I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means that they are actually perpendicular such that their dot product is zero. However, I am not sure how sine and cosine are actually orthogonal. They are 90 out of phase, but there must be a different reason why they are considered orthogonal. What is that reason? Does being orthognal really have something to do with geometry i.e 90 degree angels?
Why do we want to have orthogonal things so often in maths? especially with transforms like fourier transform, we want to have orthogonal basis. What does that even mean? Is there something magical about things being orthogonal?
Best Answer
The concept of orthogonality with regards to functions is like a more general way of talking about orthogonality with regards to vectors. Orthogonal vectors are geometrically perpendicular because their dot product is equal to zero. When you take the dot product of two vectors you multiply their entries and add them together; but if you wanted to take the "dot" or inner product of two functions, you would treat them as though they were vectors with infinitely many entries and taking the dot product would become multiplying the functions together and then integrating over some interval. It turns out that for the inner product (for arbitrary real number L) $$\langle f,g\rangle = \frac{1}{L}\int_{-L}^Lf(x)g(x)dx$$ the functions $\sin(\frac{n\pi x}{L})$ and $\cos(\frac{n\pi x}{L})$ with natural numbers n form an orthogonal basis. That is $\langle \sin(\frac{n\pi x}{L}),\sin(\frac{m\pi x}{L})\rangle = 0$ if $m \neq n$ and equals $1$ otherwise (the same goes for Cosine). So that when you express a function with a Fourier series you are actually performing the Gram-Schimdt process, by projecting a function onto a basis of Sine and Cosine functions. I hope this answers your question!