Large Analytic Functions

Abstract

Let f be a complex summable function on the circle T. f is called analytic if these equivalent conditions hold: the Fourier coefficients

$${a_n} = {(2\pi )^{ - 1}}\smallint f(x){e^{ - nix}}dx $$

((1))

all vanish for n < 0; and there is a holomorphic function F in the open disc A satisfying

$$ D\left( {{T_1}{T_2}} \right) = \left\{ {x \in {\kern 1pt} D\left( {{T_2}} \right);{T_2}x \in D\left( {{T_1}} \right)} \right\} $$

whose boundary function is f. The space H¹ is, ambiguously, the set of all such functions for F.