View source for Carathéodory-Toeplitz extension problem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100801.png" /> be given complex numbers. The Carathéodory–Toeplitz extension problem is to find (if possible) a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100802.png" />, analytic on the open unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100803.png" /> (cf. also [[Analytic function|Analytic function]]), such that

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100805.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100806.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100807.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100808.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c1100809.png" />. The problem is solvable if and only if the [[Toeplitz matrix|Toeplitz matrix]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

is positive semi-definite, and its solution is unique if and only if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008011.png" /> is singular (cf. also [[Degenerate matrix|Degenerate matrix]]). The Carathéodory–Toeplitz extension problem can be restated as a [[Carathéodory–Schur extension problem|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 [[#References|[a1]]], Chapt. 2.

Instead of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008012.png" /> satisfying a) and b), one may also seek functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008014.png" />, in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008015.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008016.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008017.png" />. (The Wiener algebra is defined as the [[Banach algebra|Banach algebra]] of complex-valued functions on the unit disc having a [[Fourier series|Fourier series]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008018.png" /></td> </tr></table>

using pointwise multiplication. The phrase  "Wiener algebra"  is also used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008019.png" /> with convolution as multiplication. There are also weighted versions; cf. [[#References|[a2]]].)

In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008020.png" /> satisfies conditions a) and b). The Wiener algebra version of the problem is of particular interest if the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008021.png" /> is required to be strictly positive on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008022.png" />. The latter version of the problem is solvable if and only if the Toeplitz matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008023.png" /> in (a1) is positive definite, and in that case there are infinitely many solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008024.png" />, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008026.png" /> is an arbitrary function in the Wiener algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008027.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008028.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008029.png" />, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008031.png" /> are uniquely determined by the data in the following way:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008032.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008033.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008034.png" /></td> </tr></table>

The central solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008035.png" />, which appears when the free parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008036.png" /> in (a2) is set to zero, is the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008037.png" /> with the additional property that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008038.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008039.png" /> is equal to zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008040.png" />, and for this reason the central solution is also referred to as the band extension. The central solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008041.png" /> is also the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008042.png" /> that maximizes the entropy integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008043.png" /></td> </tr></table>

Proofs of the above results may derived by applying the [[Band method|band method]] (see [[#References|[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|decomposition]] or Riesz spaces (cf. [[Riesz space|Riesz space]]). It refers, in fact, to a  "band pattern" , i.e. a band in a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110080/c11008044.png" />, cf. also [[Partially specified matrices, completion of|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Foias,   A.E. Frazho,   "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Gohberg,   S. Goldberg,   M.A. Kaashoek,   "Classes of linear operators" , '''II''' , Birkhäuser  (1993)</TD></TR></table>