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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2021 • Question 5

Question

Math 6210 • Real Analysis Fall 2021 • Question 5

Let \(f, g \in L^{2}(\mathbb {R})\). Show that

\[ h(x) = \int _{\mathbb {R}} f(x-y)g(y)\,dy \]

is a well-defined real number for every \(x \in \mathbb {R}\) and that \(h : \mathbb {R} \to \mathbb {R}\) is continuous.

Raw LaTeX

Let \(f, g \in L^{2}(\mathbb{R})\). Show that

\[
h(x) = \int_{\mathbb{R}} f(x-y)g(y)\,dy
\]

is a well-defined real number for every \(x \in \mathbb{R}\) and that \(h : \mathbb{R} \to \mathbb{R}\) is
continuous.

Topic Tags

Lebesgue Integration