Proving Integral Convergence with L1 Functions

In summary, the conversation is about solving a problem involving a function contained in L1([a,b]). The goal is to prove an equation involving a limit and an integral for almost everywhere x in [a,b]. The person initially thought they could use an argument involving t as an offset, but now they believe they need to use the dominated convergence theorem. They are unsure how to apply it and are seeking help. They also mention finding a similar equation in Fourier analysis but it does not fully cover the problem. They are unsure how to define h.
  • #1
scottneh
3
0
Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given.

Given:

Suppose f is contained in L1([a,b])

Prove for almost everywhere x is contained in [a,b]

limit as h goes to 0+, int (abs(f(x+t)+f(x-t)-2f(x)))dt = 0

Initially I thought that I could argue this as if t is an offset to the function and say that as h goes to zero the t would go to zero and clearly f(x+0)+f(x-0) = 2f(x), then 2f(x)-2f(x)=0

I think I have to use dominated convergence theorem but I'm not sure how to apply it.

Can someone please help me get started?

Thanks
 
Last edited:
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  • #2
You could start by defining h.
 
  • #3
The problem does not state a definition for h.

After searching around on the net I found the exact same equation, namely:

f(x+t)+f(x-t)-2f(x) in regards to Fourier analysis but it did't quite cover what the problem is stating.

How should I define h?

Thanks
 

1. What is an integral convergence proof?

An integral convergence proof is a mathematical technique used to determine whether an integral (a mathematical expression representing the area under a curve) converges or diverges. It involves evaluating the limit of a particular function as the upper and lower bounds of the integral approach infinity.

2. How do you know when to use an integral convergence proof?

An integral convergence proof is typically used when dealing with improper integrals, which are integrals with infinite bounds or integrands that are undefined at certain points. It is also used to determine the convergence of a series, which is a sequence of terms that are summed. If the terms of a series can be represented by an integral, an integral convergence proof can be used.

3. What are the different types of convergence that can be proved using this method?

The two main types of convergence that can be proved using an integral convergence proof are absolute convergence and conditional convergence. Absolute convergence means that the integral converges regardless of the order in which the terms are summed. Conditional convergence means that the integral only converges if the terms are summed in a specific order.

4. What are some common techniques used in integral convergence proofs?

Some common techniques used in integral convergence proofs include the comparison test, the limit comparison test, the ratio test, and the root test. These tests involve comparing the given integral to a known convergent or divergent integral, and using algebraic manipulations and limits to determine the convergence of the given integral.

5. Can an integral convergence proof be used for all integrals?

No, an integral convergence proof can only be used for certain types of integrals, specifically improper integrals and series representations of integrals. It cannot be used for definite integrals with finite bounds, as these can be evaluated directly using the fundamental theorem of calculus. It also cannot be used for multidimensional integrals, as these require different techniques for convergence proofs.

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