Non continuously differentiable but inner product finite

In summary, the conversation is discussing the concept of Green's function and its definition for functions that are not continuously differentiable. The author on the referred site explains that although the image of a non-differentiable function may not be square-integrable, the inner product can still be well-defined in a larger space. The example of a function with a kink is given to illustrate this. The person asking for clarification is confused about how the operator can be defined for such functions.
  • #1
Sumanta
26
0
Hello,

I was trying to understand Green's function and I stumbled across the following statements which is confusing to me.

I was referring to the following site

http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node79.html [Broken]

Here the author says the following

"What if $ u$ is not a continuously differentiable function? Then its image $ Lu$ is not square-integrable, but the inner product <v, Lu> is still well-defined because it is finite. For example, if u is a function which has a kink, then $ Lu$ would not be defined at that point and $ Lu$ would not be square-integrable. Nevertheless, the integral of $ \overline v Lu$ would be perfectly finite."

I don't understand the fact is if Lu is not defined how can u define an inner product with v at any point, ie <v, Lu>. What does it mean physically at all, is it a mathematical jugglery to move the L operator to v and then say that look it is still defined? I am totally confused.

Thanks a lot for any help in advance.
 
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  • #2
Why do you say "Lu is not defined"? If Lu is not square-integrable, then it is not in L2 but it is in some larger space, of which L2 is a subspace. The innerproduct can be defined in that larger space.
 
  • #3
Hi,

I say L is not defined because of the following. Let's give an example. Since L can be d^2/dx^2 + a(x) d/dx + b(x) and if u consider the function u s.t

for say (a< x <b), a<0, b >0

u(x) = 0 x<0,
= x x>= 0

The fn u is cont but is not differentiable at x = 0. So I am not sure how for such functions u can define the operator like this. This is my question.

Regards
Sumanta
 

1. What does it mean for a function to be "non continuously differentiable"?

Non continuously differentiable functions are those that do not have a well-defined derivative at every point in their domain. This means that the function is not smooth and has abrupt changes or breaks in its slope.

2. Can a non continuously differentiable function still have a finite inner product?

Yes, a function can be non continuously differentiable but still have a finite inner product. The inner product of two functions is a measure of their similarity and does not depend on the smoothness of the functions.

3. How is the inner product of two non continuously differentiable functions calculated?

The inner product of two non continuously differentiable functions is calculated using the same formula as for continuously differentiable functions. This involves integrating the product of the two functions over their common domain.

4. Are there any real-world applications of non continuously differentiable but inner product finite functions?

Yes, non continuously differentiable functions are commonly used in physics and engineering to model systems with abrupt changes or discontinuities. They are also used in signal processing and data analysis to represent signals with sudden changes or spikes.

5. Can a non continuously differentiable function be approximated by a continuously differentiable one?

Yes, non continuously differentiable functions can be approximated by continuously differentiable ones using techniques such as smoothing or interpolation. However, the approximation may not accurately capture the abrupt changes or breaks in the slope of the original function.

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