Proving Integral Convergence with L1 Functions

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SUMMARY

The discussion centers on proving integral convergence for functions in L1 space, specifically demonstrating that for almost every x in the interval [a,b], the limit as h approaches 0+ of the integral of abs(f(x+t) + f(x-t) - 2f(x)) dt equals 0. The Dominated Convergence Theorem is identified as a crucial tool for this proof, although the initial poster expresses uncertainty about its application. Clarification on defining the variable h is also sought, indicating its importance in the context of the problem.

PREREQUISITES
  • Understanding of L1 space and its properties
  • Familiarity with the Dominated Convergence Theorem
  • Basic knowledge of Fourier analysis concepts
  • Proficiency in real analysis, particularly limits and integrals
NEXT STEPS
  • Study the application of the Dominated Convergence Theorem in real analysis
  • Explore the properties of L1 functions and their implications in convergence
  • Review Fourier analysis techniques related to integral convergence
  • Investigate definitions and roles of variables in limit problems
USEFUL FOR

Mathematicians, students preparing for screening exams in analysis, and anyone interested in the convergence properties of integrable functions.

scottneh
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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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You could start by defining h.
 
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
 

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