Question about Smooth Interpolation

In summary, the conversation discusses finding a C^oo (C infinity) function f : R->R that is positive on the interval (-1, 1) and 0 elsewhere. Two approaches are suggested: an indirect approach using limits and a more direct approach involving finding a function with all derivatives matching up. An example function is given and its smoothness is proven using induction.
  • #1
chy1013m1
15
0
any thoughts to this question?
Give an example of a C^oo (C infinity) function f : R->R which is positive on the interval (-1, 1) and 0 elsewhere
 
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  • #2
For an indirect approach, you could try constructing one as a limit.


For a more direct approach, figure out what the obstacle is.

You know smooth functions that are zero outside of (-1, 1), and you know smooth functions that are positive inside (-1, 1).

So, what's the problem? You need all of the derivatives to match up. In other words, for your function positive inside of (-1, 1), you need all of its derivatives to be zero at 1 and at -1.

So that's your problem: you need to find a function (other than the zero function) whose derivatives are all zero at some point.
 
  • #3
just to check if i am on the right track..
let f(x ) = { e ^ (1 / (x ^ 2 - 1)) if -1 < x < 1
0 otherwise
then it is easy to show that
lim f( x) = 0 (split into 2 cases)
x->1 or -1

it is also true that
lim (f(x + h) - f(x)) / h = 0 for x = 1, -1 (show with limit proof)
h->0
then f is C1so then induction on f being Ck, show it is C(k+1)..(if i did it correctly, it should work out..)
then by induction, f is Cinf.
 
  • #4
Hrm. I find it very plausible that function will work. Incidentally, I was thinking of using the function

[tex]
g(x) := \begin{cases}
0 & x \leq 0 \\
e^{-1/x^2} & x > 0
[/tex]

as a building block, since we already know it's smooth. Actually, I think you can build your function out of this, so I'm even more convinced.
 

Related to Question about Smooth Interpolation

1. What is smooth interpolation?

Smooth interpolation is a mathematical technique used to estimate values between two known data points. It involves fitting a curve or line through the data points in a way that minimizes abrupt changes or fluctuations.

2. How is smooth interpolation different from regular interpolation?

The main difference between smooth interpolation and regular interpolation is the way in which the curve or line is fitted through the data points. Smooth interpolation attempts to minimize abrupt changes, while regular interpolation simply connects the data points with straight lines or curves.

3. What are the applications of smooth interpolation?

Smooth interpolation is commonly used in fields such as computer graphics, signal processing, and numerical analysis. It is also used in scientific and engineering applications to estimate values between data points and to smooth out noisy data.

4. What are the limitations of smooth interpolation?

Smooth interpolation can only be used for data that can be modeled by a continuous function. It also relies on the assumption that the data points are evenly spaced. Additionally, smooth interpolation may not accurately represent extreme or outlier data points.

5. What are some common methods of smooth interpolation?

Some common methods of smooth interpolation include cubic splines, polynomial interpolation, and regression techniques such as least squares. Each method has its own advantages and disadvantages, and the choice of method often depends on the specific application and data set.

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