Abstract
In the present chapter we discuss the Schur algorithm and some interpolation problems in the scalar case.We make use, in particular, of the theory of J-unitary rational functions presented in Chapter 9. We also discuss first-order discrete systems. In the classical case, interpolation problems in the Schur class can be considered in a number of ways, of which we mention:
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1.
A recursive approach using the Schur algorithm (as in Schur’s 1917 paper [261]) or its variant as in Nevanlinna’s 1919 paper [244] in the scalar case.
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2.
The commutant lifting approach, in its various versions; see Sarason’s seminal paper [258], the book [254] of Rosenblum and Rovnyak, and the book [179] of Foias and Frazho.
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3.
The state space method; see [87].
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4.
The fundamental matrix inequality method, due to Potapov, see [249], with further far-reaching elaboration due to Katsnelson, Kheifets and Yuditskii, see [226, 227].
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5.
The method based on extension of operators and Krein’s formula; see the works of Krein and Langer [233, 232] and also [25].
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6.
The reproducing kernel method; see [26, 172].
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© 2016 Springer International Publishing Switzerland
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Alpay, D., Colombo, F., Sabadini, I. (2016). First Applications: Scalar Interpolation and First-order Discrete Systems. In: Slice Hyperholomorphic Schur Analysis. Operator Theory: Advances and Applications, vol 256. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-42514-6_10
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DOI: https://doi.org/10.1007/978-3-319-42514-6_10
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Publisher Name: Birkhäuser, Cham
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