A regular analytic function of a complex variable
,
defined in the unit disc
by the finite or infinite product
where
is a non-negative integer, and
,
is a sequence of points
such that the product on the right-hand side of
(*)
converges (the convergence
condition is necessary only for an infinite product).
The Blaschke product was introduced by
W. Blaschke
[1],
who proved the following theorem: A sequence
of points
defines a function of the type
(*)
if and only if the series
is convergent. Each factor of the form
called the
Blaschke factor
for
,
defines a univalent conformal mapping of the disc
onto itself, which takes
to zero, with the normalization
.
The factors of the form
may be interpreted as Blaschke factors which correspond to the zero
with the normalization
.
The definition of Blaschke factors and Blaschke products is readily carried
over to a disc of arbitrary radius, and also
to an arbitrary simply-connected domain, which is conformally equivalent to a disc.
The sequence
(with
zeros), which is usually written out in non-decreasing order of
,
is the sequence of all zeros of the Blaschke product
(*)
(each
zero is written down as many times as its multiplicity). Thus,
Blaschke's theorem describes the sequences of zeros of all possible Blaschke
products. The product
(*)
can be regarded as
the simplest bounded holomorphic function in the disc
with a prescribed sequence of zeros. It converges absolutely and uniformly inside
,
represents a bounded holomorphic function
in
,
with angular boundary values of modulus one almost everywhere on
.
A necessary and sufficient condition for a bounded holomorphic function
in
,
,
to be a Blaschke product, is
 |
Blaschke products may be used to give a product representation
of important classes of holomorphic functions in the unit disc
.
Thus, a proof was given for the following
theorem of Blaschke:
A sequence
of points in the disc
is the sequence of all zeros of some bounded holomorphic function
,
,
in
if and only if the series
is convergent. Moreover,
can be represented as a product
where
is the Blaschke product constructed with the zeros
of the function
,
while
is a zero-free holomorphic function in
,
,
which can be represented relatively simply using an
integral formula. Apart from the bounded functions, similar
product representations may be constructed for functions of bounded form and for
Hardy classes
[2]–
[4]
(cf.
Function of bounded form).
The above theory was considerably generalized by
M.M. Dzhrbashyan
[5],
[6],
who constructed infinite products of a more general nature, which are
suitable for the factorization of much larger classes of meromorphic functions. A
solution was also found for the problem of constructing analogues
of Blaschke products and Blaschke's theorem for doubly-connected domains
[7]
and, in general, finitely-connected
[8]
domains. The solution of the problem of constructing
suitable analogues of the Blaschke product for holomorphic functions of
several complex variables is rendered very difficult by the fact
that the zeros of such functions cannot be isolated.