On the geometric deformations of functions in \(L^2[D]\) (Q365988)

The authors consider a class of diffeomorphisms \(\mathcal{W}\) defined on a closed interval \(D\subset\mathbb{R}\), and for a given function \(f\in L^2(D)\) the deformed function \(F_w (f) := f \circ w\), \(w\in \mathcal{W}\). A general relationship between the Schauder coefficients of \(f\), \(w\), and the Schauder coefficients of \(F_w (f)\) is derived. Moreover, the precise relationships between the coefficients of \(f\) and \(F_w (f)\) in the Legendre and Haar basis for linear deformations \(w\) and the Legendre basis for polynomial deformations \(w\) is computed.