Rational orthogonal systems on the plane

Abstract. In this paper we construct rational orthogonal and biorthogonal systems on the unit disc with respect to the area measure. These orthogonal systems can be considered as the planar version of the Malmquist–Takenaka systems defined on the unit circle. Similarly to the one dimensional case the starting functions are elementary rational functions and the Gram–Schmidt orthogonalization process is applied to them. Unfortunately, unlike the Malmquist–Takenaka systems there exist no explicit representation for the generated orthogonal system in the two dimensional case. We show that if the poles of the starting elementary rational functions have multiplicity 2 then the orthogonalization process reduces to an interpolation problem. Moreover the orthogonal projection can be directly calculated by means of the values of the function taken only at the so called inverse poles. At the end of the paper we present results of numerical tests that were performed for several test functions and parameters.