It is suffice to prove the following statement:
These conclusion follows immediately from the Fourier series in .
once i solve that one is the derivative of the other
but here its much harder to guess the formulla
what is the general method?
i need to find alpha beta and gama
so this expression will be minimal
i know how to solve such stuff
i have a vector and a subspace to make a projection of the vector
so i make an orthogonal basis and then i make a projection of that vector
into my space
and then the difference between that vector and the original vector is the minimal
but in order to do all that i need the
inner product formula which defines this norm.
usually i figured out the formula by guessing
but here i cant guess
so i am asking if there is a general method
Since and are both complex Hilbert Space,
Then is also Hilbert Space under the inner product defined by:
The norm is induced from the inner product.
The minimal problem is equivalent to find the distance between point and The closed subspace spanned by and and .
Indeed, Since is a free variable, you can eliminate the part, it doesn't affect the final minimal value (but it has something to do with minimal point).