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