Nearly orthonormal vectors and approximation of the pluricomplex Green function
Volume 134 / 2025
Annales Polonici Mathematici 134 (2025), 19-33
MSC: Primary 32U35; Secondary 15A60, 32-08, 15A23, 42C05, 41A10
DOI: 10.4064/ap241015-19-5
Published online: 24 June 2025
Abstract
The pluricomplex Green function of a compact set can be approximated by Bergman functions defined via orthogonal polynomial bases. We show that nearly orthogonal bases can be used instead. Studying this we look into a more general problem. Commonly used orthonormalization procedures take the Gram matrix of the vectors in question as the input when transforming those vectors into an orthonormal set. We investigate how near to orthonormal the outcome is if the Gram matrix is inexact e.g. due to numerical errors. Our approach is based on analysis of Fréchet derivatives of linear algebraic operations and can be applied to other orthonormalization procedures.
