Algebraic structures in sets of functions with singular properties.

CARANDO, DANIEL

Conferencia; Frontiers of Operator Theory; 2021

CIRM

There is an increasing interest in finding nice structures in sets of strange objects and, particularly, strange functions. In a pioneer work, Guraryi showed in 1966 that the set of continuous functions that are nowhere differentiable contains an infinite dimensional vector space (with the exception of 0). Since then and specially in the last 20 years, many authors joined the search of nice structures (infinite dimensional vector or Banach spaces, algebras, etc.) in sets of strange functions (nowhere monotone differentiable functions, continuous functions with divergent series at many points, Peano-like functions, Dirichtlet series with bad convergence properties, etc.). In this talk we will review some of these examples and comment on some recent research on bounded holomorphic functions with wild behaviour at the boundary and Dirichlet series with maximal Bohr strip. These last are part of a collaboration with Thiago Alves and Leonardo Brito (U. F. do Amazonas, Brazil)