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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 5

Question

Math 6210 • Real Analysis Spring 2009 • Question 5

Let \(f\in L^1(\mathbb R)\) such that \[ \int _{-\infty }^{\infty }|x\hat f(x)|\,dx<\infty \] Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that \(f=g\) almost everywhere.

Raw LaTeX

Let \(f\in L^1(\mathbb R)\) such that
\[
\int_{-\infty}^{\infty}|x\hat f(x)|\,dx<\infty
\]
Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that
\(f=g\) almost everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Complex Integration and Real Integral Evaluation Lebesgue Integration Lp Spaces and Sequence Spaces