Cauchy-Schwartz Inequality for Step Functions

AI Thread Summary
The discussion centers on proving the Cauchy-Schwartz inequality for step functions, specifically that the square of the integral of the product of two step functions is less than or equal to the product of their individual integrals squared. Participants express confusion about the hint provided, which involves analyzing a quadratic function defined by the integral of the sum of the step functions. The solution involves expanding the quadratic, differentiating it, and finding its minimum to establish the inequality. Clarification is sought regarding the variable of integration, confirming it should be the variable that phi and psi depend on. Overall, the conversation emphasizes the steps needed to approach the proof while addressing common misunderstandings.
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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!
 
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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.
 
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.
 
I meant integrate first. I.e.
<br /> t^2 \int \phi^{2} + 2t \int\phi \psi + \int\psi^{2} \geq 0.<br />

Now minimize that. The minimum occurs at a value of t that is a ratio of two integrals.
 
Ah yeah I got it :-) Thanks!
 
Deano10 said:
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).
 
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