I gram-schmidtt ortogonalization method false?.

[eljose79]

In fact is suposed that with gthe G-s method from any series of function you can construct and ortonormal methods..but what would happen when applied to the set of functions {x**n+y**n, with n=1,2,3...}?..in fact with this set you can not construct an orthonormal series.

 

[marcus]

Originally posted by eljose79
In fact is suposed that with gthe G-s method from any series of function you can construct and ortonormal methods..but what would happen when applied to the set of functions {x**n+y**n, with n=1,2,3...}?..in fact with this set you can not construct an orthonormal series.

this is a strange question, eljose. I think you know the answer
so tell me if what I say is right:

the G-S process allows one to construct an ON basis from a general basis.

If one starts with {1, x²,...,x²ⁿ,...}

this will result in an ON basis not of ALL the functions, say on the unit interval, but only a basis of the EVEN functions
f(-x) = f(x)
and the even functions are a vectorspace, a subspace of the whole, and you will get an ON basis of that subspace.

You always get an ON basis of the vectorspace which is SPANNED by the original set.


Likewise your set only spans a certain subspace to start with.
So GS will give you an ON basis of that subspace.

 

[eljose79]

answer?..

In fact i do not know if you take the set of functions (x**n+y**n with n=1,2,3...}..would you have an ortonormal set of functions in a given interval?.such us (-11)..well then try to expand the xy function in this set...¡it is imposbile¡..

 

[HallsofIvy]

Did you not understand what Marcus wrote? If you start with any set of functions and apply the Gram-Schmidt orthogonalization process, you will wind up with an orthonormal set of functions that spans the SAME SUBSPACE of all functions as the original set spanned.
You cannot expand "xy" with the orthonormal set because you could not do it with the original set.