Are Even and Odd Functions Orthogonal?

[theFuture]

We were doing examples in class today and showed that sin and cos were orthogonal functions. In general, is true that even and odd functions are orthogonal? I was unsure where a proof of this might begin, mostly how to generalize the notion of an even or odd function.

 

[matt grime]

This depends on what your "inner product" is.

Let's assume it is

$$<f,g> = \int_{-a}^a f(x)g(x)dx$$

an odd function is one that satisfies f(x) = -f(-x) an even one satisfies f(x)=f(-x)

1. show that the product of an even and an odd function is odd
2. show that the integral of an odd function over any interval [-a,a] is zero.

 

[theFuture]

Thanks. Now that I see it like that I can't believe I couldn't come up with that.