Generalized Fourier transform on an arbitrary triangular domain

In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach. We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions. The orthogonality and completeness of the systems are then proved. Some essential convergence properties of the generalized Fourier series are discussed. Error estimates are obtained in Sobolev norms. Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated. This work was supported by the Major Basic Project of China (No. G19990328) and National Natural Science Foundation of China (No. 60173021).