Pointwise Fourier inversion: a classical topic revisited
主 题: Pointwise Fourier inversion: a classical topic revisited
报告人: Prof. Mark Pinsky (美国西北大学)
时 间: 2006-09-15 下午 4:00 - 5:00
地 点: 理科一号楼 1418
Explicit computations show that the spherical Fourier inversion
of the indicator function of a three dimensional ball converges away from
the center of the ball, but is boundedly divergent at the center. This
applies to both the Fourier transform and to the Fourier series expansion
with respect to the Dirichlet eigenfunctions of the Laplace operator.
Beginning with this example, we are able to formulate and prove necessary
and sufficient conditions for the pointwise convergence at a pre-assigned
point for an arbitrary piecewise smooth function on Euclidean space. The
results carry over to the Fourier expansion on spheres and hyperbolic
spaces, as well as to Fourier expansions of radial functions on rank one
symmetric spaces, both of the compact and non-compact type.
另外 Mark Pinsky教授还将于9月11日下午3:00-4:00在中科院应用所703室做报告,
报告题目: Local stochastic differential geometry: feeling the shape of
a manifold with Brownian motion.
摘要: On any complete Riemannian manifold we can define the isotropic
transport process, a one-paraemeter family of Markov processes on the
tangent bundle. When the parameter tends to zero the family of processes
converge weakly to the Brownian motion, a Markov process on the original
manifold whose infinitesimal generator is a constant multiple of the
Laplace-Beltrami operator of the manifold. This observation allows one to
connect Brownian motion with the curvature tensor by means of some simple
partial differential equations involving the Laplace-Beltrami operator. In
particular we can discuss to what extent the mean exit time from small
geodesic balls determines the metric tensor of the Riemannian manifold.
