Cauchy-Schwartz Inequality for Step Functions

[kidsmoker]

Homework Statement

Let

$$\phi,\psi : [a,b] \rightarrow \Re$$

be step functions.

Prove that

$$(\int \phi\psi)^{2} \leq (\int\phi^{2})(\int\psi^{2}) .$$

Hint: Consider the quadratic function of a real variable t defined by

$$Q(t)=\int(t\phi+\psi)^2 .$$

The Attempt at a Solution

I really don't know where to start with this, and the hint only confuses me more! :p

Any help appreciated, thanks!

 

[Dick]

Q(t)>=0, since it's the integral of a nonnegative function (a square). Expand Q(t) out and differentiate with respect to t. Solve Q'(t)=0 for t and put that value of t back into the expression Q(t)>=0 and see what you get.

 

[kidsmoker]

Yeah I get a similar thing. So we get a turning point of Q at some value t=-psi/phi, and when you put this back into Q you get

$$\int0 = constant$$.

Am I being really dumb cos I can't seem to get anything like the inequality from this :((((

Cheers.

 

[Dick]

I meant integrate first. I.e.
$$t^2 \int \phi^{2} + 2t \int\phi \psi + \int\psi^{2} \geq 0.$$

Now minimize that. The minimum occurs at a value of t that is a ratio of two integrals.

 

[kidsmoker]

Ah yeah I got it :-) Thanks!

 

[Dick]

Sorry to be a pain, but I am still a little confused!

Just to check, what exactly are we integrating with respect to?

Whatever variable phi and psi are functions of. Call it x. So write psi(x) and phi(x).