Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2018 • Question 4

Question

Math 6210 • Real Analysis Fall 2018 • Question 4

Let \(H=L^2(\mathbb R/\mathbb Z)\), with the usual Lebesgue measure on \(\mathbb R/\mathbb Z\) of total measure \(1\). It is well known that this space has a Hilbert basis \(e_n=e^{2\pi i n x}\), \(n\in \mathbb Z\). Let \(f\in H\) be given by \(f(x)=x\) on \([0,1)\). Compute \(\lVert f\rVert _2^2\) directly and then using Parseval’s equality.

Raw LaTeX

Let \(H=L^2(\mathbb R/\mathbb Z)\), with the usual Lebesgue
measure on \(\mathbb R/\mathbb Z\) of total measure \(1\). It is well known that
this space has a Hilbert basis \(e_n=e^{2\pi i n x}\), \(n\in\mathbb Z\). Let
\(f\in H\) be given by \(f(x)=x\) on \([0,1)\). Compute \(\lVert f\rVert_2^2\)
directly and then using Parseval's equality.

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory