SU-MATH53 FEB252024 — Jemoka Knowledge Base

Fourier Decomposition Main idea, any induction f(x) on an interval [0, L] can be written as a sum:

\begin{equation} f(x) = a_0 + \sum_{k=1}^{\infty} a_{k} \cos \left( \frac{2\pi k}{L} x\right) + \sum_{k=1}^{\infty} b_{k} \sin \left( \frac{2\pi k}{L} x\right) \end{equation}

L-periodicity A function is L-periodic if f(x+L) = f(x) for nonzero L for all x. The smallest L > 0 which satisfies this property is called the period of the function. L-periodicity is preserved across… translation we are just moving it to the right/left dilation Suppose f(x) is L periodic and let g(x) = f(kx), then, g is also L periodic. Proof: g(x+L) = f(k(x+L)) = f(kx + kL) = f(kx) = g(x). So g would also be L periodic. However, importantly, g would also be \frac{L}{k} periodic (verified by using the same sketch as before) linear combinations Suppose f,g are L periodic and h(x) = af(x) + bg(x), then h is also L periodic. Proof:

\begin{equation} h(x+L) = af(x+L) + bg(x+L) = af(x) + bg(x) = h(x) \end{equation}

Fourier Series see Fourier Series