Local scales on curves and surfaces
arXiv:1009.4145 · doi:10.1016/j.acha.2012.03.001
Abstract
In this paper, we extend our previous work on the study of local scales of a function to studying local scales on curves and surfaces. In the case of a function f, the local scales of f at x is computed by measuring the deviation of f from a linear function near x at different scales t's. In the case of a d-dimensional surface E, the analogy is to measure the deviation of E from a d-plane near x on E at various scale t's. We then apply the theory of singular integral operators on E to show useful properties of local scales. We will also show that the defined local scales are consistent in the sense that the number of local scales are invariant under dilation.