Tony Hill

Shamir-Duduchava Factorization of Elliptic Symbols

abstract:
This paper considers the factorization of elliptic symbols which can be represented by matrix-valued functions. Our starting point is a Fundamental Factorization Theorem, due to Budjanu and Gohberg [2]. We critically examine the work of Shamir [15] together with some corrections and improvements as proposed by Duduchava [6]. As an integral part of this work, we give a new and detailed proof that certain sub-algebras of the Wiener algebra on the real line satisfy a sufficient condition for a right standard factorization. Moreover, assuming only the Fundamental Factorization Theorem, we provide a complete proof of an important result from Shargorodsky [16], on the factorization of an elliptic homogeneous matrix-valued function, useful in the context of the inversion of elliptic systems of multidimensional singular integral operators in a half-space.

Mathematics Subject Classification: 65N80, 65N12, 65N35

Key words and phrases: Fundamental solution method, adaptive cross approximation, collocation, condition numbers