Physical interpretation of Cauchy-Schwarz inequality for integrals

[maverick280857]

Hi...I've been scratching my head for this one:

If f(x) and g(x) are realvalued functions integrable over an interval (a,b) then prove that

$$|\int_{a}^{b} f(x)g(x)dx| \leq \sqrt{\int_{a}^{b}(f(x))^2dx \int_{a}^{b}(g(x))^2dx}$$

I actually don't want the proof of this inequality...I already have it (take F(x) as (f(x)- lambda g(x))^2=>integral of F(x) over (a,b) is positive as is the integrand. Then, we use properties of the quadratic function and get the desired inequality using the fact that the discriminant of the resulting quadratic must not exceed 0). However, an alternative proof would be appreciated.

But what I really want is the physical interpretation of this inequality because that is hard to come up with. The inequality reminds me of the Cauchy Schwarz Inequality and incidentally, its proof is quite similar to Cauchy Schwarz.

I would be very grateful if you could offer an explanation.

Thanks and cheers
Vivek

 

[dextercioby]

But what I really want is the physical interpretation of this inequality because that is hard to come up with. The inequality reminds me of the Cauchy Schwarz Inequality and incidentally, its proof is quite similar to Cauchy Schwarz.
I would be very grateful if you could offer an explanation.

It really doesn't have a physical interpretatation.It's a math result.It doesn't even make sense the interpret it in terms of areas.Because u have in the LHS the area of a function and in the RHS the area of other functions.Physics works with far more complicated integrals and is not always interested in mathematical subtleties and real valued functions.I should say we use complex functions far more than real ones.In Quantum Physics.

The CBS inequality is wery powerful and is used by physics,especially QM.But this mathematical formula,i haven't used it,i haven't seen it in a physics book...

Daniel.

 

[Hurkyl]

Actually, it is the Cauchy-Schwartz inequality, for the inner product:

$$f \cdot g := \int_a^b f(x) g(x) \, dx$$

It doesn't really have a physical meaning unless you have ascribed physical meanings to these integrals, or to this inner product.

 

[maverick280857]

Thanks for your help.

 

[mathwonk]

well of course like the usual cauchy schwartz inequality, it seems to express the fact that the dot product of two vectors is always less than or equal to the product of their lengths.

this is used to define the angle between two vectors in an inner product space, as arccos(f.g/|f||g|). so the meaning of this inequality should be that you can define the angle between two functions that satisfy these convergence restrictions.

so it really does seem to have a very clear geometric interpretation.