Integration and sequences of functions

In summary, the conversation discusses a problem involving a continuous function on [0,1] and its relationship to the integral of x^n times the function being equal to zero for all even natural numbers n. The Weierstrass Approximation Theorem is mentioned as a possible approach, but the conversation ultimately concludes that by extending the function to an even function on [-1,1] and showing that the integral of the squared function is zero, it can be proven that the original function is equal to zero for all values of x on [0,1].
  • #1
Hitman2-2

Homework Statement



Let f be a continuous function on [0,1]. Prove that if

[tex]

\int_{0}^{1} x^n f(x) dx = 0

[/tex]

for all even natural numbers n, then f(x) = 0 for all [tex]x \in [0,1] [/tex].

Homework Equations


The Attempt at a Solution



I'm pretty much stuck on this problem. All I know is that by the Weierstrass Approximation Theorem, there exists a sequence of polynomials Pn that converges uniformly to f, so

[tex]

\lim_{n\rightarrow 0} \int_{0}^{1} P_n f(x) dx = \int_{0}^{1} f(x) \lim_{n\rightarrow 0} P_n dx
= \int_{0}^{1} f^2(x) dx

[/tex]

but that's about as far as I can get.
 
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  • #2
How about if you extend f(x) to an even function g(x) on the interval [-1,1]? And while you are at it subtract a constant so the integral of g(x) over [-1,1] is zero. Now the integral of x^k times g(x) equals zero for all k greater than or equal zero.
 
  • #3
What would be the relationship between

[tex]

\int_{0}^{1} x^n f(x) dx

[/tex]

and

[tex]

\int_{-1}^{1} x^n g(x) dx

[/tex]
 
  • #4
Hitman2-2 said:
What would be the relationship between

[tex]

\int_{0}^{1} x^n f(x) dx

[/tex]

and

[tex]

\int_{-1}^{1} x^n g(x) dx

[/tex]

If n is even then the second one is twice the first one, right? Or zero. If n is odd then the second one vanishes because its the integral of an odd function.
 
  • #5
Hmmmm. I've been trying to show integral of g^2 over [-1,1] is zero. But I can't figure out what to do with the 'constant part' of g(x). My suggestion earlier that you just 'subtract it off' doesn't work. Maybe there's an easier way. Any new ideas?
 
  • #6
No, unfortunately. But I'll let you know if I figure anything out. Thanks again.
 
  • #7
Hitman2-2 said:
No, unfortunately. But I'll let you know if I figure anything out. Thanks again.

Do that, thanks. This is annoying me.
 
  • #9
Dick said:
Um, you wouldn't consider n=0 an even natural number, would you? See https://www.physicsforums.com/showthread.php?t=290229

For the course I'm currently in, I think so. We know 0 is an even number. In my class, {0,1,2,...} is taken to be the set of natural numbers (though other professors I've had take {1,2,3,...} to be the natural numbers).
 
  • #10
Ok, that fixes it. The 'constant part' of g(x) must be zero. Now can you show that the integral of g(x)^2 must be zero? Just as you first started to do with Weierstrass. Do you see how that proves g(x)=0, hence f(x)=0?
 
  • #11
Dick said:
Ok, that fixes it. The 'constant part' of g(x) must be zero. Now can you show that the integral of g(x)^2 must be zero? Just as you first started to do with Weierstrass. Do you see how that proves g(x)=0, hence f(x)=0?

Well, g(x) would be continuous on [-1,1] so we have a sequence of polynomials pn converging uniformly to g, hence

[tex]

0 = \int_{-1}^{1} g(x) \lim_{n\rightarrow 0} p_n dx
= \int_{-1}^{1} g^2(x) dx

[/tex]

right?
 
  • #12
Yes, if you have followed everything that went before. All the powers in p_n integrated against g are zero.
 
  • #13
all right, I think I see what's going on. Thanks again for all your assistance.
 

Related to Integration and sequences of functions

1. What is integration in mathematics?

Integration is a mathematical concept that involves finding the area under a curve. It is the inverse process of differentiation and is used to calculate the total change of a function over a given interval.

2. How is integration used to solve real-world problems?

Integration is used in many real-world applications, such as calculating the volume of a container, finding the distance traveled by an object, and determining the total revenue of a business. It is also used in physics, engineering, and economics to model and analyze various systems.

3. What are the different methods of integration?

The most common methods of integration include the use of basic integration rules, such as the power rule, substitution, integration by parts, and partial fractions. Other techniques, such as trigonometric substitutions and numerical integration, are also used to solve more complex problems.

4. What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a fundamental concept in integration that relates differentiation and integration. It states that the integral of a function can be evaluated by finding the antiderivative of the function and evaluating it at the endpoints of the interval. In other words, integration and differentiation are inverse operations of each other.

5. What are sequences of functions?

Sequences of functions are a series of functions that are indexed by natural numbers. Each function in the sequence is defined for a specific domain and is related to the other functions in the sequence by a specific rule or pattern. They are often used in calculus to approximate more complex functions and to study the behavior of functions as the input values approach a specific value.

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