First, let's establish what we mean by orthogonal. Well, with vectors in Euclidean space, the concept is pretty simple: it means that they're 90 degrees apart. One test for orthogonality is that their dot product is zero. Well, it turns out that this concept is generalizable, and can be applied to much more than just vectors in 3D euclidean space. It can also be applied to functions. So what's the equivalent of taking a dot product of a function?
First, let's review dot products for vectors. Let's say we have two vectors, V and W. We can label their components as Vx, Vy, and Vz, and Wx, Wy, and Wz. Now the dot product is
V dot W = Vx Wx + Vy Wy + Vz Wz
We just multiply each component, then add them up.
Now for functions. Let's say we have functions f(x) and g(x). The equivalent to multiplying each component is to multiply the values of the function at the same point, or f(x)g(x). Now we just add all that up, or in other words, integrate the functions. So
[latex]f(x) \cdot g(x) = \int{f(x) g(x) dx}[/latex]
Let me give you an example of two functions that are orthogonal: sin(x) and sin(x+pi/2). Those two functions are orthogonal, and ANY function of the form sin(x+c) (where c is an arbitrary constant) can be expressed as some unique linear combination of those first two orthogonal functions. This is equivalent to picking out an x and a y axis in a 2D plane, and expressing any vector as some linear combination of unit x and y vectors. We could have picked other orthogonal functions: sin(x+pi/8) and sin(x-3pi/8) also work, for example. But we can only pick two functions of this form which are mutually orthogonal, any third function cannot be orthogonal to both of them.
This is only a first step, but have you followed so far?
