On the geometric deformations of functions in $L^2[D]$

Abstract

We derive a formal relationship between the coefficients of a function expanded in either the Legendre basis or Haar wavelet basis, before and after a polynomial deformation of the function’s domain. We compute the relationship of coefficients explicitly in three cases: linear deformation with Haar basis, linear deformation with Legendre basis, and polynomial deformation with Legendre basis.