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 $⟨x,y⟩$ to each pair, $(x,y)$, of vectors in $V$ and satisfies the following conditions:

1. $⟨y,x⟩=\overline{⟨x,y⟩}$ (conjugate symmetric);
2. $⟨x,x⟩\ge 0$, with equality if and only if $x=0$ (positive definite);
3. $⟨aw+bx,y⟩=a⟨x,y⟩+b⟨x,y⟩$ for all $w$, $x$ and $y$ in $V$ and scalars $a$ and $b$ (bilinear).

Note

In a real vector space, the first axiom reduces to a straightforward symmetry: $⟨x,y⟩=⟨y,x⟩$.

Warning

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 $\{{\varphi }_0,{\varphi }_1,{\varphi }_2,\dots \}$ is orthogonal with respect to some given inner product. Then the fourier series for $f$ with respect to this set is ${\displaystyle \sum _{m=0}^{\mathrm{\infty }}{c}_m{\varphi }_m(x)}$ with $c_m={\displaystyle \frac{⟨f,{\varphi }_m⟩}{⟨{\varphi }_m,{\varphi }_m⟩}}$.

The usual fourier series for $f$ is obtained when ${\varphi }_n(x)=\{\begin{array}{ll}1& m=0,\\ \cos (\frac{m\pi x}{2l})& m\text{ even and positive}\\ \sin (\frac{(m+1)\pi x}{2l})& m\text{ odd}\end{array}$ and the inner product of two functions, $u$ and $v$, is defined as $⟨u,v⟩={\int }_{-l}^{l}u(x)v(x)dx$.