The principal goal of the present project is to pursue further developments in three major topics in mathematical analysis in the context of spaces of smooth and ultradifferentiable functions satisfying global decay estimates, i.e. Gelfand-Shilov spaces. We shall be concerned with factorization problems, Whitney extension principles, and the problem of moments. It is our intention to use abstract functional analytic tools to systematize and extend several wellestablished methods and obtain new results in each of these branches. The latter may also be considered as the unifying theme in our research proposal. Our specific objectives are to:

• Show new factorization results of Dixmier-Malliavin type and apply them to solve factorization
problems for various convolution modules of smooth and ultradifferentiable functions.
• Prove an extension theorem for smooth jets defined on unbounded closed sets that satisfy
global decay estimates. We shall also consider the related problem of continuous linear extension
in this new setting.
• Investigate multivariate moment problems for both the Schwartz space of rapidly decreasing
smooth functions and Gelfand-Shilov spaces of non-quasianalytic type.