A note on the efficient evaluation of a modified Hilbert transformation

In this note we consider an eicient datassparse approximation of a modiied Hilbert type transformation as it is used for the spacestime inite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the irst-order time derivative is symmetric and positive deinite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for datassparse representations and for acceleration of the computations. Numerical results show the eiciency in the approximation of the irst-order time derivative. An eicient realisation of the modiied Hilbert transformation is a basic ingredient when considering general spacestime inite element methods for parabolic evolution equations, and for the stable coupling of inite and boundary element methods in anisotropic Sobolev trace spaces.