Convolution


    
  • For \(\phi\) any approximate identity, \(\phi \ast f\) is generally smooth.
  • If \(f\) is uniformly continuous then \(\phi \ast f\to f\) uniformly.
  • \(f\in C^k_c\) implies \(\phi \ast f\in C^k_c\) and the first \(k\) derivatives \((\phi \ast f)^{(k)}\) converge uniformly to \(f^{(k)}\).
  • For any \(L^p\), if \(f\in L^p\) then \(\phi \ast f\to f\) in \(L^p\).