On the behavior of disk algebra bases with applications

The present paper was motivated by an article [H. Akcay, On the existence of a disk algebra basis, Signal Processing 80 (2000) 903-907] on a basis in the disk algebra. Such bases play a central role for the representation of linear systems. In this article it was shown that the Lebesgue constant of a certain set of rational orthogonal functions in the disk algebra diverges. The present paper provides a generalization of this result. It shows that for any arbitrary complete orthonormal set of functions in the disk algebra the Lebesgue constant diverges. However, even if the Lebesgue constant diverges the orthonormal set may still be a disk algebra basis. Moreover, the paper discusses some implications of the divergence result with regard to the robustness of basis representations.