Linear connections for reproducing kernels on vector bundles

Vector Bundles, Connections and Curvature

Definition 1. Let M be a differentiable manifold. A C∞ complex vector bundle consists of a family {Ex}x∈M of complex vector spaces parametrized by M , together with a C∞ manifold structure of E = ∪x∈MEx such that 1. The projection map π : E →M taking Ex to x is C∞, and 2. For every x0 ∈M , there exists an open set U inM containing x0 and a diffeomorphism φU : π −1(U)→ U × C taking a vector spac...

Some linear Jacobi structures on vector bundles

We study Jacobi structures on the dual bundle A to a vector bundle A such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic function. We prove that a Lie algebroid structure on A and a 1-cocycle φ ∈ Γ(A) induce a Jacobi structure on A satisfying the above conditions. Moreover, we show that this correspo...

Normal frames for linear connections in vector bundles and the equivalence principle in classical gauge theories

Frames normal for linear connections in vector bundles are defined and studied. In particular, such frames exist at every fixed point and/or along injective path. Inertial frames for gauge fields are introduced and on this ground the principle of equivalence for (system of) gauge fields is formulated. Bozhidar Z. Iliev: Normal frames, connections, gauge fields 1

Reproducing kernels , Bergman kernels , Poisson kernels

Favard theorem for reproducing kernels

Consider for n = 0, 1, . . . the nested spaces Ln of rational functions of degree n at most with given poles 1/αi, |αi| < 1, i = 1, . . . , n. Let L = ∪0 Ln. Given a finite positive measure μ on the unit circle, we associate with it an inner product on L by 〈f, g〉 = ∫ fgdμ. Suppose kn(z, w) is the reproducing kernel for Ln, i.e., 〈f(z), kn(z, w)〉 = f(w), for all f ∈ Ln, |w| < 1, then it is know...