Sign regular matrices and variation diminution: Single-vector tests and characterizations, following Schoenberg, Gantmacher-Krein, and Motzkin

Bounded holomorphic functions on the disk have radial limits in almost every direction, as follows from Fatou's theorem.Given a zero-measure set E in the torus T, we study the set of functions such that lim r→1 -f (r w) fails to exist for every w ∈ E (such functions were first constructed by Lusin).We show that the set of Lusin-type functions, for a fixed zero-measure set E, contains algebras of algebraic dimension c (except for the zero function).When the set E is countable, we show also in the several-variable case that the set of Lusin-type functions contains infinite dimensional Banach spaces and, moreover, contains plenty of c-dimensional algebras.We also address the question for functions of infinitely many variables.