hille's main work was on integral equations, differential equations, special functions, dirichlet series and fourier series. later in his career his interests turned more towards functional analysis. his name persists among others in the hille–yosida theorem. hille was a member of the london mathematical society and the circolo matematico di palermo. hille served as president of the american mathematical society (1937–38) and was the society's colloquium lecturer in 1944. he received many honours including election to the united states national academy of sciences (1953) and the swedish royal academy of sciences. he was awarded by sweden with the order of the polar star.