# Nehari extension problem

Let be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all satisfying the following conditions:

i) the th Fourier coefficient of is equal to for each ;

ii) the norm constraint holds true. Here, is the norm of as an element of the Lebesgue function space and is the unit circle. Instead of condition ii) one may require , and in the latter case one calls the problem suboptimal.

The Nehari extension problem is not always solvable. In fact (see [a12]), the problem has a solution if and only if the infinite Hankel matrix

induces a bounded linear operator on , the Hilbert space of all square-summable sequences, such that its operator norm is at most one, i.e., . The suboptimal version of the problem is solvable if and only if . If , either the solution of the Nehari extension problem is unique or there are infinitely many solutions. If , then the problem and its suboptimal version always have infinitely many solutions, which can be parametrized by a fractional-linear mapping.

For the suboptimal case, the set of all solutions in the Wiener algebra , i.e., when one requires additionally that , can be described as follows. In this case, it is assumed that the given sequence is absolutely summable. Let . Then the operators and are boundedly invertible on , and one can build the following infinite column vectors:

Now, consider the functions

Then, each solution of the suboptimal Nehari extension problem for the sequence is of the form

(a1) |

where and is an arbitrary element of the Wiener algebra such that for and the th Fourier coefficient of is zero for each . Moreover, (a1) gives a one-to-one correspondence between all such and all solutions . The central solution, i.e., the solution , which one obtains when the free parameter in (a1) is identically zero, has a maximum entropy characterization. In fact, it is the unique solution of the suboptimal Nehari extension problem that maximizes the entropy integral

The Nehari extension problem has natural generalizations for matrix-valued and operator-valued functions, and it has two-block and four-block analogues. In the matrix-valued case, a superoptimal Nehari extension problem is studied also. In the latter problem the constraint is made not only for the norm, but also for a number of first singular values [a13]. There exist many different approaches to treat the Nehari problem and its various generalizations. For instance, the method of one-step extensions (see [a1]), the commutant-lifting approach (see [a6] and Commutant lifting theorem), the band method (see [a10]), reproducing-kernel Hilbert space techniques (see [a5]), and Beurling–Lax methods in Krein spaces (see [a4] and Krein space). The results are used in control theory (see [a8]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional representation (a1) can be represented explicitly in state-space form (see [a9] and [a3]).

The Nehari extension problem also has non-stationary versions, in which the role of analytic functions is taken over by lower-triangular matrices. An example is the problem to complete a given lower-triangular array of numbers,

to a full infinite matrix such that the resulting operator on is bounded and has operator norm at most one. The non-stationary variants of the Nehari extension problem have been treated in terms of nest algebras [a2]. The main results for the stationary case carry over to the non-stationary case [a11], [a7].

#### References

[a1] | V.M. Adamjan, D.Z. Arov, M.G. Krein, "Infinite Hankel block matrices and related extension problems" Transl. Amer. Math. Soc. , 111 (1978) pp. 133–156 Izv. Akad. Nauk SSSR Ser. Mat. , 6 (1971) pp. 87–112 |

[a2] | W.B. Arveson, "Interpolation in nest algebras" J. Funct. Anal. , 20 (1975) pp. 208–233 |

[a3] | J. Ball, I. Gohberg, L. Rodman, "Interpolation of rational matrix functions" , Operator Theory: Advances and Applications , 45 , Birkhäuser (1990) |

[a4] | J. Ball, J.W. Helton, "A Beurling–Lax theorem for Lie group which contains classical interpolation theory" J. Operator Th. , 9 (1983) pp. 107–142 |

[a5] | H. Dym, " contractive matrix functions, reproducing kernel Hilbert spaces and interpolation" , CBMS , 71 , Amer. Math. Soc. (1989) |

[a6] | C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990) |

[a7] | C. Foias, A.E. Frazho, I. Gohberg, M.A. Kaashoek, "Discrete time-variant interpolation as classical interpolation with an operator argument" Integral Eq. Operator Th. , 26 (1996) pp. 371–403 |

[a8] | B.A. Francis, "A course in control theory" , Springer (1987) |

[a9] | K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and the -error bounds" Int. J. Control , 39 (1984) pp. 1115–1193 |

[a10] | I. Gohberg, S. Goldberg, M.A Kaashoek, "Classes of linear operators II" , Operator Theory: Advances and Applications , 63 , Birkhäuser (1993) |

[a11] | I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "The band method for positive and contractive extension problems: An alternative version and new applications" Integral Eq. Operator Th. , 12 (1989) pp. 343–382 |

[a12] | Z. Nehari, "On bounded bilinear forms" Ann. of Math. , 65 (1957) pp. 153–162 |

[a13] | V.V. Peller, N.J. Young, "Superoptimal analytic approximations of matrix functions" J. Funct. Anal. , 120 (1994) pp. 300–343 |

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Nehari extension problem.

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