Carathéodory interpolation

Let $ p ( z ) = a _ {0} + a _ {1} z + \dots + a _ {n - 1 } z ^ {n - 1 } $ be a polynomial of degree at most $ n - 1 $. Let $ H ^ \infty $ be the Hardy space (cf. Hardy spaces) formed by the set of all analytic functions $ f $ in the open unit disc whose $ H ^ \infty $- norm $ \| f \| _ \infty = \sup \{ {| {f ( z ) } | } : {| z | < 1 } \} $ is finite. One says that $ f ( z ) $ is an interpolant of $ p ( z ) $ if $ f $ is a function in $ H ^ \infty $ and $ \{ a _ {j} \} _ {0} ^ {n - 1 } $ are the first $ n $ Taylor coefficients of $ z ^ {j} $ for $ f $, that is, $ f ( z ) = p ( z ) + z ^ {n} h ( z ) $ for some $ h $ in $ H ^ \infty $( cf. also Taylor series).

The Carathéodory interpolation problem is to find the set of all interpolants $ f $ of $ p $ satisfying $ \| f \| _ \infty \leq 1 $. Of course, this set can be empty. Let $ A _ {n} $ be the $ ( n \times n ) $ lower triangular Toeplitz matrix defined by

$$ A _ {n} = \left ( \begin{array}{cccc} a _ {0} & 0 &\dots & 0 \\ a _ {1} &a _ {0} &\dots & 0 \\ \vdots &\vdots &\vdots &\vdots \\ a _ {n - 1 } &a _ {n - 2 } &\dots &a _ {0} \\ \end{array} \right ) . $$

Then there exists a solution of the Carathéodory interpolation problem if and only if $ \| {A _ {n} } \| \leq 1 $. Moreover, there exists a unique solution of the Carathéodory interpolation problem if and only if $ \| {A _ {n} } \| = 1 $. In this case the unique interpolant $ f $ of $ p $ satisfying $ \| f \| _ \infty \leq 1 $ 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) $ b _ \alpha ( z ) = { {( z + \alpha ) } / {( 1 + {\overline \alpha \; } z ) } } $ where $ | \alpha | < 1 $. By recursively unravelling this Möbius transformation, I. Schur discovered that $ \{ a _ {j} \} _ {0} ^ {n - 1 } $ uniquely determines and is uniquely determined by $ \{ r _ {j} \} _ {0} ^ {n - 1 } $, where $ \{ r _ {j} \} _ {0} ^ {n - 1 } $ forms a sequence of complex numbers now referred to as the Schur numbers, or reflection coefficients, for $ \{ a _ {j} \} _ {0} ^ {n - 1 } $. The Schur algorithm is a computational procedure, discovered by Schur, which computes $ \{ r _ {j} \} _ {0} ^ {n - 1 } $ from $ \{ a _ {j} \} _ {0} ^ {n - 1 } $, or vice versa, in about $ n ^ {2} $ computations. Moreover, $ | {r _ {j} } | < 1 $ for all $ 0 \leq j < n $ if and only if $ \| {A _ {n} } \| < 1 $. In this case the set of all solutions $ f $ of the Carathéodory interpolation problem is given by

$$ f ( z ) = b _ {r _ {0} } ( z b _ {r _ {1} } ( \dots ( z b _ {r _ {n - 1 } } ( z f _ {n} ) ) \dots ) ) , $$

where $ f _ {n} $ is an arbitrary function in $ H ^ \infty $ satisfying $ \| {f _ {n} } \| _ \infty \leq 1 $. Furthermore, $ \| {A _ {n} } \| = 1 $ if and only if $ | {r _ {j} } | < 1 = | {r _ {k} } | $ for $ 0 \leq j < k $ and $ r _ {m} = 0 $ for $ m > k $. In this case

$$ f ( z ) = b _ {r _ {0} } ( z b _ {r _ {1} } ( \dots ( z b _ {r _ {k - 1 } } ( z r _ {k} ) ) \dots ) ) $$

is the unique solution of the Carathéodory interpolation problem. If the reflection coefficients $ \{ r _ {j} \} _ {0} ^ {n - 1 } $ do not satisfy any one of the previous conditions, then $ \| {A _ {n} } \| > 1 $ and there is no solution of the Carathéodory interpolation problem; see [a4] for further details.

The Schur numbers $ \{ r _ {j} \} $ 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 $ p ( z ) $ has all its roots inside the open unit disc without computing the zeros of $ p ( z ) $; 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)