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I know the mathematical definition but I'm having a hard time understanding the utility of orthogonal polynomials. I'm not saying they are useless, far from that! It is just that I like understanding thinks from a higher level than its mathematical definition...

  • What are the interesting properties of this type of polynomials?
  • Why is it relevant that their inner product is equal to 0?
  • What is the beauty on them?

To provide some context, I'm currently working with polynomial chaos expansions and I have to explain the method to non mathematicians. I need to be able to explain it in simple words. This is why I need to understand orthogonal polynomials in simple words

Felipe Aguirre
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    I'm having a hard time understanding the utility of orthogonal polynomials - Utility and futility are just one letter apart... :-) – Lucian Feb 14 '14 at 14:03
  • @Lucian, are you implying they are useless? ;) – Felipe Aguirre Feb 14 '14 at 14:04
  • Don't worry about them until you, personally, have some reason to care about them.* (Advice applying other branches of learning, as well...) There are whole books written on the topic of Orthogonal Polynomials, so in fact they have been found useful. But of course that doesn't mean they are useful for you. [* Your department or advisor telling you that you will need them is one such reason.] – GEdgar Feb 14 '14 at 14:14
  • @GEdgar, I do think they are usefull and I'm currently using them in very interesting applications. It is just that I'm not able to explain them to non mathematicians and that is my task in this moment – Felipe Aguirre Feb 14 '14 at 14:19

2 Answers2

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Such families of orthogonal polynomials allow numerical quadratures (for example Gauss-Legendre-quadrature).

Another application is the approximation of a function by tchebycheff-polynomials.

The fact, that the inner product is zero helps to find simple formulas for the mentioned applications.

Also, these polynomials have some interesting minimality properties used to interpolate a function.

Peter
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  • thank you for your answer. However, I still have a hard time getting the grip with the concept. Perhaps I'm demanding to much? – Felipe Aguirre Feb 14 '14 at 14:05
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Take the case of $$\frac{1}{|\vec r - \vec r_0|}=\sum_n P_n (\cos \theta) \frac{r^n} {r_0^{n+1}}$$ You would have a really hard time solving electromagnetism problems in an analytical way without this property of Legendre polynomials.

Usually there is no analytical solution for what you want to achieve but an expansion in orthogonal polynomials is a really neat way to convert differential or integral relations into pure algebraic ones.

Just simplify!

Rogério Jorge
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