FIND ME ON

GitHub

LinkedIn

Convolution

🌱

AnalysisMeasureTheory

For measurable f,g:R→[0,āˆž)f,g:\mathbb{R}\to[0,\infty), their convolution is (fāˆ—g)(x)=∫Rf(t)g(xāˆ’t) dt(f*g)(x)=\int\limits _{\mathbb{R}}f(t)g(x-t) \, dt

- f,g∈L1ā€…ā€ŠāŸ¹ā€…ā€Šfāˆ—g∈L1f,g\in L^{1}\implies f*g\in L^{1} - f,g≄0ā€…ā€ŠāŸ¹ā€…ā€Šfāˆ—g∈L1f,g\ge 0\implies f*g\in L^{1} - This allows us to apply Fubini-Tonelli…

Linked from