Spline fitting discontinuous functions given just a few Fourier coefficients
Let \(f\) be a given \(p\)-periodic, piecewise \(C^1\)-function. The author uses \(p\)-periodic (quadratic) splines with equidistant nodes to approximate \(f\) when few Fourier coefficients of \(f\) and the locations of the jump discontinuities of \(f\) and \(f'\) are known. It is assumed that the locations of these discontinuities are special spline nodes. The results deal with the orthogonal projection \(P\) of the considered spline space onto the space of trigonometric polynomials. The reconstruction of \(f\) is based on inverting \(P\). It is shown that \(P\) is invertible and its inverse has a normal close to 1. A numerical example is given.
scientific article; zbMATH DE number 770302
scientific article