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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1997 • Question 14

Question

Math 6410 • ODE Fall 1997 • Question 14

Let \(\zeta \in C_0(\mathbb {R})\) be continuous with compact support such that the integral \[ \int _{-\infty }^{\infty }\zeta (x)\,dx=1. \] Show that as distributions, the convolutions \(\zeta _\epsilon *F\to F\) as \(\epsilon \to 0\), where \[ \zeta _\epsilon (x):=\frac {1}{\epsilon }\zeta \left (\frac {x}{\epsilon }\right ). \]

Raw LaTeX

Let \(\zeta\in C_0(\R)\) be continuous with compact support such that the
integral
\[
  \int_{-\infty}^{\infty}\zeta(x)\,dx=1.
\]
Show that as distributions, the convolutions \(\zeta_\epsilon*F\to F\) as
\(\epsilon\to0\), where
\[
  \zeta_\epsilon(x):=\frac{1}{\epsilon}\zeta\left(\frac{x}{\epsilon}\right).
\]

Topic Tags

Distributions and Weak Solutions Gronwall Inequality and Continuous Dependence Point, Continuous, and Residual Spectrum