Expand a function in terms of Legendre polynomials

[Logarythmic]

Problem:
Suppose we wish to expand a function defined on the interval (a,b) in terms of Legendre polynomials. Show that the transformation u = (2x-a-b)/(b-a) maps the function onto the interval (-1,1).

How do I even start working with this? I haven't got a clue...

 

[quasar987]

I do not speak english in everyday life, and so the words "maps the function onto the interval (-1,1)" seems confusing to me. I would normally interpret them as "u is such that [itex]u\circ f[/itex]:(a,b)-->(-1,1)". But that does not seem to make sense, since we do not know the form of f. What does make sense, is to understand it as "u is such that [itex]f\circ u[/itex]:(a,b)-->R is the same as f:(-1,1)-->R". In other words, you need to show that [itex]f\circ u[/itex] is a restriction of f. But that is easy, you only need to show that u is onto from (a,b) to (-1,1).

 

[Logarythmic]

Ok, I'm really not good at linear algebra so I think I'll jump this problem. ;)