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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2001 • Question 2

Question

Math 6210 • Real Analysis 2001 • Question 2

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product \(f*g\) is defined by \[ f*g(x)=\int f(y)g(x-y)\,dy, \] then \(f*g(x)\) is a continuous function which vanishes at infinity.

Raw LaTeX

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product
\(f*g\) is defined by
\[
f*g(x)=\int f(y)g(x-y)\,dy,
\]
then \(f*g(x)\) is a continuous function which vanishes at infinity.

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lp Spaces and Sequence Spaces