2016/01/13 by Andrew Ursitti, Ursitti, Andrew
Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #advanced mathematical theories #math.CA
paper · pdf · doi:10.48550/arxiv.1601.03135
12 pages
arxiv created 2016/01/13 · arxiv updated 2016/01/14
It is shown that if γ is a path of finite p variation (1≤ p< 2) in a euclidean vector space and f,g,h are Lipschitz functions on the trace of γ then s↦ F(s)=∫γfsg dh defines an entire holomorphic function provided the convex hull of the image of f does not contain zero. If in addition |log z|≤ log 2 on the convex hull of the image of f then for any s∈ C, F(s) can be computed from the nonnegative integer values \F(k)\k∈ N. If in addition to these hypotheses each of f,g,h is a polynomial, then the values F(k) are computable directly from the signature of γ thus all values of F(s) are computable from the signature. As a special case the winding number of a closed path γ around an affine submanifold of codimension two is computed from finitely many terms of the signature provided certain estimates are satisfied.