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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 3

Question

Math 6210 • Real Analysis 2006 • Question 3

Let \(\alpha \) be an irrational number. Let \(f\) be a real valued measurable function on \(\mathbb R\) such that \(f(x)=f(x+1)=f(x+\alpha )\). Use Fourier series to show that \(f\) is constant almost everywhere.

Raw LaTeX

Let \(\alpha\) be an irrational number. Let \(f\) be a real valued
measurable function on \(\mathbb R\) such that
\(f(x)=f(x+1)=f(x+\alpha)\). Use Fourier series to show that \(f\) is constant almost
everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Fourier Analysis and Convolution Measurable Functions