orthogonal functions

[dirk_mec1]

1. The problem statement, all variables and given/known data


http://img55.imageshack.us/img55/8494/67023925dy7.png (http://imageshack.us)



3. The attempt at a solution

All functions orthogonal to 1 result in the fact that: \int_a^b f(t)\ \mbox{d}t =0

Now the extra condition is that f must be continous. (because of the intersection).

But where does the fact that f(a)=f(b)=0 comes from? And why look at the deratives?

[Dick]

Remember way back in first year calc, when you learned that to do that integral you find an antiderivative F(x) and evaluate F(b)-F(a). This is that same problem in disguise.

[dirk_mec1]

Well I thought of this: \int_a^b \int_a^t f(s)\ \mbox{d}s \mbox{d}t =0

[Dick]

Well I thought of this: \int_a^b \int_a^t f(s)\ \mbox{d}s \mbox{d}t =0

Fine. What are you going to do with it? Why don't you just define
F(x)=\int_a^x f(s)\ \mbox{d}s

What are some of the properties of F(x)?