SIO239: Math Methods for Geophysics, 2008
Instructors: Glenn Ierley and Bob Parker
Fourier Series
The first topic is Fourier series. The material
is also covered in Chapters 12 and 13 of Mathematical
Methods for Physics and Engineering by Riley, Hobson and Bence,
the main text for the course. However, I will take a different
approach, and so I will
provide detailed notes that you will be able to download from this
site. The class will be given as a series of
lectures, at a time to be determined.
I treat Fourier series as a particular example of expansion in an
orthogonal basis on a Hilbert space. Without getting too abstract,
we can get a lot of insight by treating the general situtation,
mostly on a finite real interval. So we will see inner products, and
norms, Schwarz's inequality, Parseval's theorem, self adjoint
operators and their eigenfunctions. We can illustrate the general
concepts with expansions in sines and cosines (Fourier series),
orthogonal polynomials, spherical harmonics and orthogonal taper
functions used in the estimation of power spectra. But Fourier
series are still the simplest kind and so we give some explicit
series and study different modes of convergence, including a
discussion of Gibbs' phenomenon.
Fourier Transforms
Here I follow the traditional motivation that the Fourier transform
decomposes a function into components with different frequencies.
I derive or state a number elementary properties, such as 2-norm
invariance, transformation under scaling and a shift of origin.
We calculate a number of explicit transforms, for functions such as
the box car and the Gaussian. Then we discuss convolution and the
Convolution theorem and its implications for the impulse and
frequency responses of a simple seismometer. The numerical
approximation of the transform with the FFT and the trapezoidal rule
is developed, which brings in the amazing Poisson Sum Rule.
The Fourier transform in two or more dimensions is explained, with
an introduction to the special properties connected with
circular symmetry about the origin, and the Hankel transform.
Finally, if time permits, we take a peek at distribution theory and
the mathematics connected with generalized functions, such as the
infamous delta function.