Sparse solution to an underdetermined linear integral equation is the central problem for a broad range of applications - scattering, sensing, imaging, machine learning, signal and image processing, data analysis and compression, model reduction, optimal control and design. We will introduce a weak formulation of the problem and construct its sparse solution by a nonlinear process - the design of a Gaussian quadrature for the kernel of the integral equation. We will present a systematic method to solve the resulting quadrature problem by eigen decomposition.