Reconstruction of piecewise smooth functions from non-uniform grid point data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.