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  "body": "\n## Summary\n\nThe Mellin convolution of two functions $f$ and $g$ on $(0, \\infty)$ is $(f \\star g)(x) = \\int_0^\\infty f(x/y) g(y) \\, \\frac{dy}{y}$.\nIt is the [[convolution|2Z29JN3VES]] of the multiplicative group $(0, \\infty)$ with its invariant measure $dy/y$.\nThe [[Mellin transform|N5PVGRJE1T]] turns it into a product: $\\mathcal{M}(f \\star g) = \\mathcal{M}f \\cdot \\mathcal{M}g$ where both transforms converge absolutely.\nIt describes, for example, the law of a product of independent positive random variables and many integral operators with homogeneous kernels.\n\n## Explanations\n\n### L1 human\n\nOrdinary convolution averages $f(x - y)$ against $g(y)$, mixing shifts.\nMellin convolution averages $f(x/y)$ against $g(y)$, mixing rescalings instead.\nThe weight $dy/y$ treats the intervals $[1, 2]$ and $[10, 20]$ as equally large, as a logarithmic scale does.\nFor example, the indicator function of $(0, 1]$ convolved with itself gives $-\\log x$ on $(0, 1]$ and $0$ beyond.\n",
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