Boundary Values of Bounded Holomorphic Functions of Several Variables

A typical result (consequence of Theorem 1) is that if such a function tends to a limit X as x—»x° from inside an open cone with vertex at x°, then it tends "on the average" to X as x—>x° from inside any open cone with vertex at x°. In particular, if such a function tends to limits in each of two open cones with a common vertex, these limits must be equal; for n = l (when the cones are half-lines), this is a classical theorem of Pringsheim and Lindelof. Theorem 2 is a Tauberian theorem for H boundary values; specialized to one variable it yields (among other things) a new proof of Lindelöfs theorem that a bounded analytic function which tends to a limit radially does so also within an angle, as well as an apparently new relation between the average behaviour of an JÏ boundary function to the left, and that to the right, of a given point. Theorem 3 shows that a much stronger localization of uniform convergence for, say, Fejér means, is valid for if boundary functions than for bounded measurable functions generally; for example, if the restriction of an i? boundary function to a closed ball in R is continuous, the Fejér means converge to it uniformly on the closed ball, not merely on subballs of smaller radius.