and now we want to minimise the magnitude squared . Using the properties of the inner product, we find
The last line follows because orthogonality of the functions implies
Now we try to minimise with respect to each of the by setting the derivatives to zero:
so we find that
We should make sure that this is a minimum, not a maximum. This is quite simple—since there’s only one value, we know that this is the only extreme point and that if we added on some really large number to all the then the approximation would obviously be worse. This makes the single extremum a minimum for certain. We could also do the more rigorous second-derivative test, which would also show us that it is a minimum as is a property of the inner product.
This solution for is really a remarkable result. You can put in the trigonometric definitions of the and see that it retrieves the definitions of the Fourier series coefficients way up at the top of this article.
What’s more impressive, however, is that everything we did did not care what the were! In fact, they only had to be an orthogonal basis; the trigonometric functions were just one possibility.
Another valid basis we could have used is made up of polynomials; the monomials themselves aren’t orthogonal under this inner product, but there is a method called the Gram–Schmidt procedure that can be used to turn them into an orthogonal basis. If you do this, you come up with a series of polynomials called the Legendre polynomials. The first few of these are
These appear in the expansion of the electrostatic potential around a multipole in Cartesian coordinates, and consequently in the spherical harmonic functions, which turn up all over physics.
Now this basis is also orthogonal, so if we want to make a “Fourier–Legendre” series expansion of called
then we already know that the coefficients are defined by
This is why abstract concepts in linear algebra are so useful; with no additional work we gained a whole new method of series expansion!
In the next part of this series, we’ll compare how this new Legendre series expansion behaves in comparison to the Fourier and Taylor series.
This article is the second part of a series. You can find all of the rest of the articles in this series here: