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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 4

Question

Math 6210 • Real Analysis Fall 2007 • Question 4

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

Raw LaTeX

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

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lp Spaces and Sequence Spaces