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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2011 • Question 3

Question

Math 6210 • Real Analysis Spring 2011 • Question 3

Let \(f\) be a continuous function on the circle \(\{e^{i\theta }\mid 0\le \theta <2\pi \}\). Let \[ c_n:=\frac {1}{2\pi }\int _0^{2\pi } f(e^{i\theta })e^{-in\theta }\,d\theta . \] Prove or disprove: \[ f(e^{i\theta })=\lim _{N\to \infty }\left (\sum _{n=-N}^{N}c_ne^{in\theta }\right ) \] for all \(\theta \).

Raw LaTeX

Let \(f\) be a continuous function on the circle
\(\{e^{i\theta}\mid 0\le\theta<2\pi\}\). Let
\[
c_n:=\frac{1}{2\pi}\int_0^{2\pi} f(e^{i\theta})e^{-in\theta}\,d\theta.
\]
Prove or disprove:
\[
f(e^{i\theta})=\lim_{N\to\infty}\left(\sum_{n=-N}^{N}c_ne^{in\theta}\right)
\]
for all \(\theta\).

Topic Tags

Lebesgue Integration