An optimal spline-method for solving operator equations
The authors consider a spline method for approximately solving operator equations of the form \(Gx+Tx=y\), where \(G\) and \(T\) are linear operators between spaces \(X\) and \(Y\) which are any of \(2\pi\)-periodic function spaces \(L_2\) and \(W_2^1\), \(G\) and \(G+T\) are continuously invertible, and \(T\) is completely continuous. The approximate solution, by the Galerkin method, is sought using the pair \((X_n, Y_n)\) of spaces, where \(X_n\) is the space of all \(2\pi\)-periodic first order splines with \(2n+1\) equidistant nodes between \(-\pi\) and \(\pi\), and \(Y_n\) is a space of trigonometric polynomials. Overall assumptions in the paper are that the orthogonal projection \(\Phi_n\) from \(Y\) onto \(Y_n\) satisfies \(\Phi_n G=G\Phi_n\) on \(X_n\) and \(Gx+Tx=y\) is uniquely solvable for all \(y\in Y\). The authors establish unimprovable, in certain sense, error estimates and prove the optimality by order of the method. The applicability of the method obtained in the general setting to both regular and singular integral and integral-differential equation, is also established. Reviewer's remark: A good contribution in the area of numerical functional analysis.
scientific article; zbMATH DE number 1960450