View source for Carathéodory interpolation

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100601.png" /> be a polynomial of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100602.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100603.png" /> be the Hardy space (cf. [[Hardy spaces|Hardy spaces]]) formed by the set of all analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100604.png" /> in the open unit disc whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100605.png" />-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100606.png" /> is finite. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100607.png" /> is an interpolant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100608.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c1100609.png" /> is a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006011.png" /> are the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006012.png" /> Taylor coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006014.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006015.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006017.png" /> (cf. also [[Taylor series|Taylor series]]).

The Carathéodory interpolation problem is to find the set of all interpolants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006019.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006020.png" />. Of course, this set can be empty. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006021.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006022.png" /> lower triangular [[Toeplitz matrix|Toeplitz matrix]] defined 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/c110060/c11006023.png" /></td> </tr></table>

Then there exists a solution of the Carathéodory interpolation problem if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006024.png" />. Moreover, there exists a unique solution of the Carathéodory interpolation problem if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006025.png" />. In this case the unique interpolant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006027.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006028.png" /> is a [[Blaschke product|Blaschke product]].

The Schur method for solving the Carathéodory interpolation problem [[#References|[a1]]], [[#References|[a2]]] is based on the Möbius transformation (cf. [[Fractional-linear mapping|Fractional-linear mapping]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006029.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006030.png" />. By recursively unravelling this Möbius transformation, I. Schur discovered that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006031.png" /> uniquely determines and is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006033.png" /> forms a sequence of complex numbers now referred to as the Schur numbers, or reflection coefficients, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006034.png" />. The Schur algorithm is a computational procedure, discovered by Schur, which computes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006035.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006036.png" />, or vice versa, in about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006037.png" /> computations. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006038.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006039.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006040.png" />. In this case the set of all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006041.png" /> of the Carathéodory interpolation problem is 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/c110060/c11006042.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006043.png" /> is an arbitrary function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006044.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006045.png" />. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006046.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006050.png" />. In this case

<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/c110060/c11006051.png" /></td> </tr></table>

is the unique solution of the Carathéodory interpolation problem. If the reflection coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006052.png" /> do not satisfy any one of the previous conditions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006053.png" /> and there is no solution of the Carathéodory interpolation problem; see [[#References|[a4]]] for further details.

The Schur numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006054.png" /> are precisely the reflection coefficients which naturally occur in certain inverse scattering problems for layered media in geophysics. Therefore, the Schur algorithm plays an important role in geophysics and marine seismology, see [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]. Finally, it has been noted that the Schur algorithm can also be used to obtain a Routh or Jury test for the open unit disc, that is, the Schur algorithm can be used to determine whether or not a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006055.png" /> has all its roots inside the open unit disc without computing the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110060/c11006056.png" />; see [[#References|[a2]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Schur,   "On power series which are bounded in the interior of the unit circle"  I. Gohberg (ed.) , ''Methods in Operator Theory and Signal Processing'' , ''Operator Theory: Advances and Applications'' , '''18'''  (1986)  pp. 31–59  (Original (in German): J. Reine Angew. Math. 147 (1917), 205–232)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Schur,   "On power series which are bounded in the interior of the unit circle. II"  I. Gohberg (ed.) , ''Methods in Operator Theory and Signal Processing'' , ''Operator Theory: Advances and Applications'' , '''18'''  (1986)  pp. 68–88  (Original (in German): J. Reine Angew. Math. 184 (1918), 122–145)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Claerbout,   "Fundamentals of geophysical data processing" , McGraw-Hill  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Foias,   A. Frazho,   "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.A. Robinson,   S. Treitel,   "Geophysical signal analysis" , Prentice-Hall  (1980)</TD></TR></table>