Colloquium April 24

Date: Friday, April 24

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Vladimir Bolotnikov of College of William and Mary

Title: Boundary angular derivatives of analytic self-maps of the unit disk

Abstract: Let S denote the class of functions analytic and bounded by one in modulus on the open unit disk. These functions were characterized in terms of their Taylor coefficients at the origin by I. Schur as follows: the function s(z) = ∑k=0∞skzk belongs to S if and only if the lower triangular toeplitz matrix

  Sn = (
 s0 0 ... 0
 s1 s0 $ \ddots$ :
 : $ \ddots$ $ \ddots$ 0
 sn ... s1 s0
)

is contractive for every integer n≥ 0. By a conformal change in variable, a similar result can be established for an arbitrary point ζ∈D (rather than the origin).

In the talk, we will discuss a boundary analog of this result which is more interpolation nature: given a sequence {ck}k=0N of complex numbers and given a point t0 on the unit circle T, we will present necessary and sufficient conditions for the existence of a function s∈S which admits the asymptotic expansion

s(z) = c0 + c1(z - t0) + ... + cN(z - t0)N + o(| z - t0|N)

as z tends to t0 nontangentially. The latter is equivalent to the existence of nontangential boundary limits sk(t0) : = limz-> t0s(k)(z)/k! and equalities sk(t0) = ck for k = 0,..., N. The case where N = ∞ will also be considered.

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