Proving the Closure of Even Functions in the Algebra of Polynomials

In summary, the conversation discusses how to show that the closure of a subset of continuous functions with even degree terms (A) in a larger set of continuous functions (F) is equal to the set of even functions in F. It also explores how the same process can be applied to a subset of functions with odd degree terms in order to show that the closure is the set of odd functions in F. The method involves approximating functions with polynomials and using the Weierstrass Approximation Theorem.
  • #1
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Homework Statement
Let F be the set of all continuous functions with domain [-1,1] and codomain R. Let A be the algebra of all polynomials that contain only terms of even degree (A is a subset of F). Show that the closure of A in F is the set of even functions in F.

The attempt at a solution
I have to show that (i) if f in F is even, then f is in the closure of A and (ii) if f is in the closure of A, then f is even. I don't have problems proving (ii), rather I'm stuck proving (i). Here's what I have so far:

Let f in F be even. By the Weierstrass Approx. Theorem, there is a sequence of polynomials {p_n} that converge uniformly to f. Now let q_n be the polynomial derived from p_n by squaring each term so that all the degrees are even. For x in [0,1], q_n(sqrt(x)) = p_n(x), so {q_n(sqrt(x))} converges to f(x). Since f is even, {q_n(sqrt(x))} converges to f(-x). Now it would be nice to show that {q_n} converges to f, but this is not the case. If anything, {q_n} converges to f(x^2). How do I proceed from here?
 
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  • #2
If qn is converging to f(x2) maybe you want to try approximating f(sqrt(|x|)) instead?
 
  • #3
Office_Shredder said:
If qn is converging to f(x2) maybe you want to try approximating f(sqrt(|x|)) instead?
But what comes after that? I'm interested in approximating, f(x), not f(sqrt(|x|)). This is the part I can't figure out.
 
  • #4
If f(x) is even and continuous, then it's equal to F(x^2) where F is continuous on [0,1] (sure F(x)=f(sqrt(|x|))). Approximate F(x) with a polynomial p(x) on [0,1]. Then p(x^2) approximates f(x), right?
 
  • #5
Ah, OK. My error was that I started approximating f(x) first. Silly me.
 
  • #6
Now suppose we change A so that it is the set of all polynomials in F whose terms all have odd degree. Show that the closure of A is the set of odd function in F.

Again, (ii) is easy to show, but (i) is not. Let f in F be an odd function. We need to find two continuous functions g and h such that g(h(x)) = f(x). By the Weierstrass Approx. Theorem, there is a sequence of polynomials {p_n} that converge uniformly to g. We will need for p_n(h(x)) to converge to g(h(x)) = f(x), such that p_n(h(x)) is a polynomial with odd degree terms. It is not clear to me how we can make p_n(h(x)) into a polynomial with odd degree terms. For example, how do we even get rid of the constant term in p_n(h(x))?
 
  • #7
If p(x) approximates f(x) on [-1,1], then p(-x) also approximates f(-x). Can you think of a way to combine the two approximations to get another approximation with only odd powers?
 
  • #8
Smart. p(x) - p(-x) approximates f(x) - f(-x) = f(x) + f(x) = 2f(x). p(x) - p(-x) has odd degree terms. And so 1/2[p(x) - p(-x)] is a polynomial in A that approximates f(x).

What made you think about this?
 
  • #9
Hard to say. But if g(x) is any function (g(x)-g(-x))/2 is odd. If g is already odd it just gives you g back again. So if p is 'almost' g. Then (p(x)-p(-x))/2 must be 'almost' g. You could have done the even problem the same way.
 

Related to Proving the Closure of Even Functions in the Algebra of Polynomials

1. What is an even continuous function?

An even continuous function is a mathematical function that is symmetrical about the y-axis. This means that the value of the function at x is equal to the value of the function at -x. It is also continuous, meaning that there are no breaks or gaps in the graph of the function.

2. How do you graph an even continuous function?

To graph an even continuous function, you can use a graphing calculator or plot points on a coordinate plane. Remember that the graph will be symmetrical about the y-axis, so you only need to graph one side and then reflect it across the y-axis to get the complete graph.

3. What are some examples of even continuous functions?

Some examples of even continuous functions include cosine, absolute value, and parabolic functions. These functions have a symmetrical graph about the y-axis and have no breaks or gaps in their graph.

4. How do you determine if a function is even or not?

To determine if a function is even or not, you can use the symmetry test. Substitute -x for x in the function and simplify. If the resulting function is the same as the original, then the function is even. If the resulting function is the negative of the original, then the function is odd.

5. What is the importance of even continuous functions?

Even continuous functions are important in mathematics because they allow us to model and solve real-world problems. They also have many applications in physics, engineering, and economics. Additionally, they have special properties that make them easier to work with and analyze compared to other types of functions.

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