Blaschke Condition

If {a_j} subset= D(0,1) (with possible repetitions) satisfies


where D(0,1) is the unit open disk, and no a_j=0, then there is a bounded analytic function on D(0,1) which has zero set consisting precisely of the a_js, counted according to their multiplicities. More specifically, the infinite product


where B_(a_j)(z) is a Blaschke factor and z^_ is the complex conjugate, converges uniformly on compact subsets of D(0,1) to a bounded analytic function B(z).

See also

Blaschke Factor, Blaschke Factorization, Blaschke Product

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Krantz, S. G. "The Blaschke Condition." §9.1.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 118-119, 1999.

Referenced on Wolfram|Alpha

Blaschke Condition

Cite this as:

Weisstein, Eric W. "Blaschke Condition." From MathWorld--A Wolfram Web Resource.

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