Detection of edges from nonuniform Fourier data (Q657870)

Let \(f\) be a periodic, piecewise-smooth function \(f:[-\pi,\pi]\to\mathbb R,\) \([f]\) be the associated jump function. Detection of edges is a method to find the jump function from given data. Here the data are nonuniform Fourier coefficients \(\hat f(\lambda_n)=\langle f(x),\exp(i\lambda_nx)\rangle_{n\in\mathbb Z}\) when the family \(\{\exp(i\lambda_nx)\}_{n\in\mathbb Z}\) is a frame for \(L^2[-\pi,\pi].\) The authors introduce the concentration factor method to design a function \(\sigma:\mathbb R\to\mathbb C\) such that \[ T_N^{\sigma}(f)=\sum_{n=-N}^N\sigma(\lambda_n)\hat f(\lambda_n)S_N^{-1}e^{i\lambda_nx}\to[f](x) \] as \(N\to\infty,\) where \(S_N^{-1}\) is inverse to the finite frame operator \[ S_Nf=\sum_{n=-N}^N\hat f(\lambda_n)e^{i\lambda_nx}. \] Numerical examples are provided.