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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2000 • Question 5

Question

Math 6210 • Real Analysis 2000 • Question 5

Let \(f\) be a continuous positive valued function on \(\mathbb R\), and let \(g\) be the characteristic function of a finite nonempty open interval in \(\mathbb R\). Show that the function \(h=fg\) is in \(L^1(\mathbb R)\), but its Fourier transform \(\hat h\) is not.

Raw LaTeX

Let \(f\) be a continuous positive valued function on \(\mathbb R\), and
let \(g\) be the characteristic function of a finite nonempty open interval in
\(\mathbb R\). Show that the function \(h=fg\) is in \(L^1(\mathbb R)\), but its
Fourier transform \(\hat h\) is not.

Topic Tags

Fourier Analysis and Convolution Lp Spaces and Sequence Spaces