Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generalized interpolation in $H^\infty$ with a complexity constraint


Authors: Christopher I. Byrnes, Tryphon T. Georgiou, Anders Lindquist and Alexander Megretski
Journal: Trans. Amer. Math. Soc. 358 (2006), 965-987
MSC (2000): Primary 47A57, 30E05; Secondary 46N10, 47N10, 93B15
DOI: https://doi.org/10.1090/S0002-9947-04-03616-5
Published electronically: December 9, 2004
MathSciNet review: 2187641
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a seminal paper, Sarason generalized some classical interpolation problems for $H^\infty$ functions on the unit disc to problems concerning lifting onto $H^2$ of an operator $T$ that is defined on $\EuScript{K} =H^2\ominus\phi H^2$($\phi$ is an inner function) and commutes with the (compressed) shift $S$. In particular, he showed that interpolants (i.e., $f\in H^\infty$ such that $f(S)=T$) having norm equal to $\Vert T\Vert$ exist, and that in certain cases such an $f$ is unique and can be expressed as a fraction $f=b/a$ with $a,b\in\EuScript{K}$. In this paper, we study interpolants that are such fractions of $\EuScript{K}$ functions and are bounded in norm by $1$ (assuming that $\Vert T\Vert<1$, in which case they always exist). We parameterize the collection of all such pairs $(a,b)\in\EuScript{K}\times\EuScript{K}$ and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where $\phi$ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.


References [Enhancements On Off] (What's this?)

  • 1. V. M. Adamjan, D. Z. Arov and M. G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR Sbornik 15 (1971), 31-73. MR 45:7505
  • 2. D. Z. Arov and M. A. Nudelman. Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), 1-45. MR 96k:47016
  • 3. J. A. Ball and J. W. Helton,
    A Beurling-Lax theorem for the Lie group $U(m,n)$ which contains most classical interpolation theory,
    J. Operator Theory 9 (1983), 107-142. MR 84m:47046
  • 4. C. I. Byrnes, A. Lindquist, S. V. Gusev, and A. S. Matveev,
    A complete parameterization of all positive rational extensions of a covariance sequence,
    IEEE Trans. Automat. Control 40 (1995), 1841-1857. MR 96i:93015
  • 5. C. I. Byrnes, H. J. Landau and A. Lindquist, On the well-posedness of the rational covariance extension problem, in Current and Future Directions in Applied Mathematics, eds. M. Alber, B. Hu, J. Rosenthal, Birkhäuser, pp. 83-106, 1997. MR 98c:93026
  • 6. C. I. Byrnes, S. V. Gusev, and A. Lindquist,
    A convex optimization approach to the rational covariance extension problem,
    SIAM J. Contr. and Optimiz. 37 (1998) 211-229. MR 99f:93135
  • 7. C. I. Byrnes, T. T. Georgiou, and A. Lindquist,
    A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint,
    IEEE Trans. Automat. Control 46 (2001), 822-839. MR 2002c:93038
  • 8. C. I. Byrnes and A. Lindquist,
    On the duality between filtering and Nevanlinna-Pick interpolation,
    SIAM J. Contr. and Optimiz. 39 (2000), 757-775. MR 2001h:93090
  • 9. C. I. Byrnes, S. V. Gusev, and A. Lindquist,
    From finite covariance windows to modeling filters: A convex optimization approach,
    SIAM Review 43 (2001), 645-675. MR 2002k:93080
  • 10. C. I. Byrnes and A. Lindquist, Interior point solutions of variational problems and global inverse function theorems, Report TRITA/MAT-01-OS13, 2001, Royal Institute of Technology, Stockholm, Sweden, 2001.
  • 11. C. I. Byrnes and A. Lindquist, A convex optimization approach to generalized moment problems, Control and Modeling of Complex Systems: Cybernetics in the 21st Century: Festschrift in Honor of Hidenori Kimura on the Occasion of his 60th Birthday, K. Hashimoto, Y. Oishi and Y. Yamamoto, Editors, Birkhäuser, 2003, 3-21. MR 2004a:93022
  • 12. C. Carathéodory and L. Fejer,
    Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizenten und über den Picard-Landau'schen Satz,
    Rend. Circ. Mat. Palermo 32 (1911), 218-239.
  • 13. R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier, Grenoble 20 (1970), 37-76. MR 42:5088
  • 14. P. L. Duren,
    Theory of $H^p$ Spaces, Academic Press, 1970. MR 42:3552 8pt
  • 15. H. Dym and I. Gohberg, A maximum entropy principle for contractive interpolants, J. Functional Analysis 65 (1986), 83-125. MR 87h:47065
  • 16. J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981. MR 83g:30037
  • 17. T. T. Georgiou, Partial Realization of Covariance Sequences, Ph.D. thesis, CMST, University of Florida, Gainesville 1983.
  • 18. T. T. Georgiou,
    Realization of power spectra from partial covariance sequences,
    IEEE Trans. Acoustics, Speech and Signal processing 35 (1987), 438-449.
  • 19. T. T. Georgiou,
    A Topological Approach to Nevanlinna-Pick Interpolation,
    SIAM J. Math. and Anal. 18 (1987), 1248-1260. MR 88j:30076
  • 20. T. T. Georgiou,
    The Interpolation Problem with a Degree Constraint,
    IEEE Trans. Automat. Control 44 (1999), 631-635.
  • 21. T. T. Georgiou and A. Lindquist, Kullback-Leibler approximation of spectral density functions, IEEE Trans. Information Theory 49 (2003), 2910-2917.
  • 22. U. Grenander and G. Szegö, Toeplitz forms and their applications, Univ. California Press, 1958. MR 20:1349
  • 23. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, 1962. MR 24:A2844
  • 24. S. Kullback, Information Theory and Statistics, 2nd edition, New York: Dover Books, 1968 (1st ed. New York: John Wiley, 1959). MR 21:2325
  • 25. D. Mustafa and K. Glover, Minimum Entropy $H_\infty$ Control, Springer-Verlag, Berlin Heidelberg, 1990. MR 92g:93002
  • 26. R. Nevanlinna, Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmen, Ann. Acad, Sci. Fenn. Ser A 13(1), 1919.
  • 27. N. K. Nikol'skii, Treatise on the Shift Operator, Springer-Verlag: Berlin, 1986. MR 87i:47042
  • 28. G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23.
  • 29. D. Sarason, Generalized Interpolation in $H^\infty$, Trans. Amer. Math. Society 127 (1967), 179-203. MR 34:8193
  • 30. I. Schur, On power series which are bounded in the interior of the unit circle I and II, Journal fur die reine und angewandte Mathematik 148 (1918), 122-145.
  • 31. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hibert Space, North-Holland, Amsterdam, 1970. MR 43:947
  • 32. O. Toeplitz, Über die Fouriersche Entwicklung positiver Funktionen, Rendiconti del Circolo Matematico di Palermo 32 (1911), 191-192.
  • 33. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Springer-Verlag, New York, 1985. MR 90b:49005

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47A57, 30E05, 46N10, 47N10, 93B15

Retrieve articles in all journals with MSC (2000): 47A57, 30E05, 46N10, 47N10, 93B15


Additional Information

Christopher I. Byrnes
Affiliation: Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130

Tryphon T. Georgiou
Affiliation: Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

Anders Lindquist
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Alexander Megretski
Affiliation: Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307

DOI: https://doi.org/10.1090/S0002-9947-04-03616-5
Received by editor(s): October 27, 2003
Received by editor(s) in revised form: January 21, 2004
Published electronically: December 9, 2004
Additional Notes: This research was supported in part by Institut Mittag-Leffler and by grants from AFOSR, NSF, VR, the Göran Gustafsson Foundation, and Southwestern Bell.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society