complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, the Blaschke product is a bounded

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...

in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s :''a''₀, ''a''₁, ... inside the

unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...

, with the property that the magnitude of the function is constant along the boundary of the disc. Blaschke-product

Blaschke products were introduced by . They are related to

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...

Definition

A sequence of points

(a_n)

inside the unit disk is said to satisfy the Blaschke condition when :

\sum_n (1-, a_n, ) <\infty.

Given a sequence obeying the Blaschke condition, the Blaschke product is defined as :

B(z)=\prod_n B(a_n,z)

with factors :

B(a,z)=\frac\;\frac

provided ''a'' ≠ 0. Here

\overline

is the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

of ''a''. When ''a'' = 0 take ''B''(''0'',''z'') = ''z''. The Blaschke product ''B''(''z'') defines a function analytic in the open unit disc, and zero exactly at the ''a''_''n'' (with

multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using multi ...

counted): furthermore it is in the Hardy class

H^\infty

.Conway (1996) 274 The sequence of ''a''_''n'' satisfying the convergence criterion above is sometimes called a Blaschke sequence.

Szegő theorem

A theorem of

Gábor Szegő Gábor Szegő () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and T ...

states that if ''f'' is in

H^1

, the

with integrable norm, and if ''f'' is not identically zero, then the zeroes of ''f'' (certainly countable in number) satisfy the Blaschke condition.

Finite Blaschke products

Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that ''f'' is an analytic function on the open unit disc such that ''f'' can be extended to a continuous function on the closed unit disc :

\overline= \

that maps the unit circle to itself. Then "f" is equal to a finite Blaschke product :

B(z)=\zeta\prod_^n\left(\right)^

where ''ζ'' lies on the unit circle and ''m_i'' is the

of the zero ''a_i'', , ''a''_''i'', < 1. In particular, if ''f'' satisfies the condition above and has no zeros inside the unit circle, then ''f'' is constant (this fact is also a consequence of the

maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

for

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

s, applied to the harmonic function log(, ''f''(''z''), )).

References

* W. Blaschke,
''Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen''
Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig, 67 (1915) pp. 194–200 * Peter Colwell, ''Blaschke Products — Bounded Analytic Functions'' (1985), University of Michigan Press, Ann Arbor, 140 pages. * *{{eom, title=Blaschke product, first=P.M., last= Tamrazov Complex analysis

Definition

Szegő theorem

Finite Blaschke products

See also

References