One of the slogans in T. W. Körner's book Fourier Analysis that is definitely in the harmonic analyst's toolbox: The function $f*g$ has the good properties both of f and g.

An example of its use is in approximating functions by trigonometric polynomials: convolving the function with any trigonometric polynomial gives you a trigonometric polynomial, and if you pick the polynomial carefully the resulting function will have similar properties to the original one.