Understanding Functional Spaces in Mathematics

[a_Vatar]

I've been looking for some decent info on functional space but
could not find anything. Googling gives lots of defenitions, but no explanations as such.

Basically I'd like to understand why a function can be decomposed into other function, e.g. understand a meaning of inner product with respect to functions.

Or why one can or can not say decompose cos(x) into $$x^{2}$$ and $$x^{3}$$ with some coefficients.

I don't event clearly know what filed of mathematics studies functional spaces :)

Thanks.

 

[Kummer]

I've been looking for some decent info on functional space but
could not find anything. Googling gives lots of defenitions, but no explanations as such.

Basically I'd like to understand why a function can be decomposed into other function, e.g. understand a meaning of inner product with respect to functions.

Given the Sturm-Liouville Problem:
$$[p(x)y']'+\{q(x)+\lambda r(x)\}y=0 \mbox{ on }(a,b)$$
Satisfing the boundary value problems:
$$\alpha_1 y(a)+\beta_1 y'(a)=0$$
$$\alpha_2 y(b) + \beta_2 y'(b)=0$$

The question is for what $$\lambda$$ does the above differential equation has non-trivial solutions. For example, $$y(x)=0 \mbox{ on }[a,b]$$ is certainly a solution but it is trivial. Those $$\lambda$$ are called "eigenvalues" and those functions are called "eigenfunctions".

Sturm-Liouville Theory says that such $$\lambda$$ exists (given conditions on $$p(x),q(x),r(x)$$).

The solutions corresponding to distinct $$\lambda$$ are linearly independent. And furthermore if $$\phi_1(x)$$ corresponds to $$\lambda_1$$ and $$\phi_2(x)$$ corresponds to $$\lambda_2$$ then:
$$\int_a^b \phi_1(x)\phi_2(x) dx = 0$$.
Meaning the set $$\{ \phi_n (x)\}$$ is orthogonal.

Say, $$f(x) = \sum_{n=1}^{\infty}a_n \phi_n(x)$$ can be expressed by these orthogonal functions obtained from the Sturm-Loivuille problem.
Then,
$$f(x)\phi_m(x) = \sum_{n=1}^{\infty}a_n \phi_n(x) \phi_m(x)$$
Integrate both sides (we are assuming uniform convergence):
$$\int_a^b f(x)\phi_m(x) dx = a_m \int_a^b [\phi_m(x)]^2 dx$$ because of orthogonality.
Thus,
$$a_m = \frac{1}{||\phi_m(x)||^2}\int_a^b f(x)\phi_m(x) dx$$
Where $$||\phi_m(x)||^2=\left( \sqrt{\int_a^b [\phi_m(x)]^2 dx } \right)^2$$ called the L2 measure. (just shorthand notation).


The above ideas are taken from an area of mathematicas called Harmonic analysis. The idea is to express functions in terms of other functions, not just Fourier series, hence its the Generalized Fourier Series.

 

[HallsofIvy]

One cannot decompose cos(x) into "x² and x³ with some coefficients" because x² and x³ do not span a space that contains cos(x). Of course, one can decompose cos(x) into an infinite sum of powers of x: it's MacLaurin series.