L1 convolution bound
= L1 convolution bound
{c}
{title2=$\|g*h\|_1\le\|g\|_1\|h\|_1$}
<Tonelli's theorem> gives the product <integral> of $|g(x-y)h(y)|$ as $\|g\|_1\|h\|_1$. <Fubini's theorem> then defines the <convolution> <almost everywhere> and gives an integrable result with <norm> at most that product. <Multiplication> by a <complex exponential> of <modulus> one preserves <absolute integrability> and permits the usual <Fourier transform> <convolution> identity.