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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2023 • Question 5

Question

Math 6210 • Real Analysis Spring 2023 • Question 5

In this problem, let \(\lambda \) denote Lebesgue measure on \(\mathbb {R}\) and \(\widehat {\cdot }\) denote the Fourier transform (on \(\mathbb {R}\)). Show that if \(f,g\) are Lebesgue integrable and \(\hat f=\hat g\) then \(f=g\) \(\lambda \)-a.e.

Raw LaTeX

In this problem, let \(\lambda\) denote Lebesgue measure on \(\mathbb{R}\) and \(\widehat{\cdot}\) denote the
Fourier transform (on \(\mathbb{R}\)). Show that if \(f,g\) are Lebesgue integrable and
\(\hat f=\hat g\) then \(f=g\) \(\lambda\)-a.e.

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lebesgue Measure and Translation Invariance Measure Theory