Mathematical Sciences: Banach Algebras and Their Applications

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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.