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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2000 • Question 2

Question

Math 6420 • PDE Fall 2000 • Question 2

Suppose the function \(\phi (x)\) is Lipschitz continuous on the unit interval \(0<x<2\pi \) and that \(\lim _{x\to 0}\phi (x)=\phi (0^+)\) exists. Suppose \(\int _0^\infty \frac {\sin x}{x}\,dx=\alpha \) is known. Evaluate \(\lim _{k\to \infty }\int _0^{2\pi }\phi (x)\frac {\sin kx}{x}\,dx\) (with \(k\) restricted to be integer-valued).

Raw LaTeX

Suppose the function \(\phi(x)\) is Lipschitz continuous on the unit interval
\(0<x<2\pi\) and that \(\lim_{x\to 0}\phi(x)=\phi(0^+)\) exists. Suppose
\(\int_0^\infty \frac{\sin x}{x}\,dx=\alpha\) is known. Evaluate
\(\lim_{k\to\infty}\int_0^{2\pi}\phi(x)\frac{\sin kx}{x}\,dx\) (with \(k\) restricted to be
integer-valued).

Topic Tags

Fourier Transform and Sampling Theory Linear Programming and Optimization Point, Continuous, and Residual Spectrum Real Integral Evaluation by Complex Methods