Coefficient Matching for different series

In summary: So yes, the Legendre polynomials and spherical harmonics are also orthogonal to each other.In summary, the conversation discusses coefficient matching when expressing a function as a linear combination of other basis functions, specifically using power series, Legendre series, and spherical harmonics. The discussion also covers the linear independence and orthogonality of Legendre polynomials and spherical harmonics as eigenfunctions of Sturm-Liouville problems.
  • #1
CGandC
326
34

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:
upload_2018-1-8_21-53-11.png

( 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
upload_2018-1-8_21-55-17.png
, where the p's are Legendre polynomials. :
upload_2018-1-8_22-24-16.png


My question is , can I apply coefficient matching here? ( as in the next picture: )
upload_2018-1-8_21-58-19.png


Also, consider the next case:
suppose I have :
upload_2018-1-8_22-10-7.png
, where the Y's are spherical harmonics... and I decide to show 'h' as a series such as this:
upload_2018-1-8_22-11-2.png


then, matching coefficients I get:
upload_2018-1-8_22-11-32.png


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.
 

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  • #2
CGandC said:
My question is , can I apply coefficient matching here?
Yes, the Legendre polynomials are linearly independent.

CGandC said:
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.
 
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  • #3
Orodruin said:
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)
 
  • #4
They are both eigenfunctions of Sturm-Liouville problems.
 
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  • #5
Orodruin said:
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?
 
  • #6
CGandC said:
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.
 
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Related to Coefficient Matching for different series

1. What is coefficient matching for different series?

Coefficient matching is a statistical method used to compare two time series data sets to determine their level of similarity. It involves calculating the correlation coefficient between the two series and determining if there is a significant relationship between them.

2. How is coefficient matching different from other statistical methods?

Coefficient matching is different from other methods such as regression analysis or ANOVA because it specifically focuses on comparing the correlation between two series, rather than looking at relationships between variables or groups.

3. What is the purpose of coefficient matching?

The purpose of coefficient matching is to determine if two time series data sets are related and to what degree. It can help identify patterns and trends in the data and can be used to make predictions about future values of the series.

4. What are some potential applications of coefficient matching?

Coefficient matching can be used in a variety of fields, including finance, economics, and social sciences. It can be used to analyze stock prices, economic indicators, and social media trends, among other things.

5. What are the limitations of coefficient matching?

One limitation of coefficient matching is that it only measures the linear relationship between two series, so it may not capture more complex relationships. Additionally, it is important to consider the data quality and potential outliers when interpreting the results of coefficient matching.

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