Approximation of additive convolution-like operators

More about the book

Numerical analysis for equations in boundary integral equation methods has been explored in several recent publications. Key topics include various classes of singular integral equations and those related to single and double layer potentials. Establishing a mathematically rigorous foundation and conducting error analysis for approximate solutions to these equations is challenging. This difficulty arises because boundary integral operators typically do not conform to the standard forms of identity plus compact operators or identity plus operators with small norms. As a result, standard theories for the numerical analysis of Fredholm integral equations of the second kind are not applicable. Over the past 15 years, the Banach algebra technique has emerged as a powerful tool for analyzing stability in relevant approximation methods. This approach begins with the recognition that the stability problem is an invertibility issue within a specific Banach or C*-algebra. Often, this algebra is complex, necessitating the identification of relevant subalgebras to effectively apply local principles and representation theory. Additionally, various applications frequently involve continuous operators acting on complex Banach spaces that are additive rather than linear, meaning they satisfy A(x+y)= Ax+Ay for all x, y in the given space. It is evident that continuous additive operators are R-linear.