Kergin Approximation in Banach Spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges

Splitting the Curvature of the Determinant Line Bundle

It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvature form is the natural differential representative which satisifies the same splitting principle as the Chern class of the determinant line bundle.Comment: To appear in Proc. Am. Math. So

Zeta Forms and the Local Family Index Theorem

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formula for the graded Chern character. The zeta-determinant form leads to the definition of the graded zeta-Chern class form. For a family of compatible Dirac operators D with index bundle Ind(D) we prove a transgression formula leading to a local density representing the Chern class c(Ind(D))in terms of the A-hat genus and twisted Chern character. Globally the meromorphically continued zeta form and zeta determinant form exist only at the level of K-theory as characteristic class maps K(B)->H*(B).Comment: 34 page

A Laurent expansion for regularised integrals of holomorphic symbols

For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.Comment: Expanded explanations and application