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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2013 • Question 5

Question

Math 6210 • Real Analysis Spring 2013 • Question 5

Consider \(\mathbb R\) with the Lebesgue measure, and recall the definitions of convolution and the Fourier transform, for \(f,g\in L^1(\mathbb R)\): \[ f*g(x)=\int f(x-y)g(y)\,dy,\qquad \hat f(\xi )=\int f(x)e^{-2\pi ix\xi }\,dx. \] Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that \[ \widehat {f*g}=\hat f\,\hat g. \]

Raw LaTeX

Consider \(\mathbb R\) with the Lebesgue measure, and recall the
definitions of convolution and the Fourier transform, for
\(f,g\in L^1(\mathbb R)\):
\[
f*g(x)=\int f(x-y)g(y)\,dy,\qquad
\hat f(\xi)=\int f(x)e^{-2\pi ix\xi}\,dx.
\]
Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that
\[
\widehat{f*g}=\hat f\,\hat g.
\]

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces