat{f}(\lambda) e^{i\lambda x}\,d\lambda.$$
We also introduce convenient notation for expressing the effect of the Fourier transform on functions when it is unwieldy to use
the "$\widehat{(\cdot)}$" symbol. For a function $f(x)$, its Fourier transform $\widehat{f}$ is also denoted as $\cF f$:
For a function $f(x)$, its Fourier transform $\widehat{f}$ is also denoted as $\cF f$:
$$(\cF f)(\lambda) = \int_{\R} f(x) e^{-i\lambda x}\,dx.$$
For a function $g(\lambda)$, we define its inverse Fourier transform to be
$$(\cF^{-1} g)(x) = \frac{1}{2\pi} \int_{\R} g(\lambda) e^{i\lambda x}\,d\lambda.$$
Below we visualize the Fourier transform of some basic functions. You can adjust the parameters using the sliders.