On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

A persistence diagram represents the birth and death of homology classes along a filtration as a multiset of intervals, and we consider persistence diagram vectorizations to be maps $D(X,A)\to E$ sending persistence diagrams over a metric pair $(X,A)$ to values in a Banach space $E$. We prove an isometric isomorphism between Lipschitz extensions of vectorizations, modulo constants, on the Grothendieck completion $K(X,A)$ of $D(X,A)$ and the bounded $1$-cocycles for the translation action of $K(X,A)$ on ${\mathrm{\ell }}^{\mathrm{\infty }}(K(X,A),E)$. We then prove that if every bounded linear functional on $E$ maps a Lipschitz extension of a vectorization to the sum of an additive homomorphism from $K(X,A)$ to $\mathbb{C}$ and a Fourier–Stieltjes transform of a finite complex regular Borel measure on $\widehat{K(X,A)}$, then the associated cocycle generates a synthesizable variety. We extend spectral synthesis to Lipschitz vectorizations of persistence diagrams on separable metric pairs and show whether examples of persistence diagram vectorizations in the literature have Lipschitz extensions with spectral synthesis.