Abstract
The orthogonal Fourier basis plays a special role in signal representation, as it diagonalizes linear operators that are covariant by translation. These operators are called convolutions in signal processing. This lecture reviews the definition and properties of Fourier series and the Fourier integral, showing the link between the decay of Fourier coefficients and the regularity of a function.
The Nyquist-Shannon sampling theorem is presented as a linear approximation theorem in the Fourier basis, where the function is approximated by an orthogonal projection onto the low frequencies. This approximation can be rewritten in a cardinal sine basis. The effect of high-frequency errors, known as aliasing, is demonstrated.
The sampling theorem is then generalized by replacing the approximation space and the cardinal sine basis with other spaces generated by other orthogonal bases, such as the spaces of piecewise constant functions or linear splines.
