Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 2

Question

Math 6210 • Real Analysis Fall 2009 • Question 2

Recall that the Fourier transform of a function \(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by \[ \hat f(x)=\int _{-\infty }^{\infty } e^{-2\pi ixy}f(y)\,dy \] Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Raw LaTeX

Recall that the Fourier transform of a function
\(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by
\[
\hat f(x)=\int_{-\infty}^{\infty} e^{-2\pi ixy}f(y)\,dy
\]
Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Topic Tags

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