Convolution 


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