Sebastiaan's Subspace

About mathematics and certain other interests

Category: Analysis

Calculating an auto-convolution integral by Fourier transforms

Recently I came across a forum contribution, where the author Charles Link describes how to use the unitarity of the Fourier transform on $L^2(\RR)$ to compute the definite integral

I(t_0) \DEF \int_{-\infty}^{\infty}{\frac{\sin(\tau – t_0)}{(\tau – t_0)}\frac{\sin(\tau + t_0)}{(\tau + t_0)}\,d\tau},

where $t_0 \in \RR$ is a constant. In the comments it is also argued that one may alternatively use contour integration. I liked the article, but wondered whether the symmetry of the problem would perhaps admit a simpler approach.

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Continuity of eigenvalues

As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries.

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Continuity of polynomial roots

It was recently brought up how to show that the n roots of a real or complex polynomial depend continuously on the polynomial’s coefficients. Although I have used this proposition numerous times, implicitly and explicitly, I realized that I never saw a proof of it.

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