Mathematical Sciences: Banach Algebras and Their Applications

Grabiner教授将继续研究银色微积分的解析扩张，加权卷积代数，以及幂函数级数及其类似的Banach代数。罗塔在多项式的代数环境中研究的Umblar和Sheffer移位，将在整函数的空间中进行研究。关于卷积代数的工作将集中在同态、卷积半群和闭理想上。对于幂函数级数的根Banach代数，Grabiner将在不假定权增长正则性的情况下研究此类代数的结构，并尝试将关于权的计算可行条件与代数的结构联系起来。这个项目中要进行的数学研究的一个相当具体的背景是幂级数，它可以被认为是无穷次的多项式。许多常见的函数，如正弦和exp，被称为整函数，它们具有在整个复平面上有效的级数表示，通常将这些函数视为一种扩展的多项式是合理的。在另一个方向上，可以通过对系数的极限行为施加适当的约束来形成幂函数级数的Banach代数。可能发生的情况是，计算代数中所有级数的唯一方法是取常数项。在这种情况下，有一个激进的Banach代数，一个非常奇怪的物体，它的理解是Grabiner教授研究的目标之一。

Professor Grabiner will continue his research on analytic extensions of the umbral calculus, on weighted convolution algebras, and on Banach algebras of power series and their analogues. Umbral and Sheffer shifts, investigated by Rota in the algebraic setting of polynomials, will be looked at for spaces of entire functions. The work on convolution algebras will concentrate on homomorphisms, convolution semigroups, and closed ideals. For radical Banach algebras of power series, Grabiner will study the structure of such algebras without assumptions of regularity of growth of the weights, and try to relate computationally feasible conditions on the weights to the structure of the algebras. One fairly concrete setting for the mathematical research to be undertaken in this project is that of power series, which may be thought of as polynomials of infinite degree. Many familiar functions such as sine and exp, which are called entire, have power series representations valid throughout the complex plane, and it is often reasonable to treat such functions as a species of extended polynomial. In another direction, one can form Banach algebras of power series by placing an appropriate constraint on the limit behavior of the coefficients. It may happen that the only way to evaluate all of the series in the algebra is by taking the constant term. In this case, one has a radical Banach algebra, a very exotic object whose understanding is one of the goals of Professor Grabiner's research.