The original mathematical importance of positive definite Toeplitz matrices
is in the solution of the classical Carathéodory coefficient
problem, proved independently by Toeplitz and Carathéodory
in 1911. An analytic mapping 
of the unit disk
|z|<1 into the closed right half-plane 
is
said to be a Carathéodory function, or a function
in the class C.
In 1917, Schur gave a constructive proof of the Carathéodory-Toeplitz theorem
via his study of analytic functions bounded in the unit disk [35].
In particular, Schur gave a procedure for
determining when a given function 
maps |z|<1
analytically into 
. Such a function is now called a Schur
function, or a function in the class S.
Schur showed that the function class S can be parameterized by certain
sequences of complex numbers 
, known as
Schur parameters.
They determine a continued fraction representation
of the given Schur function 
.
Schur gave the following procedure for
constructing these parameters.
The initial function 
is a Schur function if and only if
one of the following holds:
for 
;
for 
, 
,
and 
.
is a rational Schur function.
It is natural to view the Schur functions generated by the algorithm as ratios of formal power series,
In terms of these power series, we can restate Schur's algorithm as
If we partially normalize the initial ratio so that 
,
then the constant terms of the denominators are related
by
so that 
is a Schur function if and only if
is a nonnegative, nonincreasing sequence that
is infinite, or finite and terminating with
.
Schur gave explicit determinantal formulas for the numerators and denominators in (2.1), and using the the correspondence
between the algebra of formal power series and the algebra of singly infinite triangular Toeplitz matrices, he concluded that
where 
and 
are the 
leading principal
submatrices of 
and 
. This gives the
following characterization of Schur functions.
The Carathéodory-Toeplitz theorem follows directly from this result.
In particular,
the power series 
of Theorem 1 is in the class C if and only if
is in the class S. By Theorem 2.2, this is true if and only if the matrix
is nonnegative definite for every n.
