Generalized Fourier transform on an arbitrary triangular domain
The authors consider generalized Fourier series on an arbitrary triangle \(\Omega\). Let \({\mathcal L}\) be a special second-order elliptic differential operator dependent on \(\Omega\). Then the eigenvalue problem \({\mathcal L}u=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) has generalized sine functions (TSin) as eigenfunctions. The corresponding eigenvalue problem with homogeneous Neumann conditions has generalized cosine functions (TCos) as eigenfunctions. The orthogonality and completeness of both systems TSin and TCos are proved. Error estimates for the truncated generalized Fourier series are given in Sobolev norms. Finally, the convergence of these generalized Fourier series is demonstrated for some bivariate polynomials of lower degree.
scientific article; zbMATH DE number 2140541
scientific article