Approximating x^2 as a sequence of simple functions

In summary, the person is looking for a sequence of functions that converge uniformly to a specific function on the interval [0,1]. They know a simple function is one that can be written as \sum^{n}_{k=1}a_{k}1_{D_{k}}(x) where {D1,...,Dn} is a collection of measurable sets and a1,...,an is a real number. They attempted to solve the problem but didn't know how to begin. They would break the range of the function into pieces, take the inverse image of these pieces and make these the D's, the measurable sets. The a's, the actual values that the simple function takes on, would be the smallest value
  • #1
Yagoda
46
0

Homework Statement


I'm looking for a sequence of simple functions fn that converges uniformly to f(x)=x2 on the interval [0,1].


Homework Equations



I know a simple function is one that can be written as [itex]\sum^{n}_{k=1}a_{k}1_{D_{k}}(x)[/itex] where {D1,...,Dn} is collection of measurable sets and a1,...,an are real numbers.

The Attempt at a Solution


I don't really know how to begin here. My general idea would be to break up the range of the function into pieces, take the inverse image of these pieces and make these the D's, the measurable sets whose characteristic functions we are looking at. The a's, the actual values that the simple function takes on, could be the smallest value in a particular piece of the range. The higher terms of the sequence would break the range into smaller and smaller pieces.

But how would I actually write down this sequence of functions?
 
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  • #2
Yagoda said:

Homework Statement


I'm looking for a sequence of simple functions fn that converges uniformly to f(x)=x2 on the interval [0,1].


Homework Equations



I know a simple function is one that can be written as [itex]\sum^{n}_{k=1}a_{k}1_{D_{k}}(x)[/itex] where {D1,...,Dn} is collection of measurable sets and a1,...,an are real numbers.

The Attempt at a Solution


I don't really know how to begin here. My general idea would be to break up the range of the function into pieces, take the inverse image of these pieces and make these the D's, the measurable sets whose characteristic functions we are looking at. The a's, the actual values that the simple function takes on, could be the smallest value in a particular piece of the range. The higher terms of the sequence would break the range into smaller and smaller pieces.

But how would I actually write down this sequence of functions?

I would pick your measurable sets to be n intervals of size 1/n that cover [0,1]. Then you just have to define a_n for each interval. Don't get intimidated by the terminology. It's really not that hard.
 
  • #3
You're right that it actually wasn't that difficult. I was still trying to wrap my head around simple functions. Thanks for your help!
 

FAQ: Approximating x^2 as a sequence of simple functions

What does it mean to approximate x^2 as a sequence of simple functions?

Approximating x^2 as a sequence of simple functions means breaking down the function x^2 into smaller, simpler functions that when combined, closely resemble the original function. This allows for easier calculations and can provide a more accurate representation of the function over a certain interval.

Why would someone want to approximate x^2 as a sequence of simple functions?

Approximating x^2 as a sequence of simple functions can be useful in situations where the original function may be difficult to work with, such as in complex integrals or differential equations. By breaking down the function into simpler components, it becomes easier to manipulate and solve for specific values.

What are some examples of simple functions that can be used to approximate x^2?

Some examples of simple functions that can be used to approximate x^2 include linear functions, quadratic functions, and trigonometric functions. These can be combined and manipulated in various ways to closely approximate the behavior of x^2 over a specific interval.

How accurate is the approximation of x^2 as a sequence of simple functions?

The accuracy of the approximation depends on the number of simple functions used and the interval over which they are combined. The more simple functions used, the closer the approximation will be to the original function. However, there may still be some error present, especially at the boundaries of the interval.

Are there any limitations to approximating x^2 as a sequence of simple functions?

Yes, there are limitations to this approach. The accuracy of the approximation decreases as the complexity of the original function increases. Additionally, the choice of simple functions used and the interval over which they are combined can also affect the accuracy of the approximation. It is important to carefully consider these factors when using this method.

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