Difference between revisions of "Carathéodory interpolation"
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Revision as of 07:54, 26 March 2012
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
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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
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where is an arbitrary function in
satisfying
. Furthermore,
if and only if
for
and
for
. In this case
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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) |
Carathéodory interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_interpolation&oldid=23220