Convolution
Infimal convolution

In convex analysis, the infimal convolution of proper (not identically + ∞ {\displaystyle +\infty }) convex functions f 1, …, f m {\displaystyle f_{1},\dots,f_{m}} on R n {\displaystyle \mathbb {R} ^{n}} is defined by: (f 1 ∗ ⋯ ∗ f m) (x) = inf x { f 1 (x 1) + ⋯ + f m (x m) | x 1 + ⋯ + x m = x }. {\displaystyle (f_{1}*\cdots *f_{m})(x)=\inf _{x}\{f_{1}(x_{1})+\cdots +f_{m}(x_{m})|x_{1}+\cdots +x_{m}=x\}.} It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the Legendre transform: φ ∗ (x) = sup y (x ⋅ y − φ (y)). {\displaystyle \varphi ^{*}(x)=\sup _{y}(x\cdot y-\varphi (y)).} We have: (f 1 ∗ ⋯ ∗ f m) ∗ (x) = f 1 ∗ (x) + ⋯ + f m ∗ (x). {\displaystyle (f_{1}*\cdots *f_{m})^{*}(x)=f_{1}^{*}(x)+\cdots +f_{m}^{*}(x).}
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