Carathéodory interpolation

Let be a polynomial of degree at most . Let be the Hardy space (cf. Hardy spaces) formed by the set of all analytic functions in the open unit disc whose -norm is finite. One says that is an interpolant of if is a function in and are the first Taylor coefficients of for , that is, for some in (cf. also Taylor series).

The Carathéodory interpolation problem is to find the set of all interpolants of satisfying . Of course, this set can be empty. Let be the lower triangular Toeplitz matrix defined by

Then there exists a solution of the Carathéodory interpolation problem if and only if . Moreover, there exists a unique solution of the Carathéodory interpolation problem if and only if . In this case the unique interpolant of satisfying is a Blaschke product.

The Schur method for solving the Carathéodory interpolation problem [a1], [a2] is based on the Möbius transformation (cf. Fractional-linear mapping) where . By recursively unravelling this Möbius transformation, I. Schur discovered that uniquely determines and is uniquely determined by , where forms a sequence of complex numbers now referred to as the Schur numbers, or reflection coefficients, for . The Schur algorithm is a computational procedure, discovered by Schur, which computes from , or vice versa, in about computations. Moreover, for all if and only if . In this case the set of all solutions of the Carathéodory interpolation problem is given by

where is an arbitrary function in satisfying . Furthermore, if and only if for and for . In this case

is the unique solution of the Carathéodory interpolation problem. If the reflection coefficients do not satisfy any one of the previous conditions, then and there is no solution of the Carathéodory interpolation problem; see [a4] for further details.

The Schur numbers 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 [a3], [a4], [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 has all its roots inside the open unit disc without computing the zeros of ; see [a2], [a4].

References

[a1]	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)
[a2]	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)
[a3]	J.F. Claerbout, "Fundamentals of geophysical data processing" , McGraw-Hill (1976)
[a4]	C. Foias, A. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a5]	E.A. Robinson, S. Treitel, "Geophysical signal analysis" , Prentice-Hall (1980)