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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 4

Question

Math 6210 • Real Analysis Spring 2008 • Question 4

Let \(f\in L^1([-\pi ,\pi ])\), and let \(c_n\) be the nth Fourier coefficient of \(f\): \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi } f(x)e^{-inx}\,dx \quad n\in \mathbb Z \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Raw LaTeX

Let \(f\in L^1([-\pi,\pi])\), and let \(c_n\) be the nth Fourier
coefficient of \(f\):
\[
c_n=\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi} f(x)e^{-inx}\,dx
\quad n\in\mathbb Z
\]
Prove that \(c_n\to0\) as \(|n|\to\infty\).

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lp Spaces and Sequence Spaces