Fourier series using other orthogonal systems

By definition, a set of functions $$S = \{\phi_1,\phi_2,...\}$$ is called an orthogonal system on the interval $$[a,b]$$ if $$\forall\phi_i,\phi_j,i\neq g: (\phi_i,\phi_j)=0$$.

Examples:

1) $$S_1 = \{1, \cos x, \sin x, \cos (2x), \sin(2x), \dots, \cos (mx), \sin(mx),...\}, m\in\mathbb{N}_0$$ is an orthogonal system on the interval $$[0,2\pi]$$.

2) $$S_2 = \{...,e^{-imx},...,e^{-ix},1,e^{ix},e^{i2x},...,e^{imx},...\}, m \in \mathbb{Z}$$ is orthogonal on the interval $$[0,2\pi]$$

Question: Can we build Fourier series without using the systems $$S_1$$ or $$S_2$$ (Roughly speaking, they are the same) but using another orthogonal system, for example, $$S_3 = \{1,x,2x,...,mx,...\}$$, which is orthogonal on the interval $$[-1,1]$$?

• I noticed your first orthogonal system is not complete (or at least its definition is difficult to understand): You need to have both functions $x \mapsto \cos(m x)$ and $x \mapsto \sin(m x)$ for every $m$ (not just even/odd $m$). – 0x539 Jan 13 at 18:10
• @0x539 Yes, you're right. I wanted to show that every even member of the system is cos and every odd is sin, but something went wrong. I'll fix it – Yauhen Mardan Jan 13 at 19:44

Yes. The underlying mechanism here is (from a certain point of view) that square-integrable functions on $$[0, 2 \pi]$$, denoted by $$L^2([0, 2 \pi])$$, form a Hilbert space $$X$$. The special thing about $$S_1 = \{x \mapsto \sin(n x), x \mapsto \cos(n x) ~|~ n \in \mathbb{N}\}$$ and $$S_2 = \{x \mapsto e^{i n x} ~|~ n \in \mathbb{N}\}$$ is that they form a complete orthonormal system (also called orthonormal basis) on $$X$$. Explicitly this means that $$S$$ is orthogonal, all elements of $$S$$ have norm 1 and that the only $$f \in X$$ satisfying $$\forall g \in S: (f, g) = 0$$ is $$f = 0$$.
If a subset $$S$$ of some $$L^2$$ space (no necessarily the $$L^2$$ space over $$[0, 2 \pi]$$) is a countable complete orthonormal system then for all $$f \in L^2$$
$$f = \sum_{s \in S} (s, f) s$$
This is exactly what you do when doing fourier series over $$[0, 2\pi]$$. The convergence of the sum is only guaranteed in $$L^2$$-Norm however.
Examples for other complete orthonormal systems are Legendre Polynomials on $$[-1, 1]$$ (Note that your set (3) is not orthogonal), $$h_n(x) = H_n(x) e^{-x^2/2}$$ on $$(-\infty, \infty)$$, where $$H_n$$ are Hermite Polynomials and spherical harmonics on the 2-Sphere.