6 Fourier Series
6.1 Vectors
6.2 Inner Products
An inner product is an abstraction of the of a scalar projection of one vector onto another. In Euclidian spaces it most often takes the form of a dot product. An inner product allows us to break vectors into orthogonal components and base approximations on projections. This in turn implies that there will be a best approximation. We’ll use this to approximate complex functions by sums of simpler functions, like polynomials, sines, and cosines. Approximating a function, or signal, by a sum of sines and cosines effectively breaks down the function into its component frequencies.
Definition 6.1 An inner product on a vector space \(V\) is a function that assigns a scalar \(\langle x,y\rangle\) to each pair, \((x,y)\), of vectors in \(V\) and satisfies the following conditions:
- \(\langle y,x\rangle = \overline{\langle x,y\rangle}\) (conjugate symmetric);
- \(\langle x,x\rangle \ge 0\), with equality if and only if \(x=0\) (positive definite);
- \(\langle a w + b x,y\rangle = a\langle x,y\rangle + b\langle x,y\rangle\) for all \(w\), \(x\) and \(y\) in \(V\) and scalars \(a\) and \(b\) (bilinear).
In a real vector space, the first axiom reduces to a straightforward symmetry: \(\langle x,y\rangle = \langle y,x\rangle\).
The notation for inner products varies depending on the application. That’s ok, so does the notation for multiplication.
6.3 The usual Fourier Series for \(f\)
6.4 A more general view of Fourier Series
Suppose \(f\) is a piecewise continuous function defined on an interval \([-l,l]\).
Further, suppose the set of function \(\{\phi_0,\phi_1,\phi_2,\dots\}\) is orthogonal with respect to some given inner product. Then the fourier series for \(f\) with respect to this set is \(\displaystyle\sum_{m=0}^\infty c_m \phi_m(x)\) with \(c_m = \dfrac{\langle f,\phi_m\rangle}{\langle\phi_m,\phi_m\rangle}\).
The usual fourier series for \(f\) is obtained when \(\phi_n(x) = \begin{cases} 1 & m=0, \\ \cos(\frac{m\pi x}{2l}) & \text{$m$ even and positive} \\ \sin(\frac{(m+1)\pi x}{2l}) & \text{$m$ odd} \end{cases}\) and the inner product of two functions, \(u\) and \(v\), is defined as \(\langle u,v\rangle = \int_{-l}^{l} u(x) v(x) \, dx\).