Generalized Fourier series. A few comments and two examples: spherical harmonics and Chebyshev polynomials. An application to PDEs: diffusion in a finite interval in one spatial dimension. More in part 2! Video. Notes.
Category Archives: Fourier series and transforms
Fourier series: [sin(x)]^n
Using the binomial theorem, I derive the Fourier series for [Sin(x)]^n in exponential form, without doing any integrals. Video. Notes.
Fourier series: exponential form
From the trigonometric (sine/cosine) form to the complex exponential form of the Fourier series, using Euler’s identity.Video. Notes.
Fourier series: example 2
A quick discussion of exp(-x^2) cos(3x), followed by an explicit derivation of the Fourier series of sin^3(x) without doing integrals.Video. Notes.
Fourier series: example 1
A first example of calculating a Fourier series explicitly. More examples in upcoming videos! Video. Notes.
The quantum free particle
Using background on complex numbers and Fourier modes, here is a first discussion of the most essential quantum mechanical system. Video. Notes.
Fourier series: first concepts
A first look at the definition and concept of Fourier series as a global expansion in a finite interval, compared with a Taylor series as a local expansion. More in upcoming videos! Video. Notes.
Diagonalizing differential operators on the lattice
How do you represent a function on a computer with discrete memory? What about its derivatives? A first look at stuff on a lattice, the matrix form of the first-derivative operator (forward, with periodic boundary conditions), and its eigenvectors and eigenvalues (Fourier modes). Video. Notes.
A very special unitary matrix. Part 2.
A second look at the matrix of the discrete Fourier transform. Proof that it is unitary. How to apply it to a function, and why it costs O(NlogN) vs O(N^2). Video. Notes.
A very special unitary matrix
Review of some properties of unitary matrices. Orthogonality and completeness. A first look at the matrix of the discrete Fourier transform. Diagonalization and efficiency. Video. Notes.