Carathéodory-Toeplitz extension problem

Let $ a _ {0} \dots a _ {p} $ be given complex numbers. The Carathéodory–Toeplitz extension problem is to find (if possible) a function $ g $, analytic on the open unit disc $ | z | < 1 $( cf. also Analytic function), such that

a) $ g ( z ) = a _ {0} + 2a _ {1} z + \dots + 2a _ {p} z ^ {p} + O ( z ^ {p + 1 } ) $, $ | z | < 1 $;

b) $ { \mathop{\rm Re} } g ( z ) \geq 0 $ for all $ | z | < 1 $. Put $ a _ {- j } = {\overline{ {a _ {j} }}\; } $ for $ j = 1 \dots p $. The problem is solvable if and only if the Toeplitz matrix

$$ \tag{a1 } \Gamma = \left ( \begin{array}{cccc} a _ {0} &a _ {- 1 } &\dots &a _ {- p } \\ a _ {1} &a _ {0} &\dots &a _ {- p + 1 } \\ \vdots &\vdots &\vdots &\vdots \\ a _ {p} &a _ {p -1 } &\dots &a _ {0} \\ \end{array} \right ) $$

is positive semi-definite, and its solution is unique if and only if, in addition, $ \Gamma $ is singular (cf. also Degenerate matrix). The Carathéodory–Toeplitz extension problem can be restated as a Carathéodory–Schur extension problem. The Levinson algorithm from filtering theory provides a recursive method to compute the solutions of the problem. For these and related results, see [a1], Chapt. 2.

Instead of functions $ g $ satisfying a) and b), one may also seek functions $ f $, $ f ( \zeta ) = \sum _ {k = - \infty } ^ \infty f _ {k} \zeta ^ {k} $, in the Wiener algebra $ {\mathcal W} $ with the property $ f ( \zeta ) \geq 0 $ for every $ \zeta \in \mathbf T $. (The Wiener algebra is defined as the Banach algebra of complex-valued functions on the unit disc having a Fourier series

$$ f ( z ) = \sum _ {n = - \infty } ^ \infty {a _ {n} z ^ {n} } , \quad \sum _ {n = - \infty } ^ \infty \left | {a _ {n} } \right | < \infty, $$

using pointwise multiplication. The phrase "Wiener algebra" is also used for $ L _ {1} ( \mathbf R ) $ with convolution as multiplication. There are also weighted versions; cf. [a2].)

In this case, $ g ( \zeta ) = f _ {0} + 2 \sum _ {k = 0 } ^ \infty f _ {k} \zeta ^ {k} $ satisfies conditions a) and b). The Wiener algebra version of the problem is of particular interest if the solution $ f $ is required to be strictly positive on the unit circle $ \mathbf T $. The latter version of the problem is solvable if and only if the Toeplitz matrix $ \Gamma $ in (a1) is positive definite, and in that case there are infinitely many solutions $ f $, given by

$$ \tag{a2 } f ( \zeta ) = { \frac{1 - \left | {h ( \zeta ) } \right | ^ {2} }{\left | {u ( \zeta ) + h ( \zeta ) v ( \zeta ) } \right | ^ {2} } } . $$

Here, $ h $ is an arbitrary function in the Wiener algebra $ {\mathcal W} $ with $ | {h ( \zeta ) } | < 1 $ for every $ \zeta \in \mathbf T $, and the functions $ u $ and $ v $ are uniquely determined by the data in the following way:

$$ u ( \zeta ) = ( x _ {0} + \zeta x _ {1} + \dots + \zeta ^ {p} x _ {p} ) x _ {0} ^ {- {1 / 2 } } , $$

$$ v ( \zeta ) = ( y _ {0} + \zeta ^ {-1 } y _ {-1 } + \dots + \zeta ^ {- p } y _ {- p } ) y _ {0} ^ {- { \frac{1}{2} } } , $$

where

$$ \left ( \begin{array}{c} x _ {0} \\ x _ {1} \\ \vdots \\ x _ {p} \\ \end{array} \right ) = \Gamma ^ {- 1 } \left ( \begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \\ \end{array} \right ) , \quad \left ( \begin{array}{c} y _ {- p } \\ \vdots \\ y _ {- 1 } \\ y _ {0} \\ \end{array} \right ) = \Gamma ^ {- 1 } \left ( \begin{array}{c} y _ {0} \\ \vdots \\ 0 \\ 1 \\ \end{array} \right ) . $$

The central solution $ f _ {\textrm{ cen } } ( \zeta ) = | {u ( \zeta ) } | ^ {-2 } $, which appears when the free parameter $ h $ in (a2) is set to zero, is the unique solution $ f $ with the additional property that the $ j $ th Fourier coefficient of $ f ^ {- 1 } $ is equal to zero for $ | j | > p $, and for this reason the central solution is also referred to as the band extension. The central solution $ f _ {\textrm{ cen } } $ is also the unique solution $ f $ that maximizes the entropy integral

$$ { \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ \pi { { \mathop{\rm log} } f ( e ^ {it } ) } {dt } . $$

Proofs of the above results may derived by applying the band method (see [a2], Sect. XXXV.3), which is a general scheme for dealing with a variety of positive and contractive (operator) extension problems from a unified point of view. (The word "band" refers to a decomposition of an algebra with involution, reminiscent of the use of bands as in the theory of decomposition or Riesz spaces (cf. Riesz space). It refers, in fact, to a "band pattern" , i.e. a band in a matrix $ \{ {( i,j ) } : {| {i - j } | \leq m } \} $, cf. also Partially specified matrices, completion of.)

The Carathéodory–Toeplitz extension problem has natural generalizations for matrix- and operator-valued functions. The problem also has a continuous analogue (with the role of the open unit disc being replaced by the upper half-plane) and non-stationary versions for finite or infinite operator matrices.

References

[a1]	C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a2]	I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators" , II , Birkhäuser (1993)