Mellin Transform Convolution
📂Fourier AnalysisMellin Transform Convolution
Definition
The convolution of the Mellin transform is as follows.
(f×g)(y)=∫0∞f(x)g(xy)xdx
Explanation
It is also called multiplicative convolution.
Proof
M(f×g)=(Mf)(Mg)
It suffices to show that the above equation holds.
M(f×g)(s)=∫0∞xs−1(f×g)(x)dx=∫0∞xs−1(f×g)(x)dx=∫0∞xs−1(∫0∞f(y)g(yx)ydy)dx=∫0∞∫0∞xs−1f(y)g(yx)ydydx=∫0∞∫0∞ys−1zs−1f(y)g(z)dydz=∫0∞ys−1f(y)dy∫0∞zs−1g(z)dz=Mf(s)Mg(s)
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