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
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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:
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Now, consider the functions
![]() |
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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 ![]() |
[a5] | H. Dym, "![]() |
[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 ![]() |
[a9] | K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and the ![]() |
[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 |
Nehari extension problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nehari_extension_problem&oldid=49323