Coefficient Matching for different series

[CGandC]

Homework Statement

Hello,
I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space.

Homework Equations

- Knowledge of power series, polynomials, Legenedre polynomials, Spherical harmonics..

The Attempt at a Solution

Say I express f(x) as a power series, and f(x) = 1+2x+3x^2 , so, I can match the coefficients as in the following picture:

( no problem there, since the series is a polynomial)

Ok..but what If I now have some function g(x) and I express it as Legendre series (in the x domain: -1 to 1 ) , and I know that

, where the p's are Legendre polynomials. :

My question is , can I apply coefficient matching here? ( as in the next picture: )

Also, consider the next case:
suppose I have :

, where the Y's are spherical harmonics... and I decide to show 'h' as a series such as this:

then, matching coefficients I get:

Eventually, I get a contradiction , on the one hand : A=2 , on the other hand A=3 , so my understanding of coefficient matching in this part is clearly wrong ( because it aint a series representing polynomials? )... why?

Note:
I was taught that coefficient matching works in polynomials and since power series is a polynomial, but Legendre series and the last series is different... yet, I was not told that coefficient matching does not work in other cases, that makes me troubled as I'm unsure if coefficient matching applies only to polynomials.

Much thanks in advance for helpers.

 

[Orodruin]

My question is , can I apply coefficient matching here?

Yes, the Legendre polynomials are linearly independent.

Eventually, I get a contradiction , on the one hand : A=2 , on the other hand A=3 , so my understanding of coefficient matching in this part is clearly wrong ( because it aint a series representing polynomials? )... why?

You have different $A$ for each combination of $\ell$ and $m$. The spherical harmonics are linearly independent.

 

[CGandC]

Yes, the Legendre polynomials are linearly independent.You have different $A$ for each combination of $\ell$ and $m$. The spherical harmonics are linearly independent.

How do I show that Legendre polynomials/Spherical harmonics are linearly independent?
( perhaps using Wronskian? or perhaps because knowing each set is orthogonal to it self, then the set of functions is automatically linearly independent)

 

[Orodruin]

They are both eigenfunctions of Sturm-Liouville problems.

 

[CGandC]

They are both eigenfunctions of Sturm-Liouville problems.

I understand... I have another question:
If I have eigenfunctions that arise from S-L problems, I understand that they are linearly independent... but are they also always orthogonal to each other?

 

[Orodruin]

I understand... I have another question:
If I have eigenfunctions that arise from S-L problems, I understand that they are linearly independent... but are they also always orthogonal to each other?

Yes, assuming that it is a regular SL-problem and you use the appropriate inner product.