Fertigungsverfahren 2
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The Derivative of a Blaschke Product
7.1 Frostman’s Theorem, Local Version Let (zn )n≥1 be a Blaschke sequence and let B(z) =
∞ |zn | zn − z . z 1 − z¯n z n=1 n
For a fixed point z ∈ D, we know that the partial products BN (z) =
N |zn | zn − z z 1 − z¯n z n=1 n
converge to B(z). Indeed, more is true. The convergence is uniform on each compact subset of an open set Ω which contains the open unit disc. Naturally, we may ask a similar question for the behavior of BN (eiθ ) as N −→ ∞. Does the limit (7.1) lim BN (eiθ ) N →∞
necessarily exist? This question is more interesting to consider when eiθ is a point of accumulation of the zeros of B. Since otherwise, we know that B is in fact analytic at this point and the convergence of BN is even uniform on a small disc around eiθ . In the general case, the answer is not affirmative. For example, if all the zeros are on the interval [0, 1), then BN (z) =
N rn − z , 1 − rn z n=1
which shows that BN (1) = (−1)N , and thus limN →∞ BN (1) does not exist. Let us look at the boundary values of a Blaschke product from a different point of view. According to Fatou’s theorem, we know that the radial limits 99 J. Mashreghi, Derivatives of Inner Functions, Fields Institute Monographs 31, DOI 10.1007/978-1-4614-5611-7 7, © Springer Science+Business Media New York 2013
100
7 The Derivative of a Blaschke Product
limr→1 B(reiθ ) exist for almost all eiθ ∈ T. But, first, the exceptional set of measure zero on which the radial limits fail to exist depends on B. Second, even for a fixed B, we do not have any intrinsic description of this set, except of course the fact that its Lebesgue measure is zero. Hence, given B and a fixed point eiθ ∈ T, Fatou’s result does not tell us anything about the existence of the radial limit at this point. The following theorem is a significant result which partially answers this question. This theorem gives a sufficient condition under which a Blaschke product has a unimodular radial limit at a given point on T. We naturally expect this value to be the limit of partial product at the boundary point. For each Blaschke sequence (zn )n≥1 , any subsequence (znk )k≥1 (finite or infinite) of (zn )n≥1 is by itself a Blaschke sequence and thus the Blaschke product |zn | zn − z k k B(z; {znk }) = znk 1 − z¯nk z k
is well-defined. The Blaschke products B(z; {znk }) are called the subproducts of B. Note that B is a subproduct of itself corresponding to the whole sequence (zn )n≥1 . For a more complete version of the following result, see Theorem 7.3. Theorem 7.1 (Frostman [23]) Let (zn )n≥1 be a Blaschke sequence, and fix eiθ ∈ T. Suppose that ∞ 1 − |zn | < ∞. iθ − z | |e n n=1
(7.2)
Then each subproduct B(z; {znk }) is convergent at the boundary point z = eiθ and, moreover, lim B(reiθ ; {znk }) = B(eiθ ; {znk }) = lim BN (eiθ ; {znk }).
r→1
N →∞
Conversely, if all subproducts of B have unimodular radial limits at eiθ then (7.2) holds. Proof. It is enough to show that the partial products BN (z) =
N |zn | zn − z , z 1 − z¯n z n=1 n
(z ∈ D),
are uniforml
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